Circular Motion, Work, Conservative/non-conservative forces

Physics 2010 Summer 2011

REVIEW FOR MIDTERM 3

1. The figure to the right is a graphic depiction of a section of a roller coaster at an amusement park. Point A

is the starting position and is 8.0 m from the ground.

A. A car (mass = 400 kg) starts from rest at point A

and passes point B with a speed of 6.4 m/s. If the

distance from point A to point B along the track is

20.0 m, how large is the average fictional force

retarding the motion of the car?

B. Assume the track is frictionless from point B to

point C. What is the speed of the car at point C?

2. Starting from rest, a locomotive pulls a series of boxcars up

an incline of 3.0°. After the train has moved 2.40 km it is

traveling at a speed of 45.0 km/h. Assume that the train has a mass of 6.4 × 10 kg.5

A. How much work is done by the locomotive?

B. What fraction of the work is done against gravity?

C. Assuming uniform acceleration, how long does it take the train to reach this speed?

3. The designer of a roller coaster wants the riders to experience weightlessness as they round the top of one

hill. The radius of curvature is 30.0 m. How fast must the car be going?

4. A curve on a road has a 60.0 m radius and is banked so that no frictional force is required for a car going at

25.0 m./s to safely negotiate the curve. At what angle should it be banked?

5. Tarzan (weight 890 N) grabs a vine, which is initially

horizontal, and attempts to swing to the ground. The

breaking strength of the vine is 1780 N. As Tarzan is

swinging, he is surprised that the vine breaks. At what

angle (2) does it break?

6. Air bags are used in cars to decelerate the occupants slowly when a car is suddenly decelerated in a crash.

(a) Compute the work done by the decelerating force acting on a 55.0 kg driver if the car is brought to

a rest from an initial speed of 20.0 m/s.

(b) Find the minimum thickness of the air bag if the average decelerating force is not to exceed

8900 N (2000 lb).

7. A rubber ball of mass 30.0 g is dropped from a height of 2.00 m onto a floor. The velocity of the ball is

reversed by the collision with the floor and the ball rebounds to a height of 1.50 m. Find the impulse that

was applied to the ball during the bounce.

8. A baseball catcher moves his glove away from a 45.0 m/s fastball as he catches it. The glove moves 20.0

cm upon catching the ball. Find the average decelerating force the catcher exerts on the ball. The ball has a

mass of 0.150 kg.

9. As you ride on a Ferris wheel your apparent weight is different at the top than at the bottom. Calculate your

apparent weight at the top and bottom of a Ferris wheel given that the radius is 7.2 m and it completes 1

revolution every 28.0 s. Your mass is 55.0 kg.

10. A popular amusement ride is shown in the drawing. In this ride people sit in a swing that is

suspended from a long rotating arm. Riders are at a distance of 12.0 m from the axis of rotation

and move with a speed of 25.0 mi/hr.

A. Find the centripetal acceleration of the riders.

B. Find the angle, 2, the supporting wires make with the vertical.

11. A 1.9 kg block slides down a frictionless ramp. The top of the ramp is

1.5 m above the ground. The bottom of the ramp is 0.25 m above the

ground. The block leaves the ramp horizontally a distance d away.

Find d.

12. A small truck with a mass of 2300 kg is moving at 16.0 m/s. The truck collides with a stationary car whose

mass is 1800 kg. They lock together and the wheels lock.

A. What is the velocity just after they collide?

B. Find the impulse (both magnitude and direction) that acts on the skidding car/truck combination

just after they collide.

C. If the coefficient of kinetic friction is 0.80, determine how far the cars skid before coming to rest.

13. (a) How high a hill can a car coast up (engine disengaged) if friction is negligible and its initial speed

is 110 km/h?

(b) A 750 kg car with an initial speed of 110 km/h is observed to coast up a hill to a vertical height of

22.0 m above its starting point. How much thermal energy was generated by friction?

(c) What is the average force of friction if the hill (in part b) has a slope of 2.5° above the horizontal?

(d) What is the work done by the normal force in the process?

14. A man swings on a rope to pick up a lady friend. He scoops her into his arms right at the bottom of his

swing and takes her to a ledge. The mass of the man is 70.0 kg and the mass of the lady is 48.0 kg. The

ledge from which he starts is 8.00 m above the low point of the swing.

(a) After starting from rest, what is his velocity just before he scoops up the lady?

(b) What is the velocity just after scooping her up?

(c) How much kinetic energy is lost in the collision?

(d) What is the highest ledge they can swing to?

15. A 60.0 kg gymnast is hanging by his arm from a bar and is swinging in a vertical circle. As an

approximation assume that the radial distance between the bar and the point where the gymnast's mass is

located is 100.0 cm. As he swings through the lowest point on the circle, he has a speed of 2.8 m/s.

(a) Find the magnitude of the centripetal acceleration.

(b) What is the tension in his arm?

(c) What is the tension in his arm when he makes an angle of 40° with the vertical?

16. A 75.0 kg person is riding in a car which is moving at 20.0 m/s when the car hits a bridge abutment.

(a) Calculate the average force acting on the person if a padded dashboard stops him in 85 ms.

(b) Calculate the average force acting on the person if an air bag stops him in 750 ms.

17. A 0.075 kg toy airplane is tied to the ceiling with a string.

When the airplane’s motor is started it moves with a constant

speed of 1.21 m/s in a horizontal circle of radius 0.44 m

(a) Find the angle the string makes with the vertical.

(b) Find the tension in the string.

18. The water slide shown ends at a height of 1.5 m above the

pool. If the person starts from rest at point A and lands in the

water at point B, what is the height of the water slide?

19. If a roadway is banked at the proper angle, a car can round a corner without any assistance from friction

between the tires and the road. Find the appropriate banking angle for a 900 kg car traveling at 20.5 m/s in

a turn of radius 85.0 m.

20. A small mass m slides without friction along the

looped apparatus shown. If the object is to remain on

the track at the top of the circle, from what minimum

height h must it be released?

21. A motorcycle (mass of cycle + rider = 2.50 × 10 kg)2

travels up a hill at a steady speed to point B. From point

B to point C he coasts up the hill in neutral. Assume no

air resistance and negligible friction.

(a) If he wants to feel weightless at point C, what

must his speed have been at point B?

(b) What average power was necessary to sustain

him at that constant velocity found in (a) as he

traveled from point A to point B?

22. A popular amusement ride is shown in the drawing. In

this ride, people sit in chairs suspended by a cable that swings in a

circle. The radius of the circle is 10 m long and the riders move at a

constant speed of 25.0 mi/hr.

(a) Find the angle 2.

(b) The mass of the rider is 50 kg and the mass of the chair is 10

kg. Assuming the cable is massless, find the tension.

(c) If the person was sitting on a massless scale during the ride,

what would the scale read? (Friction is negligible.)

23. A small object of mass m slides without friction along a

looped track of radius r, as shown. Your answers should be

expressed in terms of r, g and m.

(a) If the object is to remain on the track at the top of the

circle, from what minimum height h can it be

released?

(b) What velocity does the object have after completing the loop when released from the minimum

height h?

24. A 5.0 kg block slides down a frictionless ramp. The top of the

ramp is 2.0 m above the ground. The bottom of the ramp is

0.50 m above the ground. The block leaves the ramp

horizontally and lands a distance d away. Find d.

25. A child is twirling a 0.02 kg ball on a string at a constant speed

in a horizontal circle whose radius is 0.25 m, as shown in the drawing. The

ball travels once around the circle in 0.400 s.

(a) Find is the work done by gravity on the ball during 1 revolution.

(b) What is the centripetal acceleration of the ball?

(c) Find the magnitude of the centripetal force and list all the forces

contributing to it..

(d) Determine the tension in the string.

26. A massless spring of spring constant 200 N/m is placed on

vertically on a horizontal table as shown (Fig. A). A

block is placed on top of the spring such that the weight of

the block compresses the spring 0.02 m from its unstrained

length (Fig. B).

(a) What is the mass of the block?

The block is pushed down a little farther (Fig. C) by a 100 N force then released.

(b) How fast is the block moving as it comes off the spring (Fig. D)?

(c) What is the maximum height above the launch point the block reaches?

27. A 500 kg roller coaster has a speed of 5 m/s at

point A. A chain mechanism exerts a force of 300

N on the roller coaster as it moves it up the hill

from point A to point B, a distance of 80 m, in 10

seconds. From A to C the track is not frictionless.

The work done by friction from A to C is -2 × 104

J. From C to E the track is frictionless. The roller

coaster barely stays on the track at point D. The

radius of the loop is r = 15 m.

(a) What is the average power of the chain mechanism?

(b) Find the speed of the roller coaster at point D.

(c) Determine the height (h) of the roller coaster at point A.

28. Nils pulls a sled (mass = 20 kg) up a hill at an incline of 20° for

30 m at a constant speed of 1.20 m/s. The angle between the

rope and the hill is 15°. Due to friction, the tension is constant

at 90 N.

(a) Draw free body diagram showing all the forces the sled

feels. Label all the forces.

(b) List all the nonconservative forces the sled feels.

(c) List all the conservative forces the sled feels.

(d) Which forces do work (non-zero) on the sled?

(e) Find the change in the total mechanical energy of the

sled.

(f) What is the magnitude of the work done by friction on the sled?

29. A. Objects A and B are placed on the spring as shown. Object A has twice as much mass as object B.

If the spring is depressed and released, propelling the objects into the air, object A will:

(a) rise one fourth as high as object B.

(b) rise half as high as object B.

(c) rise to the same height as object B.

(d) rise twice as high as object B.

(e) rise four times as high as object B.

B. A spring powered dart-gun fires a dart 1 m vertically into the air. In order for the dart to go 4 m,

the spring would have to be depressed:

(a) 2 times the distance. (d) 8 times the distance.

(b) 3 times the distance. (e) 9 times the distance.

(c) 4 times the distance.

C. A winch is used to lift heavy objects to the top of a building under construction. A winch with a

power of 50 kW was replaced with a new winch with a power of 100 kW. Which of the following

statements about the new winch is NOT true.

(a) The new winch can do twice as much work in the same time as the old winch.

(b) The new winch takes twice as much time to do the same work as the old winch.

(c) The new winch takes twice as much time to do four times the work as the old winch.

(d) The new winch can raise objects with twice as much mass at the same speed as the old

winch.

(e) The new winch can raise objects with the same mass at twice the speed of the old winch.

D. A motorcycle (mass of cycle plus rider = 2.50 × 10 kg) is traveling at a steady speed of 20.0 m/s.2

The force of air resistance acting on the cycle and rider is 1.00 × 10 N. Find the power necessary2

to sustain this speed if the road is sloped upward at 37.5° with respect to the horizontal.

(a) 2,000 W (d) 31,829 W

(b) 31.829 W (e) 40,874 W

(c) 40.874 W

E. A 100 N force is applied as shown to a 10 kg object for two seconds. If the

object is initially at rest, what is its final velocity. (Ignore friction)

(a) 1.00 m/s (d) 17.3 m/s

(b) 5.89 m/s (e) 34.6 m/s

(c) 8.66 m/s

30. A popular amusement ride is shown in the drawing. For this ride, people sit in

chairs suspended by a cable that swings in a circle. The radius of the circle is 15 m

long and the riders move at a constant speed of 20.0 mi/hr. {1 mile = 1609 m)

(a) Find the angle 2.

(b) The mass of the rider is 50 kg and the mass of the chair is 10 kg.

Assuming the cable is massless, find the tension.

(c) How much work is done by the tension in ½ a revolution?

(d) List all the forces that contribute to the centripetal force the rider feels and

find the magnitude of the centripetal force.

(e) If the person were sitting on a massless scale during the ride, what would

the scale read? (Friction is negligible.)

31. Nils pulls a sled (mass = 20 kg) up a hill at an incline of 20° for 10 s at

a constant speed of 1.20 m/s. The angle between the rope and the hill

is 15°. The coefficient of kinetic friction between the sled and the

snow is 0.15.

(a) List all the nonconservative forces the sled feels.

(b) List all the conservative forces the sled feels.

(c) Which forces do work (non-zero) on the sled?

(d) Find the change in the total mechanical energy of the sled.

(e) What is the magnitude of the work done by tension on the

sled?

32. A small object of mass m slides without friction along a looped track of radius r, as shown. The object is

released from a height h. The object barely remains on the track at the top of the circle and later encounters

an unstrained spring with spring constant k. Your answers

should be in terms of one or more of the following

variables: h, g, m and k.

(a) What is the value of r?

(b) What is the magnitude of the Normal force that the

object feels at point B?

(c) What is the speed of the object at point C?

(d) By how much does the object compress the

spring?

33. A The designer of a roller coaster wants the riders to experience weightlessness as they round the top

of a hill. The radius of curvature is 20 m. How fast must the car be going?

B. A 0.25 kg block is dropped straight downward onto a vertical spring. The spring constant of the

spring is 58 N/m. The block sticks to the spring and the spring compresses

0.2 m before coming to a momentary halt. What is the speed of the block just

before it hits the spring?

C. A popular amusement ride is shown in the drawing. In this ride people sit in a

swing that is suspended from a long rotating arm. Riders are at a distance (r)

of 10 m from the axis of rotation and move with a speed of 10 m/s. Find

angle, q, the supporting wires make with the vertical.

D. A baseball catcher moves his glove away from a 50 m/s fastball as he catches it. The glove moves

15 cm upon catching the ball. Find the average decelerating force the catcher exerts on the ball.

The ball has a mass of 0.15 kg.

34. A. At amusement parks, there is a popular ride where the floor of a rotating

cylindrical room falls away, leaving the backs of the riders “plastered” against

the wall. Suppose the radius of the room is 3.3 m and the speed of the wall is

10 m/s when the floor falls away

(a) What is the coefficient of kinetic friction that must exist between a

rider’s back and the wall if the 55 kg rider is to accelerate at 1 m/s 2

(magnitude only) when the floor drops away? Assume static friction

is not enough to keep the rider in place.

Now assume that the coefficient of static friction is just barely large enough to keep the rider in

place

(b) ________ By what factor would the coefficient of static friction need to change to keep the

rider in place if the radius was doubled?

(c) ________ By what factor would the coefficient of static friction need to change to keep the

rider in place if the velocity was cut in half?

(d) ________ By what factor would the coefficient of static friction need to change to keep the

rider in place if the mass of the rider tripled?

B. (a) What is the work done by the tension (T = 470 N) in

moving this block 3 meters?

k(b) There is friction between the block (100 kg) and table (:=0.1). If the block was initially moving at 1 m/s, what is

its speed after it has moved 3 meters to the right? [Use

work at least once.]

i 35. A. A block is pushed, with an initial speed of v = 7.3 m/s,

down a frictionless ramp. At the initial instant, the center of

mass of the block is 4.2 m (h) above the bottom of the

ramp. At the bottom of the ramp is a 1 meter long flat

k1 section with friction (m = 0.3). After traveling through this

section, the block goes into a very short curved runway that

sends the block traveling vertically upward only.

2Determine the maximum height, h , above the bottom of the

ramp that the block attains.

B. In the situation above, after leaving the friction section, the

block instead goes into another flat filled section also with

k2 friction (m = 0.7). How fast is the block moving after

traveling 2 meters in this new section?

36. A large object of mass 988 kg slides without friction along a

looped track of radius 10 m as shown. The object just barely

remains on the track at the top of the circle.

(a) From what minimum height must the object have been

released from rest?

(b) what is the maximum distance that the block compress

the spring (k = 250,000 N/m)?

(c) Now the object is released from a height (H) of 100 m

and goes off a small ramp which is 5 meters tall (h). Find

horizontal distance (L) of the block from the end of the

ramp to where it hits the ground?

37. A. A car is traveling at a constant speed around a circular track whose radius is 2.5 km. The car goes

half-way around the track in 205 s. What is the magnitude of the centripetal acceleration of the

car?

B. A roller coaster at an amusement park has a curve that tops out in a vertical circle of radius

r = 22.0 m. A passenger feels the seat of the car pushing upward on her with a force equal to one

quarter of her weight as she goes over the top. How fast is the roller coaster traveling at the top of

the curve?

C. A projectile of mass 0.550 kg is shot straight up with an initial speed of 19.0 m/s. If the projectile

rises to a maximum height of only 12.5 m, determine the magnitude of the average force due to air

resistance.

D. A skier slides horizontally along the snow for a distance of 21 m before coming to rest. The

kcoefficient of kinetic friction between the skier and the snow is : = 0.059. Initially how fast was

the skier going?

38. A. A particle of mass 3 kg, starting from point A in the drawing at a

0height h = 3.0 m, is projected down the curved runway. Upon leaving the runway at

point B, the particle is traveling straight upward and reaches a

fheight h = 4.7 m above the floor before falling back down. If !45

J of work is done by friction on the particle, find the speed of the

particle at point A.

B. A 9.5 kg monkey is hanging by one arm from a branch and is swinging on a vertical circle. As an

approximation, assume a radial distance of 90 cm between the branch and the point where the

monkey's mass is located. As the monkey swings through the lowest point on the circle, it has a

speed of 3.2 m/s. What is the magnitude of the tension in the monkey's arm?

C. You are trying to lose weight by working out on a rowing machine. Each time you pull the rowing

bar (which simulates the "oars") toward you, it moves a distance of 1.3 m in a time of 1.6 s. The

readout on the display indicates that the average power you are producing is 82 W. What is the

magnitude of the force that you exert on the handle?

39. A. A "swing" ride at a carnival consists of chairs that are swung in a circle by 12.0

m cables attached to a vertical rotating pole at an angle of 2 = 63.5° , as the

shown in the drawing. The total mass of the chair and its occupant is 248 kg.

1. Find the speed of the chair.

2. How much work is done by the tension in ¼ of a revolution?

B. A 1.8 kg object is suspended from a vertical spring whose spring constant is

135 N/m. The object is at rest. The object is then pulled straight down by an additional distance

of 0.15 m and released from rest. Find the speed with which the object passes through its original

position on the way up.

40. A. A 1.20 × 10 kg crate is being pushed across a horizontal floor by a force that makes an angle2

of 30.0° below the horizontal. The coefficient of kinetic friction is 0.300. What should be the

magnitude of , so that the net work done by it and the kinetic frictional force is 100 J when the

block has moved 2 meters?

B. A 47.0 g golf ball is driven from the tee with an initial speed of 56.0 m/s and rises to a height of

24.0 m.

1. Neglecting air resistance, determine the kinetic energy of the ball at its highest point.

2. What is its speed when it is 8.0 m below its highest point?

Solutions to selected questions