Faculty of Engineering

ENGG152

ENGINEERING MECHANICS

Laboratory Handbook

Weeks 5 to 11

Spring Session 2011 Student Name:

Student Number:

TABLE OF CONTENTS  PAGE  Information about Pre‐Labs  3 General Instructions  4 General Laboratory Health and Safety Procedures  6  Experiment 1 : “Real World Impacts” ‐ Billiard Ball Motion  9 Experiment 2 : Polygon of Forces  17 Experiment 3 : Energy Balance  21 Experiment 4 : Centrifugal Motion  27 Experiment 5 : Determination of Coefficient of Friction  31 Experiment 6 : Governor Rig  35    All experiments will be held in the SMART Infrastructure Building 6G09N, shown in the figure below.  

 

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  You must bring this lab manual each time you attend the laboratory to record your experimental findings.    Updated by David Hastie, August 2011 

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Information about Pre‐Labs Pre‐labs  have  been  developed  for  all  6  experiments  and  are  accessible  via  the ENGG152  eLearning  site.  These pre‐labs provide  valuable  information  on  how  to conduct  the  experiments  and  carry  out  the  required  analysis.  These  pre‐labs  also include an online multiple choice quiz. You must review the Pre‐labs and complete the online quizzes BEFORE your experiments, scoring at least 90% (14/15). Multiple attempts are allowed.  You MUST bring your student card to the lab with you as there will be an electronic scanner  to  check  that you have  successfully  completed  the pre‐lab quizzes  for  the experiments you are scheduled to do. Failure to score 90% will result  in you (as an individual)  incurring  a  penalty  of  50%  of  the  group  mark  you  obtain  for  that particular experiment.   Please read all of the information contained in this handbook. It has been compiled to assist you in your learning of statics and dynamics. Assessment in this section of the subject will assume that you are aware of the information provided here. Do not hesitate  to  raise  any  questions  you  may  have  with  the  coordinator  or  lecturers involved.  

 YOU ARE NOT PERMITTED TO PRINT FROM THE 

SMART LAB. 

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General Instructions It  is generally considered  that  learning and understanding Statics and Dynamics  is greatly  enhanced by way of  experiments or  through  experience. The most notable reminder of this phenomenon is the experience incurred by Newton which resulted in  the  subsequent great  contribution by Newton  to  the  field of Dynamics. To  this end,  the Faculty of Engineering has  introduced a major  laboratory component  into the assessment of 100 level Statics and Dynamics. It is also the intention that students enjoy and gain greatest benefit  from  this effort and experience. To ensure greatest benefit,  it  is  recommended  that  each  student  comply  with  the  following housekeeping details.  • Each  student  enrolled  in  ENGG152  is  required  to  complete  SIX  specified 

experiments, between weeks 5 and 11. These experiments will be conducted  in the new SMART Infrastructure Building (6.G09N). 

 • Students will be assigned  to  lab groups by  the subject coordinator  to carry out 

these experiments, and will attend three 2 hour sessions. Two experiments will be completed during each session. Make sure you are familiar with the timetable and schedule for your group (available on the eLearning site for ENGG152). 

 • The  need  to  complete  two  experiments  in  one  session  demands  every  group 

commence  punctually,  work  effectively  and  leave  equipment  in  the  same condition as they found it, within the allocated time period. 

 • Noting the demand for time efficiency, it is recommended that all students pre‐

read and familiarise themselves with the experiment details before attending the laboratory session. This should be achieved by reading  the relevant sections of this  laboratory manual  and  completing  the pre‐labs  and  corresponding  online quizzes.  Obviously  poorly  prepared  students  will  experience  difficulty completing the experiments in the allocated time, generate laboratory congestion and subsequently eliminate opportunity  to complete  the  follow‐on experiment. Failure to complete the particular set tasks will also result in lower marks for the experimental report. 

 • Other  than  in  exceptional  circumstances,  students  must  complete  the 

experiments in the specified laboratory sessions. If you miss your scheduled lab class,  you DO NOT  need  to  lodge  an  academic  consideration  but  you MUST contact the subject coordinator as soon as possible to allocate you an alternative lab time.  DO NOT JUST TURN UP TO ANOTHER LAB CLASS ! 

 

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Lab Assessment Requirements There will be three components that will make up the lab assessment component of ENGG152.  1. Successful completion of pre‐lab quizzes As  previously mentioned,  you will  need  to  complete  the  pre‐labs  and  associated quizzes before attending your lab session to complete the experiments. Your student card will be scanned as you enter the lab and the score you obtained in the pre‐lab quiz will  be  recorded.  If  you  fail  to  achieve  a  score  of  at  least  14/15,  you  (as  an individual)  will  incur  a  penalty  of  50%  of  the  group  mark  you  obtain  for  that experiment.  2. Experiments 1, 2, 3 and 6 You will be required to submit a group lab report for experiments 1, 2, 3 and 6. This will  involve  completing  the  relevant  report  submission  sheets  in  this  document. Where  written  answers  are  required,  use  only  the  space  provided.  One  group member will be required  to submit  the report  to  the EEC,  the details can be  found below.  3. Experiments 4 and 5 To  reduce  the  number  of  submitted  reports,  the  report  submission  sheets  for experiments 4 and 5 must be completed during the scheduled lab session and will be marked immediately by the demonstrator supervising the experiments. Your group will be required to complete one set of calculations for each experiment and then you can use the supplied spreadsheets to enter the remaining data. This will be checked by the demonstrator as well as any required written responses.  Submission of Lab Reports Every  group  report must  include  a  barcoded  cover  sheet, printed  via  the CoverIt webpage  on  the  Engineering  homepage.  A  link  to  this  page  is  provided  on  the ENGG152 eLearning site and the subject outline. Note that for experiment 2, you will have  additional  A3  sheets  that  must  be  included  with  your  submission.  It  is suggested  that you  submit your  experiment  2  lab  report  inside  a plastic  sleeve  to avoid sheets going missing. Also ensure that all group members’ names are listed on the coversheet to ensure marks are awarded to all concerned.  Date Due: The lab reports are due one week (7 days) after the scheduled completion of  your  last  lab  class  (check  the  due  date  on  the  CoverIt  sheet).  They must  be submitted to the Engineering Enquiry Centre (EEC). If your lab is scheduled in week 9, you will have until week 10 to submit your lab report.  Late Submission Penalties: Late submission of reports will attract a 5% penalty per working day for each group member. 

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General Laboratory Health and Safety Procedures  1. INTRODUCTION  Laboratories and work areas contain many potential safety hazards. However, with proper  control  these  hazards  can  be  eliminated.  The  following  guidelines  are intended to outline basic laboratory and work area safety requirements.  The student must also be aware of the University of Wollongong safety and security rules that are given on notices displayed throughout the University.  2. GENERAL PROCEDURES: LABORATORY AND WORK AREA SAFETY  • Practical jokes and unauthorised experiments are forbidden. • Smoking in laboratories is not permitted. • Eating or drinking is not permitted in the laboratory. • Persons working in a laboratory must wear suitable clothing. • Suitable  footwear  that  fully  encloses  the  feet must  be worn  in  laboratories  and 

work areas at all times. Sandals or thongs are not permitted. • Safety glasses must be worn in laboratories or any other work area where there is 

a risk of dangerous substances splashing into eyes or of objects impacting with the eye. 

• Long hair should be tied back when working near moving equipment. • Other protective clothing (PPE) must be worn where appropriate. • Bags  must  not  be  placed  on  the  benches  but  stored  in  the  space  provided 

underneath.  3. REPORTING OF ACCIDENTS AND POTENTIAL HAZARDS  Any accidents  that occur must be reported  to  laboratory staff and supervisors who are  then  required  to  inform  the Head  of  School. Direct  reporting  to  the Head  is required  in  the  absence  of  the  appropriate  laboratory  staff  or  the  supervisor. Students also have a duty to report any operational procedures which are considered to be unsafe or potentially hazardous, using the reporting sequence outlined above.  4. EMERGENCY PROCEDURES  In the event of fire or other emergencies  in Building 6 that may endanger staff and students, the following procedures apply –  1.  (a)  In the event of a fire, alert others in the immediate area 

(b) Notify  security  by  dialling  extension  4555.  Security will  then  contact  the relevant emergency services (if required)  

 

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2.  When an alarm sounds, follow the directions of the Building Warden(s). • The priority is to leave the building immediately via the nearest exit • A safe exit from the building requires an orderly and prompt response • In the case of an emergency, do not use the lift, use the stairwell and proceed to 

Assembly Point A as shown below. • Watch out for traffic as you cross the road(s) to the assembly point.  

 

A

  • Staff  and  students  are  not  to  re‐enter  the  building until  advised  by  building 

wardens (they will be wearing orange vests) or Security that it is safe to do so   Also, familiarize yourself with the Standard Fire Orders which should be displayed in every room around campus   Please note that some mobile phone carriers have limited service within the SMART Infrastructure building and you may need to exit the building if you need to make a phone call.  

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Experiment 1 : “Real World Impacts” - Billiard Ball Motion Aim: To explore oblique impacts of spheres, the principles of the conservation of energy and momentum by examining the motion of colliding billiard balls. Significant Properties/Phenomena: Potential Energy, Kinetic Energy (linear and rotational), Conservation of Energy, Conservation of Momentum, Mass, Inertia, Friction, Oblique Impacts and Rebound Angles, Rolling Resistance, Elastic Impacts, Coefficient of Restitution. General Procedure: One billiard ball (the “impact” ball) will be released from a known height on the chute. It will impact with a second ball (the “target” ball) placed on the trajectory table. The time of the collision and the subsequent impacts of both balls, with the fence around the trajectory table, will be recorded via a microphone and a LCD voltmeter. The distance the balls travel after the collision and the angles of impact will also be noted. The experiment is to be repeated for a number of different release heights and impact locations. The order of impact with the fence (target ball or impact ball first) should also be noted to ensure that the correct times are allotted. In addition, the offset of the target ball required to cause the target ball, following the collision, to roll through a "gate" positioned at a specified angle will be found by trial and error. Nomenclature: g Acceleration due to gravity (9.81 m s-2) Ι Mass moment of inertia of the billiard ball (units) m Mass of the billiard ball (0.141 kg) R Radius of the billiard ball (0.02615 m) V Velocity of the billiard balls (m/s) Vc Corrected velocity of the billiard ball at chute exit (m/s) Vm Measured velocity of the impact ball at chute exit (m/s) V’

I Average velocity of the impact ball after the collision (m/s) V’

T Average velocity of the target ball after the collision (m/s) Δh Release height of the ball on the chute (distance above the trajectory table) (mm) ω Angular velocity of the billiard ball (radians/second) Analysis: The main aim of this experiment and subsequent analysis is to estimate the energy available at the collision, and compare it to the subsequent energy stored in the balls (indicated by the sum of their kinetic energies as they reach the fence). Some insight into the practical operation of the conservation of energy principle, and the magnitudes and mechanisms of energy losses should be gained as a result. Using the conservation of energy principle, it is possible to estimate the velocity of the impact ball prior to the collision, as the potential energy (m.g.Δh) is converted to kinetic energy. It is noted that the kinetic energy of a rolling billiard ball is stored in 2 components: – a linear component (0.5 m V2) and a rotational component (0.5 I ω2). The rotational component accounts for the energy in the spinning ball and is analogous to linear kinetic energy. “I” is the mass moment of inertia of the ball. It is a parameter representing the resistance of the body to being rotated (in the same way that the mass of a body is a parameter representing the resistance of a mass to being accelerated). It can be

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found by carrying out the following integration, or (the course probably favoured by most engineers) by looking up an engineering handbook.

Eq 1 I r dR

= ∫ 2

0

m

For a solid sphere, the result is: I = 2/5 m R2 For a rolling sphere the angular velocity (ω) is directly related to the velocity and radius of the sphere (i.e. ω =V/R), therefore, with appropriate substitutions of this and the inertia result, the following expression relating the initial potential energy and the subsequent kinetic energy components can be obtained: This equation can be used to determine a theoretical velocity for the impact ball resulting from the release height (Δh).

2 2 2 2

2

7 or 2 5 10 1

mV mV R mV Vm g h g hR

⋅ ⋅Δ = + = ⋅Δ =27

0 Eq 2

However there are some further complications. Measured velocities of the ball exiting the chute are significantly less than those determined from Equation 2. This is a result of friction and rolling resistance in the chute, and the fact that whilst descending the chute the ball is rolling about a smaller radius (it sits “down” in the rails). This second factor causes more energy to be stored in the rotational component than would be the case for pure rolling (think of a yo-yo descending the string as an extreme case of this effect). To account more accurately for the energy distribution in this experiment it is necessary to determine a “corrected” velocity as this excess rotational velocity is converted to linear velocity. Ideally you would measure and determine these parameters as part of this experiment. However, in this case it will be necessary to make use of the results presented in Figure 1.1.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Change in Height (m)

Bill

iard

Bal

l Vel

ocity

(m

/s) Theoretical

Measured

Corrected

Figure 1.1: Impact Ball Velocities at Chute Exit

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For a particular release height, the corrected velocity curve from Figure 1.1 can be used to estimate the impact ball velocity prior to the collision. This velocity can also be substituted into Equation 2 (i.e. Vc in place of V) to estimate the energy in the system immediately prior to the collision. The energy in the system after the collision may be estimated by determining the average velocities of the impact and target balls (V’

I and V’T) from the distance travelled and the time

between the collision and impact with the wall. These can then be used in Equation 2 to find the kinetic energy in each ball, and added to estimate the total energy in the system post collision. This energy, as a proportion of the initial theoretical energy (m.g.Δh) and the system energy prior to the collision, calculated as suggested in the previous paragraph, can then be found. Equipment: Felt surfaced semi circular trajectory table fitted with a ball chute. Tektronix digital storage oscilloscope Microphone and power supply Two billiard balls Tape measure “Bulls Eye” level Perspex target ball placement guide gate Procedure: • Check that the trajectory table is level, adjust as required • Turn on the Microphone Power Supply • Conduct some trials confirming trigger setting on oscilloscope, and that the impacts are

being recorded Place the target ball at a known offset on the trajectory table using the target ball placement guide (the offset is at the discretion of the group). Note the offset used. Place the impact ball at one of the marked positions up the chute. NB a number of “dots” have been placed along the chute, such that Δh = 15, 60, 80, 100, 120, 140 and 160mm (the impact balls must be placed in "tangential alignment" with these dots to achieve the listed Δh). Ensure that the oscilloscope is “ready” to trigger. Release the impact ball. Note and record the angular position of the balls as they reach the fence. Note the time of travel post collision from the oscilloscope display. Reposition the balls at the impact point and measure the distance of travel for each ball. Repeat the above procedure for a total 5 different impact and target ball settings and record the results in Table 1.1 of the report submission sheets. For each test, carry out the analysis outlined in the “Analysis” section. Record the results in Table 1.3 of the report submission sheets. The tutor will nominate two angles (between 10° and 50°) for your group to place the "gate". Centre the gate on the specified angle and then determine by trial and error the offset required to cause the target ball to roll through the gate following a collision. For each specified angle repeat this for both a clockwise and anti-clockwise angular offset of the gate. Record the results in Table 1.2 of the report submission sheet, including the offset you calculate would be required to achieve this outcome.

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(NB consideration of Figure 1.2 indicates that: sin (φ) = Offset / ( 2 R ).

Offset

R26.15 mm

φ

Figure 1.2: Oblique Impact Geometry Respond to the questions and issues raised in the report submission sheets.

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Experiment 1 : “Real World Impacts” - Billiard Ball Motion REPORT SUBMISSION SHEET: Group Members 1: _____________________ 2: _____________________ 3: _____________________ Lab Group: __________ Date: ______________ Tutor: _____________ Aim: (1 mark): Table 1.1: Recorded Test Data (2 marks):

Test No.

Release Height (mm)

Target Ball

Location

Post Collision Travel Details

(eg 10 mm Left)

Impact Ball Target Ball

Time (ms)

Angle at Fence (deg)

Distance (mm)

Time (ms)

Angle at Fence (deg)

Distance (mm)

1

2

3

4

5

Gate Angles specified by Demonstrator: Angle 1: ………… Angle 2: ………… Table 1.2: Target Ball Offset Found (4 marks):

Target Ball Offset Required (mm) Gate Location Clockwise angle offset of gate Anti-clockwise angle offset of gate

Calculated Offset Required (mm)

Angle 1: …..

Angle 2: …..

Observations of Impact Ball Behaviour at and after the collision (1 mark):

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Observations of Target Ball Behaviour at and after the collision (1 mark): NB the “energy accounted for” in Table 1.3 is to be expressed as a percentage of the initial energy, i.e. for the theoretical energy case: = (Energy Post Collision + Rolling Resistance Loss) / (Theoretical Energy) x 100 The rolling resistance loss may be estimated from the height up the chute that the ball must be released from to just reach the fence. (The 15mm Δh marking is approximately this value). Table 1.3: Analysis of Results (5 marks)

Energy Prior to Collision Energy Post Collision Energy accounted for, based on:

Impact Ball Target Ball

Test No.

Theoretical (Nm)

K.E. (from “Corrected”

Velocity) (Nm)

V’I

(ms-1) K.E. (Nm)

V’T

(ms-1) K.E. (Nm)

Rolling Resist. Energy Loss (Nm)

Theoretical energy

(%)

Corrected energy

(%)

1

2

3

4

5

Due to the number and repetitive nature of the calculations involved in completing Table 1.3 it is strongly recommended that you set up a spreadsheet for this purpose. What is the average proportion of energy that you could account for over these tests ? (1 mark) Theoretical Basis: Corrected Energy Basis:

What aspects of the experiment and or analysis procedure may contribute to any energy losses not accounted for ? (2 marks)

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How and where would you expect frictional forces to act significantly ? (1 mark)

Do the rebound angles and corresponding target ball offsets (experimental and calculated)

noted in Table 1.2 match your expectations? – Comments (2 marks)

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Experiment 2 : Polygon of Forces  Aims: To be able to use graphical methods and analytical methods to solve 2D static equilibrium problems. To use the polygon of force vectors to determine an unknown mass. To demonstrate the vector addition of two forces. Significant Properties/Phenomena: • Equilibrium of coplanar forces on a particle (Chapter 3.3) • Free body diagrams (examples 3.2 and 3.3) • Vector addition of forces (Chapter 2.3 and

2.4) Experimental Procedure: Equipment: 1 pegboard stand with three 50 gram hangers, 11 off 50 gram masses, 1 unknown mass (approx 150 gram), 1 cord with two rings attached, 1 cord with one ring attached, two pulleys. In addition 1x A3 drawing board, 2x A3 sheets of white paper, 30cm ruler, 250mm 45 deg set square, protractor.

Figure 2.1 Task 1 Unknown weight Task 1: Determine the weight of an unknown mass Use the cord with two rings and attach the end loops to the bolts at A and D as shown in Figure 2.1. Hang a 50 gram hanger with a 50 gram mass (100 gram total) from the ring at B. Hang the “unknown” mass (it is approximately 150 grams) from the ring at C. Attach an A3 sheet on the white face of the pegboard behind the cords and masses. Using a ruler, and taking care to view each line of the cord normal to the board (to avoid parallax errors), draw the direction of each vector (cord). Draw the vertical vectors directions also. You should have 5 vector directions. Mark a horizontal reference line on the A3 sheet from the pegboard holes. Transfer the A3 sheet to the drawing board taking care to line up the horizontal and vertical directions correctly. Using the T square, set square and protractor, measure the angle of each vector to the horizontal. Label each line and note the angles. Now draw a vertical line in the middle of the sheet at a scale of 1mm = 1 gram. This represents the vertical force at B (W1 = 0.1 x 9.81 = 0.981N). From the bottom of this line draw the vector for the force FAB at the correct angle. From the top of the vertical line draw a line parallel to FBC. Form the triangle W1, FAB and FBC. Use the ruler to measure the length of each side to get the forces (remember 1mm = 1 gram force = 0.00981N). Note the three forces and their directions on a free body diagram for the point B.

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The line for FBC is the common one between the two free body diagrams for B and C. Draw a vertical line at the right hand end of FBC for the mass of W2. Draw a line through the left hand end of FBC parallel to FCD. Complete the new triangle of vectors FCD, FBC and W2. Measure the lengths to find the forces. Note the forces and directions on a free body diagram of the point C. Note the mass in grams for the unknown. Also solve the equilibrium of point B analytically by resolving the forces into i and j components. Similarly solve for the equilibrium of point C to find W2. Compare this with the answer obtained graphically. Task 2: Demonstration of vector addition. Hang 200 grams (including hanger) over the right hand pulley, E, 250 grams over the left hand pulley, F, and 200 grams from the ring, G, between the two pulleys. Attach an A3 sheet on the white face of the pegboard behind the cords and masses. Carefully draw lines parallel to the three vectors at G. View the cords normal to the pegboard so as to avoid parallax errors. Make sure that the masses are not touching the peg board. Note the relative locations of the points E, F and G. Transfer the A3 sheet to the drawing board and carefully measure the angles of each vector relative to the vertical and horizontal. Draw a triangle at a scale of 1mm = 1gram. Use the vector GE and the vector GF. The third side is equal but opposite to the vector sum of the two diagonal forces. Compare the length of this side and its direction to the known value of the central force.

200g

F E

G

200g 250g

Figure 2.2 Masses over pulleys E and F Analysis for Tasks 1 and 2 From the geometry of the triangles of forces, calculate the theoretical force value for the unknown mass in Task 1 and calculate the resultant of the forces FGE and FGF in Task 2. Compare both of these theoretical results with the values obtained graphically.

*****

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Experiment 2 : Polygon of Forces  REPORT SUBMISSION SHEET: Group Members 1: _____________________ 2: _____________________ 3: _____________________ Lab Group: __________ Date: ______________ Tutor: _____________ Submit your two A3 drawings from tasks 1 and 2, including the original vector directions, and the polygons of forces. (5 marks each) Accurate free body diagrams need to be drawn for particles B and C in task 1 and G in task 2. Task 1 Analysis

B

Equilibrium of point B: (2 marks) FBA = N = i + j N

FBC = N = i + j N

ΣFx = 0 :

ΣFy = 0 :

Figure 2.3 Free body diagram for point B

Equilibrium of point C: (2 marks)

C

Figure 2.4 Free body diagram for point C

FCD = N = i + j N

Force W2 = N

Mass of W2 = g

ΣFx = 0 :

ΣFy = 0 :

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Task 2 Analysis Equilibrium of point G: (2 marks)

FGE = N = i + j N

FGF = N = i + j N

Resultant = i + j N

G

ΣFx = 0 :

ΣFy = 0 :

Figure 2.5 Free body diagram for point G

Difference from graphical value from experiment = i + j N Discuss the sources of error in this experiment and the magnitude of each. Estimate the overall effect of these errors. (2 marks) Comment on the graphical versus analytical results of your experiments. (2 marks)

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Experiment 3 : Energy Balance  Aims: To examine the transfer of energy between linear and rotational components within an apparently closed system through the application of the principle of conservation of energy. To obtain an appreciation for balancing energy in a practical system. Significant Properties/Phenomena: Conservation of energy, rotational and linear kinetic energy, potential energy, mass moment of inertia, measurement. Requirements: In this experiment you have to determine the energy balance for 2 states of the system indicated in Figure 3.1, and to compare them. The first state is with the drop mass and the rotor stationary and the drop mass situated approximately 1 metre above the floor. The second state is immediately prior to the drop mass hitting the ground after falling 1 metre.

h1 ≈1metre

Floor Datum

Drop Mass

string

Attachment lugs

Separation Point

Rotorω ω1 2

Small hub diameter

(a) Rotor Release Configuration(state 1)

(b) Configuration at string disengagement (state 2)

Rotation Axis

ion Axis

Figure 3.1 System States

Nomenclature and given information: Parameter Description Assumed values Vrotor Potential energy of the rotor (J) Trotor Rotational kinetic energy of the rotor (J) Vmass Linear potential energy of the drop mass [mgΔh] (J) Tmass Linear kinetic energy of the drop mass (J) g Acceleration due to gravity 9.81 m s-2 Izz Mass moment of inertia of the rotor 0.1638 kg m2 Md Mass of the drop mass 0.5, 1 and 3 kg MR Mass of the rotor 20.312 kg dR Diameter of rotor 0.254 m dS Diameter of shaft 0.038 m ρ Density of steel 7860 kg m-3 h1 Release height of the drop mass Approximately 1 m v Linear velocity of drop mass (m s-1) ω Angular velocity of rotor (rad s-1)

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General Problem: The principle of conservation of energy states that the sum of all energy forms is constant for a closed system. When considering 2 states of a closed mechanical system, this may be stated as, simply:

V1 + T1 = V2 + T2 In the case of the system concerned:

(Vrotor + Trotor)1 + (Vmass + Tmass)1 = (Vrotor + Trotor)2 + (Vmass + Tmass)2 The process of accelerating or decelerating a mechanical system, containing rotating elements, requires an understanding of the energy required to change the angular velocity of the system. This relationship between rotational kinetic energy, T, and angular velocity, ω is governed by the mass moment of inertia about the polar axis of rotation, Izz, where:

T Izz=12

2ω Eq 1

This can be compared with the equation for linear kinetic energy:

2

21 νmT = Eq 2

(where v = r ω ) For regular shapes the value of Izz can be calculated by well-known formulae derived from first principles. In particular, rotating-solid-discs of radius r, have a mass moment of inertia about their polar axis given by:

Im r

zz =× 2

2 Eq 3

It may also be useful to represent the rotational behaviour of a body by a concentrated mass operating at a particular 'radius of gyration', kg. In that case: Eq 4 I m kzz g= × 2

Test Procedure 1. The experiment calls for you to attach drop masses of varying mass, being 0.5kg, 1kg

and 3kg and to observe the different behaviour of the system. In particular you have to calculate an energy balance for the system between states 1 and 2, for each of the drop masses.

2. Just for fun, before you do the experiment, make a prediction as to the trend in ω1 as the

drop mass is increased from 0.5 to 3kg; and the difference when the string is wrapped around the shaft and the outside of the rotor. Give the reason for your prediction (this will not be marked!!)

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3. Do the experiment: General method: a) Attach and hold string to outer lug (the one on the major rotor diameter) and rotate the

rotor anti-clockwise to raise the drop mass approximately 1 metre off the tray. (NB - If there is not enough room to raise it 1m, then try 0.9m. Ultimately, the height used does not matter, as long as you record the actual value in the table and use it for later calculations.)

b) Ensuring that the mass will not hit your foot, and that the lug does not catch your hands, release the rotor.

c) Record the maximum speed of the rotor with the tachometer. (Maximum speed will occur when the drop mass hits the floor, however, the tachometer is slow to update so you may find the maximum speed occurs a short period after the drop mass hits the floor).

d) Repeat steps (a) to (c) utilising all 3 drop masses provided and using the lugs on the shaft and the outside of the rotor to attach the string. There will be 6 results in all.

4. Noting the change in potential energy of the system between states 0 and 1, calculate the

theoretical kinetic energy in the system as the drop mass hits the ground. Compare this with that obtained by utilising the measured speeds and equations 1 and 2. Comment on any differences found. Use the report submission sheet.

*****

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Experiment 3 : Energy Balance  REPORT SUBMISSION SHEET: Group Members 1: _____________________ 2: _____________________ 3: _____________________ Lab Group: __________ Date: ______________ Tutor: _____________ Test Procedure Just for fun, before you do the experiment, make a prediction as to the trend in ω2 as the drop mass is increased from 0.5 to 3kg; and the difference when the string is wrapped around the shaft and the outside of the rotor. Give the reason for your prediction (this will not be marked!!) Determining the rotor speed

(2 marks) (2 marks)

Test String Position

Drop Mass (kg)

Release Height (m)

Rotor Speed (RPM)

Rotor Speed (rad/s)

1 Rotor 0.5

2 Shaft 0.5

3 Rotor 1.0

4 Shaft 1.0

5 Rotor 3.0

6 Shaft 3.0

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Noting the change in potential energy of the system between states 0 and 1, calculate the theoretical kinetic energy in the system as the drop mass hits the ground. Complete the table below showing the results of these calculations.

(2 marks) (2 marks) (2 marks) (2 marks) (2 marks) (2 marks) Test Initial

Potential Energy (J)

Rotor angular velocity (rad/s)

Mass Final Velocity

(m/s)

Rotor Final KE

(J)

Mass Final KE

(J)

% Error ( ) 100PE KE

PE−

×

1

2

3

4

5

6

Compare this with that obtained by utilising the measured speeds and equations 1 and 2. Comment on any differences found. (4 marks)

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Experiment 4 : Centrifugal Motion  

Aim: To investigate the dynamics of an object moving in a circular motion. Introduction: Rotational motion is a fundamental aspect of the universe. Our model of the atom is one in which electrons orbit a nucleus. The earth rotates about the sun, which in turn rotates and orbits the galaxy. Many mechanical devices involve some form of rotational motion. In all rotating systems a body only moves in a circle because a constant force acts to accelerate it towards the centre of that circle. Should this centripetal force cease, the moving body will fly off at a tangent. Theory: If a particle is to move in uniform circular motion it must experience a net acceleration directed towards the centre of the circle. The force that produces this acceleration is called a centripetal force. The acceleration a has a magnitude given by:

2va

r=

where v is the instantaneous velocity of the particle, and r is the radius of the orbit. The angular velocity is given by ω = v/r. The central force giving the acceleration may be provided by a string or a rod attached to the particle. If this force is F, then F equals the mass, m, times the acceleration i.e.

2

1vF ma mr

= =

The period, T, of the uniform circular motion of the particle is given by

2 circumference 2tangential velocity

rTvπ π

ω= = =

Thus in terms of the period of the motion, which is more easily measured than the velocity, the centripetal force is given by

2

2 2

4m rFTπ

= Eq 1

Experimental Setup and Procedure: Part 1: Disconnect the spring from the suspended mass, m, and balance the top horizontal rod by adjusting the position of the countermass. Then align the radius indicator with the tip of the suspended mass so that they are all vertically aligned. Re-attach the spring and attach the mass carrier to the suspended mass and place the string over the pulley. Add small masses to the mass carrier so that when there is a convenient mass on the mass carrier, the tip of the suspended mass lines up with the radius indicator (as shown in Figure 4.1). Record the force necessary to stretch the spring by this amount, F1.

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Part 2: Detach the mass carrier and string from the suspended mass. Leave the spring attached and rotate the vertical shaft until a sufficient angular velocity is attained so that the suspended mass passes exactly over the previously set indicator at a constant angular speed. Record the time required to complete 10 revolutions. Once this has been done, Equation 1 can be used to determine the centripetal force, F2.

From Equation 1 it will be seen that a determination of the centripetal force responsible for this particular circular path of the suspended mass can be made by measuring the period of this motion and the mass of the suspended body. Compare this computed force with the force required to extend the spring the same amount when the system is not rotating. Are the results in agreement within experimental error?

Stationary system

springMass, m

position over whichmass passes whensystem is rotating

Mass, M F = Mg

Figure 4.1

Carry out the tests and analysis as outlined in Part 1 and Part 2 above for three different settings of the radius indicator and record the results in Table 4.1. For each radius indicator position, measure the time for 10 revolutions three times to determine the average.

*****

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Experiment 4 : Centrifugal Motion   REPORT SUBMISSION SHEET: Group Members 1: _____________________ 2: _____________________ 3: _____________________ Lab Group: __________ Date: ______________ Tutor: _____________ Table 4.1 (6 marks – 2 marks for each column) Observation Number 1 2 3

Mass for Spring Extension M (kg)

Spring extension force, F1

(N)

Suspended mass m

(kg)

Orbit radius

(m)

1

2

Time for 10 revolutions of mass centre

(s) 3

AVERAGE

Time period T

(s/revolution)

Centripetal force F2 (N)

Percentage error

1001

21 ×−F

FF

Discussion of Results: On the next page, add the forces to the Free Body Diagrams of (a) the suspended mass “m”, in the Part 1 set up arrangement (spring being stretched due to the hanger masses) (2 marks) (b) the suspended mass “m”, in the part 2 set up arrangement (spring being stretched due to rotation) (2 marks)

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(a) Part 1 (b) Part 2 If the suspended mass, m, does not hang vertically down, what effect will this have on the experiment ? (4 marks) Discuss the results of your experiment. (3 marks) Comment on the accuracy and the variation in percentage error. (3 marks)

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Experiment 5 : Friction  Aim: The following experiment illustrates a few techniques that may be used to determine the coefficients of friction under different circumstances. Introduction: Friction is a feature of all real mechanical interactions. It is an integral part of out everyday life. Many of the greatest problems involve the elimination of friction and yet most of the apparatus we use depends upon it. Friction is the cause of most wear and often results in unnecessary amounts of energy being wasted. For these reasons we often wish to eliminate it. In other applications, though, friction is a vital quantity for it is the most common means of transmission of mechanical power. It is the friction between the tyres of a car and the road that enables a motor vehicle to move and the friction between the rubber and the metal that allows pulley systems to transmit power. For all of these reasons it is obvious that an understanding of friction and a knowledge of measurement techniques involving this physical phenomenon are important. Friction is a very difficult quantity to measure accurately but by means of a few simplifying assumptions, reasonable values can be obtained. Nomenclature: fs Maximum value of static friction (N) g Acceleration due to gravity (9.81 m s-2) L Length of incline (mm) m Mass of block A (plus additional 50g masses etc as appropriate) N Normal force between bodies A and B (N) h Height of incline (mm) θ Angle of incline (radians) μs Coefficient of static friction Determination of coefficient of static friction Theory: If a force, F, is applied to a body A resting on body B as shown below, then body A will remain at rest below a certain critical value of F.

F A

B

Figure 5.1 This can be explained by the fact that a force of friction acts to oppose the applied force F. The force of friction reaches a maximum just before movement occurs. It is found experimentally that the friction force may be closely represented by the following formula: s sf Nμ= Eq 1

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where fs is the maximum value of static friction μs is the coefficient of static friction N is the normal force of body B upon body A The coefficient of static friction, μs, is found to be dependent on the nature of the contact surfaces of A and B. We investigate whether μs depends upon the load (i.e. force per unit area) on the surfaces in contact. Consider the block resting on an inclined plane in Figure 5.2.

θ

L

hmg

N fs

Figure 5.2

When the block is just about to move sin smg fθ = Eq 2 cosmg Nθ = Eq 3 then because s sf Nμ=

( )2 2

tancossh h

L L hμ θ

θ= = =

− Eq 4

Where 1sin hL

θ − ⎛ ⎞= ⎜ ⎟⎝ ⎠

Experimental a) Using the metal slide and blocks supplied, determine the maximum angle for which the

blocks can sit on the incline without slipping. Measure the height (h) and the length of the slope (L) of the incline. From this determine the coefficient of static friction. Obtain a mean value for this quantity by repeating this measurement five times. Repeat the procedure for each block (aluminium, brass, wood and steel).

b) Add 50g, 100g, 150g and 200g masses to the block, and again obtain values for μs. Carry out one reading only for each mass in this case. From these results determine whether μs is independent of mass (hence the magnitude of the normal force between the two bodies).

Equipment metal slide aluminium, brass, wood and steel sliding blocks set of weights (50g, 100g, 150g and 200g) measuring rule

*****

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Experiment 5 : Friction  REPORT SUBMISSION SHEET: Group Members 1: _____________________ 2: _____________________ 3: _____________________ Lab Group: __________ Date: ______________ Tutor: _____________ Aim: (1 mark)

Table 5.1: Static Friction - Aluminium Sliding Block: (L = mm)

Test Number Height (mm) θ (radians) μs 1 2

Mean value of μs

3 4 5

(2 marks)

Table 5.2: Static Friction - Brass Sliding Block: (L = mm)

Test Number Height (mm) θ (radians) μs 1 2

Mean value of μs

3 4 5

(2 marks)

Table 5.3: Static Friction - Wood Sliding Block: (L = mm)

Test Number Height (mm) θ (radians) μs 1 2

Mean value of μs

3 4 5

(2 marks)

Table 5.4: Static Friction - Steel Sliding Block: (L = mm)

(2 marks) Test Number Height (mm) θ (radians) μs

1 2

Mean value of μs

3 4 5

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Table 5.5: Static Friction – Adjusting the Normal Force, Data: (L = mm)

(2 marks) 50g Load 100g Load 150g Load 200g Load Material Height

(mm) Height (mm)

Height (mm)

Height (mm)

Aluminium Brass Wood Steel

Table 5.6: Static Friction – Adjusting the Normal Force, Results: (L = mm)

No Load (Tables 1-4)

50g Load 100g Load

150g Load

200g Load

Average

Material μs μs μs μs μs μs Aluminium Brass Wood Steel

(2 marks)

What do you conclude regarding the effect of normal force on the maximum static friction force? (2 marks)

What relation do you use to determine the coefficient of static friction? (2 marks)

Give three everyday examples where a low coefficient of friction is important? (1½ marks)

- - - Give three everyday examples where a high coefficient of friction is important? (1½ marks)

- - -

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Experiment 6 : Governor Experiment 

In 1788 Matthew Boulton and James Watt invented a regulating system with a governor to control the speed of a steam engine, see Figure 6.1. This device made it possible to effectively control the speed of steam engines. A similar type of governor is still being used for a number of high pressure fuel injection pumps of diesel engines.

Shaft driven by engine

To steamvalve

Ball Ball

Pivot Points

Slides on shaft

Figure 6.1 Steam engine governor

Aim: In this laboratory experiment we will investigate the operation of a governor. The static behaviour of the governor, that is, the governor height versus rotational speed is to be examined, comparing measurements from the experiments with a theoretically calculated value. Significant Properties/Phenomena: Centrifugal Force, Gravitational Force. General Procedure The governor will be rotated at a certain angular velocity by adjusting the voltage control knob on the power supply. The height of the sliding collar will then be determined using the scale on the rotating shaft. The measurement results will then be tabulated, plotted and compared to a theoretically determined value.

Equipment: • Governor driven by a DC motor • Power supply with voltage control knob to regulate the angular velocity of the governor • Tachometer to measure the rpm of the governor and determine the frequency at which the

governor rotates, thus the angular velocity • Small wooden set square • 30cm metal rule (or similar)

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Experimental Procedure (relating to Tasks 2 and 3) Before turning the power supply on, adjust the peg marker on the support leg to line up with the sliding collar. Measure the distance from the base plate to this marker and record the measurement as h0 in Table 6.1. The power supply and tachometer need to be used in unison for this part of the experiment. Referring to Table 6.1 and starting with the highest rpm, use the tachometer to measure the rpm of the governor and adjust the power supply voltage dial to obtain a reading as close to the ‘target n’ as possible. Record this value in the ‘actual n’. At this setting, adjust the peg marker to again align with the sliding collar (now further up the central rod). Measure the distance from the base plate to the marker and record the measurement as h1 in Table 6.1. Determine the change in height of the sliding collar Δh = h1 – h0. Repeat the above for the other angular velocities of Table 6.1, reducing the rpm each time. Turn the power off to the Governor power supply when you are finished with your measurements.

*****

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Experiment 6 : Governor Experiment  REPORT SUBMISSION SHEET: Group Members 1: _____________________ 2: _____________________ 3: _____________________ Lab Group: __________ Date: ______________ Tutor: _____________ Task 1 (2 marks) Derive the formula relating the angular velocity of the rotating shaft n in revolutions per minute [rpm] with the frequency of rotation f in [Hz], and derive the formula which relates f [Hz] with the angular velocity ω in [rad/s]. Formula to compute f from n: Formula to compute ω from f ……………………………… ………………………………… Task 2 (4 marks) Follow the procedure as discussed previously in “Experimental Procedure” to fill in columns 2, 5, 6 and 7. Use the equations derived in Task 1 to fill in the columns for f and ω.

Table 6.1, Measurement results

Initial height, h0 [mm] = ________ Target n

[rpm] Actual n

[rpm] f

[Hz] ω

[rad/s] Voltage

[V] h1

[mm] Δh

[mm] 125

150

175

200

225

250

300

350

400

450

Task 3 (2 marks) Determine experimentally n [rpm] for which the angles of the arms are approximately 45 [deg] with the rotational shaft. Note: use the small wooden set square as an aid for this task. n =

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Task 4 (4 marks) Plot the results of Table 6.1 in Figure 6.2. Indicate your measured values with a ‘*’ and draw a smooth line through the points. Label the axes appropriately.

d

dM

m, rm, r

g

l

l

0

100

200

300

400

500

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Figure 6.2 Plot of results Figure 6.3 Model of System Table 6.2, Data for Model Variable Description Value d Distance of arm joints 26 [mm] g Gravitational acceleration 9.81 [m/s2] l Length of arms 65 [mm] M Total mass of sliding collar 90 [g] m Mass of one ball 60 [g] r Radius of ball 0 [mm] The governor can be approximated by a model as shown in Figure 6.3. The data relating to this model is given in Table 6.2. Note that the mass of the arms is assumed to be negligible and the balls are modelled as point masses.

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Task 5 (4 marks) In Figure 6.4, a free body diagram with the forces acting on the upper and lower left arms of the model are shown. The arms are at an angle of 45° with the rotating axis. Compute the following forces using this free body diagram. Write down the equation(s) that you used to compute these forces. If you are uncertain of the FBD analysis, review the online Pre-Lab for this experiment.

g

d/2

d/2

Fv2

Fh1

Fh2

m

FvcFhc

Fh3

Fv1

Fv3

Computation of Fvc: Ans:

Computation of Fhc: Ans: Computation of Fv2: Ans:

Computation of Fh2: Ans:

Figure 6.4, Free Body Diagram Task 6 (4 marks) Given that Fh2 = Fcentrifugal - Fh1 , with Fcentrifugal = ω2 × r × m and r [m] the distance from the rotating shaft axis to the ball, compute n [rpm] for which the arms will be at an angle of 45° to the rotational shaft axis. Compare this value of n with the value measured in Task 3. Computation of n:

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