Web viewWhen she packs this case, she puts the ... The frequency of the swinging pendulum is 0.5...
80
P3 06 Centre of Mass 156 minutes 154 marks Q1. (a) The diagram shows three similar toys. Each toy should be able to balance on a narrow rod. The arrows show the direction in which the weight of the toy acts. Only one of the toys balances on the rod, the other two fall over. Which one of the toys is balanced? Explain the reason for your choice. .................................................................. .............................................................. ..... .................................................................. .............................................................. .....
-
Upload
hoangkhuong -
Category
Documents
-
view
218 -
download
0
Transcript of Web viewWhen she packs this case, she puts the ... The frequency of the swinging pendulum is 0.5...
P3 06 Centre of Mass
156 minutes
154 marks
Q1. (a) The diagram shows three similar toys. Each toy should be able to balance on a narrow rod. The arrows show the direction in which the weight of the toy acts.
Only one of the toys balances on the rod, the other two fall over. Which one of the toys is balanced? Explain the reason for your choice.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................(3)
(b) The diagram shows a simple toy. Different animal shapes can be positioned so that the 50 cm rod balances horizontally.
(i) Use the following equation to calculate the moment exerted by the elephant shape of weight 2N about the pivot P. Show clearly how you work out your answer and give the unit.
moment = force × perpendicular distance from pivot
..........................................................................................................................
..........................................................................................................................
Moment = ...................................(3)
(ii) Use the following relationship to calculate the weight of the monkey shape.
total clockwise moment = total anticlockwise moment
..........................................................................................................................
..........................................................................................................................
Weight = ................................ N(2)
(c) The graph shows how the length of the spring changes as the total weight of the different animal shapes change.
Use the graph to find how much the spring extends when the elephant shape and the monkey shape are hung from the rod. Show how you get your answer.
.....................................................................................................................................
.....................................................................................................................................
Extension of spring = ................... cm(2)
(Total 10 marks)
Q2. (a) The diagram shows a lifebelt. It is hanging freely from hook Y.
(i) On the diagram, mark with an X the point where you think the centre of mass of the lifebelt will be.
(1)
(ii) Explain why you have chosen this point.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................(2)
(b) The drawing shows Susan on a diving board. She is 1.5 metres from point P and she weighs 500 N.
Calculate her moment (turning effect) about point P.Show clearly how you work out your answer and give the unit.
.....................................................................................................................................
.....................................................................................................................................
Moment about P = ................................................(3)
(c) Susan has a case with wheels.
When she packs this case, she puts the heaviest items at the end where the wheels are.This means that the heaviest items are less likely to crush the other contents and it helps her to find things when she opens the case.
Explain another advantage of packing her case in this way.
To gain full marks in this question you should write your ideas in good English. Put them into a sensible order and use the correct scientific words.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................(4)
(Total 10 marks)
Q3. (a) Every object has a centre of mass. What is meant by the centre of mass?
...................................................................................................................................
...................................................................................................................................(1)
(b) The drawing shows a thin sheet of plastic. The sheet is 250 mm wide. Two holes, each with a radius of 2 mm, have been drilled through the sheet.
Describe how you could use:
• a clamp and stand• a steel rod 100 mm long and with a radius of I mm• a weight on a thin piece of string (= a plumb line)• a ruler• a pen which will write on the plastic sheet
to find the centre of mass of the plastic sheet.
To gain full marks in this question you should write your ideas in good English. Put them into a sensible order and use the correct scientific words.
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................
...................................................................................................................................(5)
(c) There is a trapdoor in the ceiling of a house.The trapdoor weighs 44 N.The drawing shows a side view of the trapdoor.
(i) Complete the three spaces to give the equation which is used to calculate the turning effect of a force.
......................... = ......................... × perpendicular between ......................... line of action and pivot
(1)
(ii) Calculate the turning effect, about the hinge, due to the weight of the trapdoor.
Show clearly how you work out your final answer and give the unit.
.........................................................................................................................
.........................................................................................................................
Turning effect = ..............................................(3)
(Total 10 marks)
Q4. (a) The diagram shows a lampshade hanging from the ceiling. Draw an X on the diagram so that the centre of the X marks the centre of the mass of the lampshade.
(1)
(a) Complete the sentence using the correct word or phrase from the box.
above below to the left of to the right of
A suspended object will come to rest with its centre of mass directly
................................................. the point of suspension.(1)
(c) The diagrams show equipment that a student uses to find the centre of mass of a thin sheet of card.
Arrange these sentences in the correct order to describe how the student can find the centre of mass of the card.
The sequence starts with sentence D and finishes with sentence E.
A A line is drawn on the card marking the position of the string.
B The pin is put through one of the holes in the card and held in the boss.
C This is repeated using the other hole.
D Two holes are made in the card with each hole near to the edge of the card.
E The centre of mass is where the lines cross on the card.
F The weight is tied to the string and then the string is hung from the pin.
D E(3)
(Total 5 marks)
Q5. (a) The diagram shows a rectangle made out of a sheet of cardboard.
Draw an X on the diagram so that the centre of the X is at the centre of mass of the rectangle.
(1)
(b) The drawing shows a car tyre.
(i) Where is the centre of mass of the tyre?
...........................................................................................................................(1)
(ii) Explain your answer to (b)(i).
...........................................................................................................................
...........................................................................................................................(1)
(Total 3 marks)
Q6. (a) The diagram shows a child’s mobile. The mobile hangs from point P on the ceiling ofthe child’s bedroom.
(i) Mark the position of the centre of mass of the mobile by drawing a letter X on the diagram. Do this so that the centre of the X marks the centre of mass of the mobile.
(1)
(ii) Explain why you have chosen this position for your letter X.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................(2)
(b) The diagram shows a device which helps to prevent a ladder from falling over.
Use the term centre of mass to explain why the ladder, in the situation shown, is unlikely to topple over.You may add to the diagram to illustrate your explanation.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................(3)
(Total 6 marks)
Q7. The diagram shows a small mobile crane. It is used on a building site.
The distance, d, is measured to the front of the cab.
The table shows information from the crane driver’s handbook.
Load in kilonewtons (kN) Maximum safe distance, d, in metres (m)
10 6.0
15 4.0
24 2.5
40 1.5
60 1.0
(a) What is the relationship between the load and the maximum safe distance?
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................(2)
(b) The crane driver studies the handbook and comes to the conclusion that a load of 30 kN would be safe at a distance, d, of 2.0 metres.
Is the driver correct?
Explain your answer.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................(2)
(c) What is the danger if the driver does not follow the safety instructions?
.....................................................................................................................................
.....................................................................................................................................(1)
(d) How should the data in the table have been obtained?
Put a tick ( ) in the box next to your answer.
average results from an opinion poll of mobile crane drivers
copied from a handbook for a similar crane
results of experiments on a model mobile crane
results of experiments on this mobile crane
(1)(Total 6 marks)
Q8.(a) A pendulum is a device that can be used for timing. Some clocks rely on the swing of a pendulum to keep time.The pendulum shown in the diagram below is suspended from point Y and swings from Ato B, through the centre point Z.
The displacement x of the pendulum bob was plotted against time as shown in the graph below.
By analysing the evidence in the graph, find the amplitude of the oscillation of the pendulum and the time period of the pendulum.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................(2)
(b) A student carried out an investigation to find out how the time period of the pendulum depends on the length of the pendulum.
During the investigation she kept the mass of the pendulum bob and the angle of release constant. Her data is recorded in Table 1.
Table 1
Length of pendulumin
metresTime for 10 swingsin
secondsTime
periodin seconds
1 0.20 9.2 0.92
2 0.40 12.8 1.2
3 0.60 15.0 1.50
4 0.80 18.0 1.80
5 1.00 20.0 2.00
(i) Explain why the student timed ten swings, rather than just timing one swing, for each length of pendulum.
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................(2)
(ii) In this question you will be assessed on using good English, organising information clearly and using specialist terms where appropriate.
Describe the steps that the student would take to obtain the data shown in Table 1.
In your description, comment on the number of decimal places and significant figures the student has used in each column.
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................(6)
(iii) The student also carried out two more pendulum investigations.During the second investigation she kept the length of the pendulum and the angle of release constant. The data for this investigation is recorded in Table 2.
Table 2
Mass of pendulumbob in grams
Time for10 swings in seconds
Time periodin seconds
1 2.5 20.0 2.00
2 5.0 20.3 2.03
3 7.5 20.1 2.01
4 10.0 20.0 2.00
5 12.5 20.2 2.02
During the third investigation she kept the length of the pendulum and the mass of the pendulum bob constant. The data for this investigation is recorded in Table 3.
Table 3
Angle of releasein
degreesTime for 10 swingsin
secondsTime
periodin seconds
1 2 20.4 2.04
2 4 20.2 2.02
3 6 20.0 2.00
4 8 20.3 2.03
5 10 20.1 2.01
What conclusions can be made from the data recorded in Table 1, Table 2 andTable 3?Your answer should include a comment on the quality of the evidence.
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................(3)
(c) The student replaced the pendulum bob with a light paper cone as shown in the diagram.
She plotted the displacement x of the pendulum bob against time as shown in the graph below.
The student concluded that the frequency of this pendulum decreased with time.Does the graph support her conclusion?
Explain the reason for your answer.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................(2)
(Total 15 marks)
Q9.Diagram 1 shows a horse rider in a jumping arena.
Diagram 1
The jump has a pole support that can be in either position A or position B, giving two different jumping heights, as shown in Diagram 2.
Diagram 2
(a) (i) Diagram 3 shows a front and side view of the pole support in position A.
Draw an X on both views in Diagram 3 so that the centre of the X is at the centre of mass of the pole support.
Diagram 3
(1)
(ii) Explain why the pole support is more stable in position B than in position A.
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................(2)
(b) The combined mass of the horse and rider is 480 kg. As they approach the jump they are moving at 4 m/s.
(i) Calculate the kinetic energy, in joules, of the horse and rider as they approach the jump.
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
Kinetic energy = .................................................. J(2)
(ii) Calculate the maximum height, in metres, that the centre of mass of the horse and rider can be lifted off the ground when the horse approaches the jump at 4 m/s.
acceleration of free fall = 10 m/s2
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
Maximum height = ............................................. m(3)
(Total 8 marks)
Q10. The drawing shows a car tyre which is hanging from the branch of a tall tree.
(a) Draw an X on the diagram to mark the centre of mass of the tyre.(1)
(b) Some children use the tyre as a swing. Pulling the tyre to one side and letting it go makes the tyre swing backwards and forwards like a pendulum.
The time it takes the tyre to swing from one side to the other and back again is called the time period.
(i) What is the unit for time period?
...............................................................................................................(1)
(ii) How would using a shorter rope change the time period of the swing?
...............................................................................................................
...............................................................................................................(1)
(Total 3 marks)
Q11. The diagrams show two concrete mixers.
Concrete mixer A Concrete mixer B
On each diagram, the centre of the white X marks the centre of mass of the concrete mixer and its contents.
(a) Complete the sentence to explain what the term centre of mass means.
The centre of mass of a concrete mixer and its contents is ........................
........................................................................................................................
........................................................................................................................(1)
(b) Both diagrams are drawn to the same scale.
Concrete mixer B is more stable than concrete mixer A.
The two features which make concrete mixer B more stable are:
1 ......................................................................................................................
........................................................................................................................
2 ......................................................................................................................
........................................................................................................................
(2)
(c) Use the terms ‘line of action of the weight’ and ‘resultant moment’ to explain why a stable concrete mixer does not fall over when it is given a small push.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................(2)
(Total 5 marks)
Q12. Two children visit a playground.
(a) The diagram shows them on a see-saw. The see-saw is balanced.
Complete the following sentences by drawing a ring around the correct word or line in the box.
force.
(i)
The turning effect of the girl’s weight is called her load.
moment.
(1)
balance
(ii)
Point P is the axis of rotation of the see-saw.
turning
(1)
the boy moves nearer to point P.
(iii) To make end A of the see-saw go up, the girl moves nearer to point P.
the girl moves nearer to end A.
(1)
(b) In another part of the playground, a tyre has been suspended from a bar.
(i) Draw an X on the diagram so that the centre of the X marks the centre of mass of the tyre.
(1)
(ii) Complete the sentence by using the correct word or phrase from the box.
above below to the left of to the right of
If the suspended tyre is pushed, it will come to rest with its centre of mass
directly .................................................. the point of suspension.(1)
(Total 5 marks)
Q13.The diagram shows how a student can find the centre of mass of a thin flat sheet of plastic.
Part of his equipment is a plumb line. This is a weight fastened to one end of a piece of string.
He hangs the sheet and the plumb line from a pin through hole A.
(a) Mark an X on the diagram so that the centre of the X marks the likely position of the centre of mass of the plastic sheet.
(1)
(b) The dashed lines on the diagram below show the position of the plumb line from each hole when the student uses a different plastic sheet.
Point P is on both the dashed lines.
Complete the following sentence by drawing a ring around the correct line in the box.
axis
Point P shows the centre of mass
of the plastic sheet.
moment
symmetry
(1)
(c) Complete the following sentence by drawing a ring around the correct word in the box.
curved
A plumb line always hangs so that it is horizontal
parallel
vertical
(1)(Total 3 marks)
Q14. The centre of mass of an object is where the mass of the object may be thought to be concentrated.
(a) Use a word or phrase from the box to complete the sentence below.
above below to the side of
A hanging object will come to rest with its centre of mass directly
............................................................................................ the point from which it hangs.(1)
(b) The diagram shows the equipment that a student uses to find the centre of mass of a sheet of cardboard.She intends to draw two lines on the sheet. The centre of mass of the sheet will be where these lines cross.
Use words from the box to complete the sentences below.
boss cardboard nail stand string weight
The nail is put through hole X in the cardboard sheet. The nail is held in the
......................................................... . The string is tied to the weight and then the other
end of the string is hung from the ..................................................... . A line is drawn on
the cardboard sheet to mark the position of the .................................. . This is repeated
using hole Y.(3)
(c) The diagram below shows a plastic rectangle.
(i) Use a ruler to draw an axis of symmetry on the rectangle.
(1)
(ii) Draw an X on the diagram so that the centre of the X marks the centre of mass of the rectangle.
(1)(Total 6 marks)
Q15. The diagram shows a fork-lift truck with a load of 2.4 kN. The clockwise moment caused by this load is 2880 Nm.
(a) Use the equation in the box to calculate the distance d.
moment = force × perpendicular distance from the line ofaction of the force to the axis of rotation
Show clearly how you work out the answer and give the unit.
........................................................................................................................
........................................................................................................................
........................................................................................................................
Distance d = .......................................................(3)
(b) This warning notice is in the driver’s cab.
Explain in terms of moments why the maximum load must not be exceeded.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................(2)
(Total 5 marks)
Q16. The drawing shows a plastic toy which can stand on its feet.
(a) (i) Draw an X on the diagram so that the centre of the X marks the likely position of the centre of mass of the toy.
Photograph supplied by Hemera/Thinkstock(1)
(ii) Explain the reason for your choice in part (a)(i).
...............................................................................................................
...............................................................................................................(1)
(b) Suggest two ways in which the design of the toy could be altered to make the toy more stable.
1 ......................................................................................................................
........................................................................................................................
2 ......................................................................................................................
........................................................................................................................(2)
(Total 4 marks)
Q17.A student was asked to find the centre of mass of a thin sheet of card. The diagram shows the result of the student’s experiment. The student drew two lines onto the card. The centre of mass is where the two lines cross.
(a) Describe how the student found the correct positions to draw the two lines.
You may include a labelled diagram in your answer.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................(3)
(b) Explain how the student can check that the position found for the centre of mass is accurate.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................(2)
(Total 5 marks)
Q18.The drawing shows a plastic toy which can stand on its feet.
(a) (i) Draw an X on the diagram so that the centre of the X marks the likely position of the centre of mass of the toy.
(1)
(ii) Explain the reason for your choice in part (a)(i).
...............................................................................................................
...............................................................................................................(1)
(b) Suggest two ways in which the design of the toy could be altered to make the toy more stable.
1 .....................................................................................................................
........................................................................................................................
2 .....................................................................................................................
.........................................................................................................................(2)
(Total 4 marks)
Q19.The diagram shows a man standing in an airport queue with his wheeled bag.
(a) The man applies an upward force to the handle of his bag to stop the bag from falling. The moment of this force about the pivot is 36 Nm.
Calculate the upward force the man applies to the handle of his bag.
Use the correct equation from the Physics Equations Sheet.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
Force = ............................................................ N(2)
(b) When the man lets go of the bag handle, the bag falls and hits the floor.
Explain why.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................(2)
(c) During his holiday the man visits the Foucault Pendulum in Paris, France.The pendulum makes 10 complete swings every 160 seconds.
Calculate the frequency of the pendulum and give the unit.
Use the correct equation from the Physics Equations Sheet.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
Frequency = ..........................................................(3)
(Total 7 marks)
Q20.(a) The diagram shows a pendulum.
Draw an X on the diagram above, so that the centre of the X marks the centre of mass of the pendulum bob.
(1)
(b) A large clock keeps time using the swing of a pendulum.
(i) The frequency of the swinging pendulum is 0.5 hertz.
Calculate the periodic time of the pendulum.
Use the correct equation from the Physics Equations Sheet.
...............................................................................................................
...............................................................................................................
...............................................................................................................
Periodic time = .............................. seconds(2)
(ii) Calculate the number of complete swings the pendulum would make in 60 seconds.
Use your answer from part (b)(i) in your calculation.
...............................................................................................................
...............................................................................................................
...............................................................................................................
Number of swings in 60 seconds = ..............................(2)
(c) The diagram shows a clock on a trolley.
The trolley is being used to move the clock.
Calculate the moment of the 64 N force about the pivot.
Use the correct equation from the Physics Equations Sheet.
........................................................................................................................
........................................................................................................................
........................................................................................................................
Moment of the force = .............................. Nm(2)
(d) The design of the trolley is now changed to make it taller.
How does making the trolley taller affect the moment produced by the 64 N force about the pivot?
........................................................................................................................
........................................................................................................................(1)
(Total 8 marks)
Q21.Forces have different effects.
(a) (i) Use the correct answer from the box to complete the sentence.
slowing stretching turning
The moment of a force is the ............................................................... effect of the force.
(1)
(ii) What is meant by the centre of mass of an object?
................................................................................................................
................................................................................................................(1)
(b) Some children build a see-saw using a plank of wood and a pivot.The centre of mass of the plank is above the pivot.
Figure 1 shows a boy sitting on the see-saw. His weight is 400 N.
Figure 1
Calculate the anticlockwise moment of the boy in Nm.
Use the correct equation from Section A of the Physics Equations Sheet.
.........................................................................................................................
.........................................................................................................................
Anticlockwise moment = ........................................ Nm
(2)
(c) Figure 2 shows a girl sitting at the opposite end of the see-saw. Her weight is 300 N.
Figure 2
The see-saw is now balanced.
The children move the plank. Its centre of mass, M, is now 0.25 m from the pivot as shown in Figure 3.
Figure 3
The boy and girl sit on the see-saw as shown in Figure 3.
(i) Describe and explain the rotation of the see-saw.
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................(3)
(ii) The boy gets off the see-saw and a bigger boy gets on it in the same place. The girl stays in the position shown in Figure 3. The plank is balanced. The weight of the plank is 270 N.
Calculate the weight of the bigger boy.
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
Weight of the bigger boy = ........................................ N(3)
(Total 10 marks)
Q22.A student carries out an investigation using a metre rule as a pendulum.
(a) Diagram 1 shows a metre rule.
Diagram 1
(i) Draw, on Diagram 1, an X to show the position of the centre of mass of the rule.(1)
(ii) State what is meant by the ‘centre of mass of an object’.
................................................................................................................
................................................................................................................(1)
(b) The student taped a 100 g mass to a metre rule.
She set up the apparatus as shown in Diagram 2.
She suspended the metre rule from a nail through a hole close to one end, so she could use the metre rule as a pendulum.
The distance d is the distance between the nail and the 100 g mass.
Diagram 2
(i) Draw, on Diagram 2, a Y to show a possible position of the centre of mass of the pendulum.
(1)
(ii) The student carried out an investigation to find out how the time period of the pendulum varies with d.
Some of her results are shown in the table.
Time for 10 swings in seconds
d in cm Firsttest Secondtest Thirdtest MeanvalueMean time for1 swing inseconds
10.0 15.3 15.4 15.5 15.4 1.54
30.0 14.7 14.6 14.7 14.7 1.47
50.0 15.3 15.6 15.4 15.4 1.54
70.0 16.5 16.6 16.5
Complete the table.
You may use the space below to show your working.
................................................................................................................
................................................................................................................(3)
(iii) In this question you will be assessed on using good English, organising information clearly and using specialist terms where appropriate.
Describe how the student would carry out the investigation to get the results in the table in part (ii).
You should include:
• any other apparatus required
• how she should use the apparatus
• how she could make it a fair test
• a risk assessment
• how she could make her results as accurate as possible.
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................(6)
(c) A graph of the student’s results is shown below.
Distance d in cm
(i) Describe the pattern shown by the graph.
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................(2)
(ii) The student thinks that the measurements of time for d = 10 cm might be anomalous, so she takes a fourth measurement.
Her four measurements are shown below.
15.3 s 15.4 s 15.5 s 15.3 s
State whether you consider any of these measurements to be anomalous.
Justify your answer.
................................................................................................................
................................................................................................................
................................................................................................................
................................................................................................................(2)
(Total 16 marks)
M1. (a) Z1
weight or mass acts through pivotaccept rod or base for pivotaccept centre of gravity in line with pivot
1
no (resultant) (turning) momentaccept clockwise moment equals anticlockwise momentdo not accept same weight on each side of rod
1
(b) (i) 30allow 1 mark for 2 15or 2 0.15
2
N cm
orfor full credit the unit must be consistent with the numerical answer
0.3
Nmdo not accept joules
1
(ii) 1.5 (N)allow 1 mark for correct transformationallow 2 marks ecf their part (b)(i)/20 (ecf only if correct physics)
2
(c) 5 (cm)allow 1 mark for 6.0 (cm)allow 1 mark for a subtraction of 1 from a value clearly obtained from the graphallow 2 marks for correct ecf using an incorrect value for (b)(i) 0.2cm
allow 1 mark for clearly showing correct use of graph using an incorrect value for (b)(ii)
2[10]
M2. (a) (i) X at the centre of the lifebeltmeasuring from the centre of X, allow 2 mm tolerance in any direction
1
(ii) any two from:if X is on vertical line below the hanger (but not at centre) can gain the first point only
below the point of suspensionaccept ‘(vertically) below Y’
at the centre (of the lifebelt)accept ‘in the middle’
(because) the lifebelt / it is symmetricalor (because) the mass / weight is evenly distributed
2
(b) Nm or newton metre(s)accept Newton metre(s)do not accept any ambiguity in the symbol ie NM, nM or nm
1
750(moment) = force (perpendicular) distance (between line of action and pivot)or (moment) = 500 1.5 gains 1 mark
2
(c) Quality of written communication:for 2 of the underlined terms used in the correct context
1
any three connected points from:
low(er) centre of mass / gravityor centre of mass / gravity will be close(r) to the wheels / axle / ground
(more) stableor less unstable
less likely to fall overaccept ‘less likely to overturn’do not accept ‘will not fall over’
the turning effect / moment (of the weight of case) is lessor so less effort is needed to hold the caseignore references to pulling the case
so the pull on her arm is less3
[10]
M3. (a) point at which its mass (seems to) act or point at which gravity (seems to) actaccept ... its weight actsaccept correct statements if the intent is clear e.g.. .. if suspended, the centre of gravity will be directly under the point of suspensione.g.... (if the object is symmetrical), the centre of gravity is on theor an axis (of symmetry)do not credit just 'it is a point'
1
(b) The answer to this question requires good English in a sensible order with correct use of scientific terms. Quality of written communication should be considered in crediting points in the mark scheme
maximum of 4 marks if ideas not well expressed
any five from:
clamp (steel) rod (horizontally)no marks if method quite unworkable
hang plastic / sheet by rod through (one) hole
hang plumb line from rod
mark ends of plumb line on the sheet anduse the ruler to draw a straight line
repeat with other hole
centre of mass is where the lines cross
check by balancing at this pointmaximum of 3 marks if no 'repeat with other hole'
5
(c) (i) (turning) effect or momentforce
distanceall three correctaccept weightaccept length
1
(ii) 17.6allow 44 x 0.4 or 0.4 x 44 for 1 mark
2
Nm or newton metre(s)do not accept N/m or N/cm1760 Ncm gains all 3 marks
1[10]
M4. (a) centre of X should appear to be on the continued line of the flex and in thebody of the lamp as judged by eye
example
1
(b) below1
(c) (D)→B→F→A→C→(E)all four correct for 3 marksor any two correct for 2 marksor just one correct for 1 mark
3[5]
M5. (a) centre of X at the point where the axes crossto within 1 mm in any direction
1
(b) (i) (at / in the) centre (of the tyre)or unambiguously shown on the diagram
1
(ii) (this is) where axes of symmetry (of the tyre) cross / intersect / meetor point at which the mass of the tyre seems to be (concentrated)
1[3]
M6. (a) (i) centre of X directly below P and between the model aeroplanesas judged by eye but between centre of propeller of top aeroplane and canopy of bottom aeroplaneexample
1
(ii) the centre of mass is (vertically) below the point of suspension / P1
the centre of mass is in the middle of the aeroplanesaccept the centre of mass is level with the aeroplanes
1
(b) centre of mass of the worker and the ladder (and device)1
line of action of the weight is inside the baseaccept the centre of mass is above / within / inside the base (of the ladder and device)
1
so there will not be a (resultant) momentaccept so he / it / the ladder will not topple even if he leans over
or it will (only) topple over if the line of action of the weight / thecentre of mass is outside the base
accept each point, either on the diagram or in the written explanation, but do not accept the point if there is any contradiction between them
1[6]
M7. (a) any two from:
• inversely proportional
• as the load gets biggerthe (maximum safe) distance gets lessallow ‘as the mass increases the distance decreases’accept an unspecified response e.g. ‘big load at a short distance’ for (1)
• load × distance = 60 (kNm)2
(b) yes, because 30 × 2 = 60 (2)accept for (1) a correct but insufficiently explained responsee.g. ‘yes because it’s safe’accept for (2) a correct response which is sufficiently explainede.g. ‘yes, because 60 (kNm) at 1 metre is safe and 30 (kNm) is half the load at twice the distancedo not accept ‘no’ and do not accept just ‘yes’do not accept ‘yes, because 30 is between 24 and 40 and 2 is between 2.5 and 1.5’do not accept ‘the crane/ cable may break’ or other dangers
2
(c) the crane may/will topple over/fall over/forward1
(d) results of experiments on this mobile craneaccept any unambiguous indication
1[6]
M8.(a) amplitude = 8 (cm)1
period = 4(s)1
(b) (i) same reading error for 10 swings as 1 swing1
so reduces (%) error in timingsorreduces error in an individual time period
accept it makes timing errors less significantaccept increases reliability / precisionignore increases accuracy
1
(ii) Marks awarded for this answer will be determined by the quality of communication as well as the standard of the scientific response.
0 marksNo relevant content
Level 1 (1–2 marks)There is a basic description of the experimental steps but no correct reference to dps or sfs.
Level 2 (3–4 marks)There is a clear description of the experimental steps and correct reference to either dps or sfs
Level 3 (5–6 marks)There is a clear and detailed description of the experimental steps and correct reference to both dps and sfs
examples of the physics points made in the response
• measure the length of the pendulum with a ruler
• pull the bob to one side, measure the angle of release and release the pendulum bob
• time 10 swings / oscillations
• change the length of the pendulum and repeat
• divide each recorded time by 10
• number of decimal places for raw data depends on resolution of measuring device
• number of sf for Time period (1.80) depends on number of sfs for time for 10 swings (18.0)
6
(iii) as the length of the pendulum increases the time period increases, and this relationship is non-linear
do not accept they are directly proportional
1
the time period does not depend on the mass of the pendulum bob or the angle of release
1
or
there is no relationship between either the mass of the pendulum bob or the angle of release and the time period
a specific statement illustrating that the conclusions are only valid within the limits of experimental uncertainty
eg there are always random uncertainties within any experiment. For Table 2 data repeating the experiment 5 times for the same mass of pendulum bob, you would expect a small variation in times between 20.0 s and 20.3 s.
1
or
a statement that the conculsions are only valid within the ranges measured for each variable
(c) No
time period does not change / always 4 seconds because1
frequency is constant / frequency is related to period (T =1 /f) and hence if period is constant frequency will be constant
1[15]
M9.(a) (i) in both views centre of X in geometric centre1
(ii) in position B the support has a lower centre of mass1
and a wider / bigger base (area)1
(b) (i) ½ × 480 × 42
1
3840 (J)correct answer with or without working gains 2 marks
1
(ii) statement or symbols showing
KE = GPE1
or 3840 = mgh
or their (b)(i) = mgh
1
0.8 (m)correct answer with or without working gains 3 marks
1[8]
M10. (a) X drawn at the centre of the tyrejudge by eye
1
(b) (i) second1
(ii) decreases it1
[3]
M11. (a) the point at which the (total) mass seems to act / appears to be concentratedaccept ‘weight’ for ‘mass’accept the point at which gravity seems to actdo not accept a definitive statement eg where (all) the mass is
1
(b) wider / larger basemarks are for a correct comparison
1
lower centre of massaccept lower centre of gravity / c of g
1
(c) line of action (of the weight) lies / falls inside the basein each case the underlined term must be used correctly to gain the
mark1
the resultant moment returns mixer to its original positionaccept there is no resultant moment / resultant moment is zeroaccept resulting moment for resultant momentdo not accept converse argument
1[5]
M12. (a) (i) moment1
(ii) rotation1
(iii) the girl moves nearer to point P1
(b) (i) X drawn in the centre of the space enclosed by the tyrejudge by eye
1
(ii) below1
[5]
M13. (a) centre of X
on the plumb lineandbetween the level of the captions ‘plastic sheet’ and ‘hole B’example
1
(b) centre of massaccept any unambiguous indication
1
(c) verticalaccept any unambiguous indication
1[3]
M14. (a) below1
(b) boss1
naildo not allow boss
1
string1
(c) (i) line from mid-point of either side to the mid-point of the opposite sideintention correct as judged by eye, use of ruler not essentialdo not allow either diagonal
1
(ii) X in the centre of the rectangle
1[6]
M15. (a) 1.2allow 1 mark for conversion of 2.4 kN to 2400 Nor for correct transformation without conversionie d = 2880 ÷ 2.4
2
metre(s)/m1
(b) any two from:
• as the load increases the (total) clockwise moment increases
• danger is that the fork lift truck / the load will topple / tip forward
• (this will happen) when the total clockwise moment isequal to (or greater than) the anticlockwise moment
accept moments will not be balanced
• (load above 10.0 kN) moves line of action (from C of M)outside base (area)
2[5]
M16. (a) (i) centre of X above the feet and in the bodya vertical line from their X falls between two lines in diagram - judged by eye
1
(ii) where the mass seems to be concentratedaccept it’s above the base (area)
accept because otherwise it would toppleaccept line of action (of weight) passes through the basedo not accept where the mass is concentrated
1
(b) any two from:
• make (the area of) feet / base bigger
• make feet wider apart
• makes legs shorter / heavier
• make head smaller / lighter
• make tail touch the ground / make the tail longeraccept ‘make centre of mass / gravity lower’
2[4]
M17.Resource currently unavailable
M18.(a) (i) centre of X above the feet and in the bodya vertical line from their X falls between two lines in diagram – judged by eye
1
(ii) where the mass seems to be concentratedaccept it’s above the base (area)accept because otherwise it would toppleaccept line of action (of weight) passes through the base
do not accept where the mass is concentrated1
(b) any two from:
• make (the area of) feet / base bigger
• make feet wider apart
• make legs shorter / heavier
• make head smaller / lighter
• make tail touch the ground / make the tail longeraccept ‘make centre of mass / gravity lower’
2[4]
M19.(a) 60allow 1 mark for correct substitution (with d in metres),ie 36 = F × 0.6an answer of 0.6 or 6 gains 1 mark
2
(b) the line of action of the weight lies outside the base / bottom (of the bag)accept line of action of the weight acts through the sideaccept the weight (of the bag) acts outside the base / bottom(of the bag)
1
a resultant / overall / unbalanced moment acts (on the bag)accept the bag is not in equilibriumdo not accept the bag is unbalanced
1
(c) 0.0625
allow 1 mark for correct substitution, ie 16 = an answer of 0.00625 gains 1 mark
2
hertz / Hzdo not accept HZ or hz
1[7]
M20.(a) centre of X drawn at centre of pendulum bobjudged by eyeaccept dot drawn at centre of circle
1
(b) (i) 2
allow 1 mark for correct substitution, ie provided nosubsequent step shown
2
(ii) 30or60 ÷ their (b)(i) correctly calculated
allow 1 mark for
or or 0.5 × 60provided no subsequent step shown
2
(c) 51.2allow 1 mark for correct substitution, ie 64 × 0.8 provided no subsequent step shown
2
(d) it increases (the moment)must be comparativeaccept 1 mark for calculation of the moment = 64 (Nm)
1[8]
M21.(a) (i) turningaccept turning ringed in the box
1
(ii) point at which mass (or weight) may be thought to be concentratedaccept the point from which the weight appears to actallow focused for concentrateddo not accept most / some of the massdo not accept region / area for point
1
(b) 600 (Nm)400 × 1.5 gains 1 mark provided no subsequent steps shown
2
(c) (i) plank rotates clockwiseaccept girl moves downwardsdo not accept rotates to the right
1
(total) CM > (total) ACMaccept moment is larger on the girl’s side
1
weight of see-saw provides CM
answer must be in terms of momentmaximum of 2 marks if there is no reference to the weight of the see-saw
1
(ii) W = 445 (N)W × 1.5 = (270 × 0.25) + (300 × 2.0) gains 2 marksallow for 1 mark:total CM = total ACM either stated or impliedor(270 × 0.25) + (300 × 2.0)if no other marks given
3[10]
M22.(a) (i) X placed at 50 cm mark1
(ii) point at which mass of object may be (thought to be) concentrated1
(b) (i) Y placed between the centre of the rule and the upper part of mass1
(ii) 16.5allow for 1 mark(16.5 + 16.6 +16.5) / 3
2
1.65value consistent with mean value givenonly penalise significant figures once
1
(iii) Marks awarded for this answer will be determined by the quality of communication as well as the standard of the scientific response. Examiners should apply a ‘best-fit’ approach to the marking.
0 marksNo relevant content
Level 1 (1 – 2 marks)A description of a method which would provide results which may not be valid
Level 2 (3 – 4 marks)A clear description of a method enabling some valid results to be obtained. A safety factor is mentioned
Level 3 (5 – 6 marks)A clear and detailed description of experiment. A safety factor is mentioned. Uncertainty is mentioned
examples of the physics points made in the response:
additional apparatus
• stopwatch
use of apparatus
• measure from hole to centre of the mass• pull rule to one side, release• time for 10 swings and repeat• divide mean by 10• change position of mass and repeat
fair test
• keep other factors constant• time to same point on swing
risk assessment
• injury from sharp nail• stand topple over• rule hit someone
accuracy
• take more than 4 values of d• estimate position of centre of slotted mass• small amplitudes• discard anomalous results• use of fiducial marker
6
(c) (i) initial reduction in T (reaching minimum value) as d increases1
after 30 cm T increases for higher value of d1
(ii) (no)
any two from:
• fourth reading is close to mean• range of data 0.2 s / very small• variation in data is expected
2[16]
E1. Foundation Tier
In part (a) most candidates could identify the balanced toy but found difficulty explaining the reason for their choice, even though the concept of moments was in the stem of part (b)(i). Although many candidates calculated the moment correctly in part (b)(i), the unit was often incorrect, N/cm was a favourite error. Many candidates were unable to correctly complete the subsequent calculation of the weight. In part (c) the original length of the spring was often not
subtracted to produce a correct value for the extension. However, the better candidates were able to correctly complete all the calculations demonstrating a sound understanding of the topic.
Higher Tier
In part (a) most candidates could identify the balanced toy but found difficulty explaining the reason for their choice, even though the concept of moments was in the stem of part (b)(i). Although many candidates calculated the moment correctly in part (b)(i), the unit was often incorrect, N/cm was a favourite error. Most candidates were able to complete the subsequent calculation of the weight. In part (c) the original length of the spring was often not subtracted to produce a correct value for the extension.
E2. Only a small minority were able to give the correct position of the centre of mass. Usually positions in the body of the lifebelt either above or below the hook were suggested and very few earned any marks in part (ii). Incorrect responses to part (i) made a rational explanation more difficult in part (ii) but responses which could still gain credit, such as ‘below the point of suspension’ and ‘the lifebelt is symmetrical’, were only given by a minority of candidates.
Susan’s moment was usually numerically correct but the unit was often omitted or incorrect.
In part (c) it was rare for candidates to gain more than one mark. Where credit was gained it was usually for ‘low centre of mass’, ‘less likely to fall over’ or ‘easier to hold’. Very few made correct statements in relation to the moments involved and the communications mark, which was given for the correct use of terms, was rarely awarded.
E3. Foundation Tier
(a) In this part the examiners were looking for a correct statement which would explain, or help to explain, the term centre of mass. The candidate did not have to account for cases in which the centre of mass lies outside of the object but, even so, appropriate responses, for example ‘the object will balance if it is supported at its centre of mass’, were rare. ‘The centre of the object’ and ‘the mass will be the same on both sides’ were fairly popular, but incorrect, responses.
(b) The specification states that ‘candidates should be able to describe how to find the centre of mass of a thin sheet of material’. Some invalidated their responses either by clamping the sheet or by suspending it through one hole and hanging the plumb line from the other hole. Others ignored any sensible consideration of the size of the sheet and bent it so that the 100 mm long rod could go through both holes. However, there was a minority of candidates who seemed to understand what was required.
(c) Only a minority of candidates were able to recollect the equation to calculate the turning effect of a force. A large proportion injected the words mass and/or height into their
erroneous responses. Some were able to find the product of 44 and 0.4 but some spoilt their efforts by increasing or decreasing by a factor of 10 or by doubling the distance to 0.8 m. If there was an attempt at a unit it was usually to suggest N/m.
Higher Tier
(a) In this part the examiners were looking for a correct statement which would explain, or help to explain, the term centre of mass. The candidate did not have to account for cases in which the centre of mass lies outside of the object but, even so, appropriate responses, for example ‘the object will balance if it is supported at its centre of mass’, were not too frequent. ‘The centre of the object’ and ‘the mass will be the same on both sides’ were fairly popular, but incorrect, responses.
(b) The Specification states that ‘candidates should be able to describe how to find the centre of mass of a thin sheet of material’. Many were able to do this, and to express themselves clearly, so gained all five marks. A small minority invalidated their responses by clamping the sheet or by suspending it through one hole and hanging the plumb line from the other hole. Some others ignored any sensible consideration of the size of the sheet and bent it so that the 100 mm long rod could go through both holes.
(c) Most candidates were able to recollect the equation to calculate the turning effect of a force. Some injected the words mass and/or height into their erroneous responses. Most candidates were able to find the product of 44 and 0.4 but some spoilt their efforts by increasing or decreasing by a factor of 10 or by doubling the distance to 0.8 m. A significant minority gave the unit as N/m. It could be that some correctly knew that their answer was in newton-metres but incorrectly thought that this was the way to write its symbol.
E4. (a) Just less than half of the candidates gained the mark. Those that did not often failed to place the X inside the lampshade. A favoured incorrect position was at the junction of the lampshade and the cable.
(b) The majority of candidates identified that the centre of mass is directly below the point of suspension.
(c) A third of candidates gained all 3 marks by identifying the correct sequence for the experiment.
E5. (a) This question was well-answered with the great majority of candidates placing the centre of mass at the point where the axes of symmetry cross.
(b) (i) A majority of candidates were able to identify the centre of the tyre as the location of and 1(b)(ii) the centre of mass, though those who did were generally unable to offer an appropriate explanation.
(ii) Some candidates erroneously stated that an object with an empty space in the middle does not have a centre of mass.
E6. (a) (i) Only a minority of candidates located the centre of mass on the part of the vertical line beneath P and between the planes. A common error was to select the junction of the supporting strings as the location of the centre of mass.
(ii) Though a third of candidates were able to gain a mark for explaining that X must be vertically beneath the point of suspension, only a small minority explained the correct location.
(b) Only a very small minority of candidates realised that it’s the combined mass of worker, ladder and device which needs to be considered. However ,over half secured one mark. This was for expressing the idea that the line of action of the weight falls within the base.About a quarter of candidates obtained their second mark by going on to explain what would cause the ladder to topple.
E7. Foundation Tier
(a)(b) Many candidates realised that the data shows that as the load increases the maximum safe distance decreases but only a minority of these could explain why the crane driver’s conclusion is correct.
(c) Several hazards were mentioned with a minority of candidates stating that the mobile crane may topple over, or words to that effect.
(d) Most candidates were able to select ‘results of experiments on this crane’ as the appropriate source for the data in the table.
Higher Tier
(a)(b) A large majority realised that the data shows that as the load increases the maximum safe distance decreases but fewer could explain why the crane driver’s conclusion is correct.
(c) Several hazards were mentioned with the majority of candidates stating that the mobile crane may topple over, or words to that effect.
(d) A large majority of candidates correctly selected ‘results of experiments on this crane’ as the appropriate source for the data in the table.
E11. (a) Many candidates failed to gain marks here as their responses were definite ie it is where all the mass is.
(b) Just over half of the candidates were able to make two clear comparative statements with a further quarter able to make one correct clear statement.
(c) Many responses were not worth a mark because candidates had not used the terms as instructed in the stem of the question. It was encouraging to note that the fuller answers often used the phrase ‘resultant moment restores equilibrium’ or words to that effect.
E12. (a) Over three fifths of candidates answered parts (i) and (iii) correctly but only just over a quarter correctly named the axis of rotation, part (ii), most opting for axis of balance.
(b) (i) Only just under a half of candidates were able to mark the centre of mass of the tyre correctly, most wrongly putting it somewhere on the tyre itself.
(ii) A large majority of candidates gave the correct answer.
E13. (a) Three-quarters of the candidates were able to mark the centre of mass on the plastic sheet in an acceptable place.
(b) Nearly all candidates were able to name the centre of mass of the triangle.
(c) Three-quarters of the candidates correctly chose that a plumb line hangs vertically.
E14. (a) (a) This was a well answered question, with four fifths of candidates giving the correct answer.
(b) Most candidates did not read the paragraph carefully enough. Less than a third of candidates gained all three marks although nearly a third of candidates inserted two of the three words correctly.
(c) (i)&(ii) These part questions were answered well with nearly all of candidates managing to
mark the centre of mass correctly. However part (c)(i) was marred by candidates drawing a plethora of lines some of which were incorrect and some of which may have been construction lines but were not identified as such and therefore a mark could not be awarded.
E15. (a) The calculation was generally well done with nearly two-thirds of candidates gaining full marks, including that given for the unit.
(b) Many candidates appreciated the possibility of the fork lift truck ‘toppling’, however fewer were able to give a correctly worded explanation as to why this should happen. Of those who attempted this, some candidates only referred to a single moment which was not identified as the resultant; others referred to more than one moment but did not identify the direction of action.
A significant number of candidates did not appreciate that part (b) was linked to part (a) and did not read the instruction to explain in terms of moments. This resulted in many responses linked to other vehicles, most often a taxi. Answers in terms of momentum and the problems in cornering and the weight of the passengers were then common.
E16. (a) (i) Most candidates located the ‘X’ in the correct region.
(ii) Many responses just said that this was the CoM location with no reason given.
(b) Over half of the candidates described two clear changes and gained both marks. The most common answers being ‘lower the centre of mass’ and ‘make the base area larger’. However, once again, poor use of language or vague answers, such as, ‘make the toy smaller’ stopped many candidates from gaining marks.
E17. (a) For a question testing recall of a standard experiment given in the specification, the responses produced were very disappointing. A few students gave a sufficient explanation to score three marks. Just over a tenth of students did not attempt it at all and a further two fifths of students did not score any marks. Most students did not describe the correct experiment and of those who tried to do so the descriptions were often superficial and inadequate. There was some confusion as to what is meant by a plumb line and a number of students do not know the difference between vertical and horizontal. Some students
scored marks by drawing a good, labelled diagram.
(b) This question was not attempted by a few students and a further half of students did not score any marks. Many students incorrectly suggested repeating the experiment with another piece of card or comparing their results with other groups.
E18.(a) (i) This was well answered with the majority of the students gaining the mark.
(ii) This was very poorly done: Three quarters of the students scored zero and of those that did score a mark, credit was usually given for the “toppling” mark rather than a correct definition regarding centre of mass.
(b) The majority of students scored at least one mark.
E19.(a) Less than half of answers gained 2 marks. The most common mistake was failing to convert 60 cm into 0.6 metres.
(b) The majority of candidates failed to gain any marks. Most failed to mention line of action, and simply restated what they were seeing in the diagram.
(c) Less than one third of candidates gained all 3 marks. Many did not give a unit.
E20.(a) The majority of candidates correctly located the centre of mass of the pendulum bob. Many of the wrong responses were at the point of attachment of the string to the support or the bob.
(b) Most candidates were able to calculate the period in (b)(i) and many correctly calculated the number of oscillations per minute in (b)(ii).
(c) The vast majority of candidates scored both marks. Of the remainder, some appeared to not have access to a calculator so were unable to carry out the multiplication. Some correctly wrote the substituted equation and then performed a division on the calculator.
(d) Around half of the candidates gained the mark here. Many responses indicated that candidates had not read the question properly. In the correct responses, candidates successfully made the link that the moment increases due to the increase in the perpendicular distance. Some candidates failed to gain the mark by not being comparative in their answer.
E21.(a) (i) Nearly all students knew that the moment of a force is the turning effect of the force.
(ii) Less than half of the students were able to state what is meant by centre of mass of an object. Many referred to a region within the object rather than a point.
(b) Almost all students were able to calculate a moment of a force.
(c) (i) Very few students scored the three marks for describing and explaining the movement of a previously-balanced plank whose pivot had been moved away from the centre of mass of the plank. The idea that the weight of the plank now provided a
moment was not understood.
(ii) This high-demand calculation was successfully performed by about a quarter of the students.
