TITLE : Simple pendulum

OBJECTIVE : To show that the period of vibration for a pendulum is independent of the mass of the bob, and to determine the gravitational acceleration, g.

THEORY :

Newton’s 2nd law

Hence, ……………………………….………………. ( Equation 1 )

Since , Equation 1 ……………….……. ( Equation 2 )

Rearrange Equation 2 ….………….…..... ( Equation 3 )

For small θ, Equation 3 …..……………………… ( Equation 4 )

where

Period of vibration, …………..……….. ( Equation 5 )

APPARATUSTwo Pendulum bob; Thread; Ruler; Stopwatch; G-clamp; boss, clamp and retort stand

EXPERIMENTAL PROCEDURE1. Attach the thread to the smaller bob

2. Set the length, L to 1 m, and then tie it to the pendulum setup as shown in figure.

3. Displace the bob slightly to the side and then release it, and obtain the time t1 for 20 oscillations.

4. Without disturbing the oscillations of the simple pendulum, obtain again the time t2

for 20 oscillations.

5. Repeat the above procedures for 6 more sets of reading, each time decreasing the length by 10 cm.

6. Repeat Steps 2 – 5 with the bigger pendulum.

RESULTS & DISCUSSIONTabulate L, t1, t2, average t, T (period of vibration), and T2 for the two different pendulum bob.

Comment on the period of vibration for pendulum with different pendulum bob.

Plot the graph of T2 vs. L and obtain the gradient of the graph

Calculate the gravitational acceleration, g, using Equation 5, and comment on the possible errors involved in the experiment.

REFERENCE1. Hibbeler, R.C., Engineering Mechanics Dynamics, 11th Edition, Prentice-Hall International.

2. Meriam J. L. and Kraige L. G., Engineering Mechanics Dynamics, 6th Edition, John Wiley & Sons, Inc.

TITLE : Friction on an inclined plane

OBJECTIVE : To determine the coefficient of static and kinetic friction for various materials.

THEORY

Figure 1. A block of weight W moving down an inclined plane

For a block on the verge of sliding down an inclined plane, the static friction is given by µs = Tan max (Equation 1)

For a block sliding down an inclined plane at a uniform velocity, the kinetic friction is given by µk = Tan (Equation 2)

Figure 2. A block of weight W being pulled up an inclined plane.

For a block of weight (W) on the verge of being pulled up an inclined plane, the pulling force is given by Pmax = W (Sin + µs Cos ) (Equation 3)

For a block on the verge of being pulled up an inclined plane at a uniform velocity, the pulling force is given by P = W (Sin + µk Cos ) (Equation 4)

APPARATUSAn inclined plane with a pulley at the top edge; Two blocks – one of wood and the other of brass; A protractor at the hinged end of the inclined plane that allows the slope plane to be measured.

EXPERIMENTAL PROCEDURECoefficient of Static Friction:

1. Select the wooden block and lay the block with its smooth edge in contact with the plane.

2. Slowly increase the angle of inclination of the plane.

3. Note the angle of inclination max at the moment when the block first begins to slide down the plane, and hence calculate the coefficient of static friction (refer to Equation 1).

4. Repeat the procedure with the other side of the block that has been fitted with a rough paper and with the brass block.

Coefficient of Kinetic Friction:

1. Measure the weight of the wooden block and lay with its smooth edge in contact with the plane.

2. Connect the cord to the wooden block over the pulley, and attach the weight container to the cord at the other end.

3. Set the incline plan to a predetermined angle such that the block is sliding down the plane.

4. Add the weights into the container until the block begins to move up the plane at a constant speed.

5. Repeat Steps 3 – 4 with another angle of plane inclination in order to determine the average value for the coefficient of kinetic friction.

6. Repeat the procedure with the other side of the wooden block and with the brass block.

RESULTS & DISCUSSIONTabulate the results and determine the coefficients of static and kinetic friction for the different materials.

Discuss on the coefficients of friction obtained.

Comment on the possible errors involved, and ways to improve the experimental result.

REFERENCE1. Meriam J. L. and Kraige L. G., Engineering Mechanics Statics, 4th Edition, John Wiley &

Sons, Inc.

2. Hibbeler, R.C., Engineering Mechanics Statics, 12th Edition, Prentice-Hall International.

TITLE : Mass Moment of Inertia of Flywheel

OBJECTIVE : To determine the mass moment of inertia of a flywheel.

THEORYFlywheel is a mechanical device with significant moment of inertia used as a storage device for rotational energy. Flywheels resist changes in their rotational speed, which helps steady the rotation of the shaft when a fluctuating torque is exerted on it.

For a thin solid disk of flywheel, the mass moment of inertia is shown in Equation 1, where m = mass, r = radius of the flywheel

(Equation 1)

For rotational motion, Newton’s second law (see Equation 2) can be adopted to describe the relation between the applied torque, T and angular acceleration, .

T = I. (Equation 2)

Note that for constant angular acceleration, the angular displacement of a rotating object can be obtained from Equation 3

= t + ½ t2 (Equation 3)

APPARATUSFlywheel apparatus; A set of weights; A stopwatch and ruler.

EXPERIMENTAL PROCEDURE1. Record the measurements of the radius of torque pulley (rp) and flywheel (r), as well as the

mass of the flywheel (m).

2. Wound a cord around the torque pulley and take a load hanger of known weight and hang it at the free end of the cord.

3. Place a load on the load hanger and hold the load in position

4. Adjust the flywheel so that the arrow marked on it aligns with the arrow marked on the rig.

5. Set the stopwatch to zero.

6. Release the load while simultaneously pressing the stopwatch button.

7. After 1 revolution, stop the flywheel and the stopwatch simultaneously.

8. Record the time taken for the flywheel to rotate 1 revolution.

9. Repeat the experiment twice to get an average value of time taken for the flywheel to rotate 1 revolution.

10. Repeat Steps 3 – 9 for another 4 different sets of load.

11. Repeat the experiment by attaching the small disk and the ring to the flywheel.

Total load, W on torque pulley (N)

Applied Torque (Nm) = W * rp

Time taken (Sec)

t1 t2 t3 Average t

RESULTS AND DISCUSSIONTabulate the results, showing the angular acceleration of the pulley for different loading.

Plot the graph of torque vs. angular acceleration (both with and without the small disk) and obtain the experimental value for the mass moment of inertia. Comment on the difference between the two values.

Compare the theoretical value of the mass moment of inertia of flywheel (Equation 1) with that obtained without the small disk. Give your comment on the discrepancy obtained, and ways to improve the experimental result.

REFERENCE1. Meriam J. L. and Kraige L. G., Engineering Mechanics Dynamics, 6th Edition, John Wiley &

Sons, Inc.

2. Hibbeler, R.C., Engineering Mechanics Dynamics, 11th Edition, Prentice-Hall International.

TITLE : Shear Force Measurement of Simply Supported Beam

OBJECTIVE : To show that the shear force at a cut section of a beam is equal to the algebraic sum of the forces acting to the left or right of the section.

THEORY L2

L1 W1 W2 W3 X

L3 X

RA RB

Shear force at section X-X is

S.F x-x = W1 + W2 + W3 - RA = R b

APPARATUSA pair of simple supports; A special beam with a cut section; A set of weights with several load hangers.

EXPERIMENTAL PROCEDURE1. Switch on the digital indicator that is connected to the transducer. For stability of the reading,

the indicator must be switched on 10 minutes before taking readings.

2. Hang the three load hangers to the beam.

3. Note the reading of the locations of the load hangers, i.e., L1, L2 and L3.

4. Note the indicator reading. If it is not zero, press the tare button on the indicator.

5. Place a desire load on each load hanger and record the value of W1, W2 and W3.

6. Record the indicator reading (i.e., shear force at the cut section).

7. Repeat Steps 4 – 7 for another 4 different sets of loading condition.

Load and its distance from the left support Shear force –Experimental, N

W1 L1 W2 L2 W3 L3

RESULTS & DISCUSSIONCalculate the theoretical value of shear force at the cut section for each loading condition and tabulate it together with the results obtained experimentally.

Comment on the difference between the theoretical and experimental results, and the possible errors involved in the experiment.

Draw the shear force diagram of the simply supported beam for any two loading conditions.

REFERENCE1. Benham, P. P. Crawford, R. J. and Armstrong, C. G., Mechanics of Engineering Materials,

2nd Edition, Longman.

2. Hibbeler, R.C., Mechanics of Materials, 7th Edition, Prentice-Hall International.

TITLE : Deflection of Simply Supported Beam

OBJECTIVE : To understand the relationship between deflection of simply supported beam and the applied load.

THEORY :The design of beams falls into two parts (i) the consideration of the stress in bending and shear and (ii) the deflection of the beam under load. It is frequently the case that the design of beam is dictated by the permissible deflection.

L/2 W L/2

P Q

Figure above shows a simply supported beam loaded at mid span with a concentrated load. The

deflection at the mid span is given by

Where W = applied load, NL = Length of the beam, mE = Young’s modulus of elasticity of the beam, N/m2

I = second moment of area, m4

APPARATUSA steel channel base with two simple supports. A set of weights with load hanger. A 25mm (wide) by 5mm (thick) by 1.4m (long) mild steel (E = 200 GPa) beam, and a dial gauge to measure the deflection of the beam.

EXPERIMENTAL PROCEDURE1. Measure the dimensions of the beam and note the reading.

2. Place the beam onto the supports, and then record the length, L between the two supports.

3. Position the ‘C’ hook and load hanger at the mid span of the beam.

4. Set the dial gauge with a flat anvil on the hook.

5. Assume the load hanger as zero load, adjust the dial gauge to get zero reading.

6. Load the beam with 1.0 N and note the corresponding deflection.

7. Repeat the experiment twice to get an average value of the deflection under such loading.

8. Repeat Steps 5 – 7 up to 5.0 N.

Load (N)Deflection, mm

1 2 3 Average

1.0

2.0

3.0

4.0

5.0

RESULTS & DISCUSSIONPlot the graph of deflection, against load, W and obtain the gradient of the graph.

Calculate the second moment of area of the beam, and then obtain the value of (i.e.,

gradient of against W). Comment on the discrepancy of the results obtained.

Discuss a few beam designs that are good for bending purposes.

REFERENCE1. Benham, P. P. Crawford, R. J. and Armstrong, C. G., Mechanics of Engineering Materials,

2nd Edition, Longman.

2. Hibbeler, R.C., Mechanics of Materials, 7th Edition, Prentice-Hall International.

TITLE : Strain Measurement of Cantilever Beam

OBJECTIVE : To understand the relationship between strain and the applied load on cantilever beam

THEORY

Bending stress, σ, at distance y from the neutral axis of the beam is given by

σ = My / I ……………………………………………………………... (1)Where M = Bending moment, N.mm

y = Distance from the neutral axis, mm I = Second moment of area, mm4

Since the Elastic Modulus, E = σ / ε ……………………………………... (2)

where ε = strain

Substitute equation (2) into equation (1), we get M = ε.(EI / y) …….... (3)

APPARATUSA clamp support; A steel beam; A set of weights; A 8 channel data acquisition system

EXPERIMENTAL PROCEDURE1. Note the dimensions of the cantilever beam and the location of strain gauge.

2. Switch on the computer and the data acquisition module. For stability of the reading, the data acquisition must be switched on 10 minutes before taking readings.

3. Run the WINviewCP32 software on the desktop.

4. Change the ‘Specific Time Interval’ from 10 to 2 second per sample.

5. Click the ‘Setup’ button to turn on ‘Module 1’, and the channels that connect to the strain gauge. When the setting is complete, save the setting by clicking ‘OK’ button

6. Press the ‘Start’ button and choose the ‘Overwrite the file’ option before clicking the ‘OK’ button to measure the initial reading of the strain gauge.

7. Allow approximately 20 seconds of readings to be captured before pressing the ‘Stop’ button. (The values displayed in the boxes of chosen channels are the strains in the members measured in millistrains.)

8. Place a 5N load on the load hanger to provide bending on the cantilever beam.

9. Click ‘Start’ button on the menu page and choose the ‘Append to the file’ option. Then, click the ‘OK’ button to measure the reading of the strain gauge under the loading condition. (Allow 20 second of reading to be captured.)

10. Unload the 5N weight and then repeat Steps 6 – 9, up to 25N. (Note that the measurements of the strain gauge, i.e., ‘data1’ (Excel file) should first be copy from the ‘WVCP32V3’ folder in Local Disk D-drive to your pen-drive, each time before repeating the experiment !!!)

RESULTS & DISCUSSIONTabulate the measurements of strain gauge and obtain the average value for each loading condition. (Multiply the average value by 1000 as strains are normally expressed in microstrains)

Calculate the strains of the beam at both the surfaces (i.e., difference between the final and the initial measurements of the strain gauge) and comment on the difference of these values.

Calculate the bending moment, M, at the strain gauge, and then plot the graph of M vs. ε.

Calculate the second moment of area of the steel beam, then estimate the Young’s modulus of the beam from the graph (Equation 3) and comment on the value obtained.

REFERENCE1. Hibbeler, R.C., Mechanics of Materials, 7th Edition, Prentice-Hall International.

2. Benham, P. P. Crawford, R. J. and Armstrong, C. G., Mechanics of Engineering Materials, 2nd Edition, Longman.