Mechanical Vibration Lab Manual

LIST OF APPARATUSES

1) Simple Pendulum

2) Mass Spring System

3) Universal Vibration System

4) Bifilar/Trifilar Suspension

5) Torsional Vibration System

6) Shaft Whirling Apparatus

LIST OF EXPERIMENTS

To study un-damped free vibrations of a simple pendulum by determining the natural frequency and time period of oscillation, and comparing the experimental results with theoretical expectations.
To study un-damped free vibrations of a vertical spring mass system by determining the natural frequency and time period of oscillation, and comparing the experimental results with theoretical expectations.
To study the role of different parts of universal vibration system and develop an understanding of free and forced & damped and un-damped vibrations.
To determine the time period and frequency of free un-damped vibrations of a spring-dashpot system and compare the experimental results with theoretical outcomes.
To determine the damping ratio (ζ), actual damping co-efficient (C) and critical damping co-efficient (C_c)for free damped vibrations of spring-dashpot system.
To determine the amplitude and magnification factor of forced un-damped vibrations due to rotating unbalance and draw corresponding resonance curve.
To experimentally determine the range of frequency ratio for region of vibration amplification and vibration isolation for forced vibrations using base excitation in Universal Vibration Syste.
To determine the moment of inertia of a hollow cylinder using Trifilar Suspension.
To determine the moment of inertia of a solid cylinder using Trifilar Suspension.
To determine the moment of inertia of a square beam using Bifilar Suspension.

EXPERIMENT 1

OBJECTIVE:

To study un-damped free vibrations of a simple pendulum by determining the natural frequency and time period of oscillation and comparing the experimental results with theoretical expectations.

APPARATUS:

Simple Pendulum, Steel Rule

THEORY:

A pendulum is an object that is attached to a pivot point so it can swing freely. This object is subject to a restoring force that will accelerate it toward an equilibrium position. When the pendulum is displaced from its place of rest, the restoring force will cause the pendulum to oscillate about the equilibrium position. In other words, a weight attached to a string swings back and forth.

A basic example is the simple gravity pendulum or bob pendulum. This is a weight (or bob) on the end of a massless string, which, when given an initial push, will swing back and forth under the influence of gravity over its central (lowest) point.

The regular motion of pendulums can be used for time keeping, and pendulums are used to regulate pendulum clocks. A simple pendulum is an ideality involving these two assumptions:

· The rod/string/cable on which the bob is swinging is massless and always remains taut.

· Motion occurs in a plane.

Under the above assumptions, the equation of motion of simple pendulum can be written as (see Figure 1):

(Small angle approximation)

Where;

l: the length of string up to the center of the bob in meter

θ: swing angle in radian.
I: second moment of inertia about pivot in kg.m²

From the equation of motion, one can find the natural frequency as follows:

PROCEDURE:

Set the string length at 30cm by using the measuring tape, the length is measured from the end of fixing nut to the center of the ball. Displace the ball to a certain level and release.
Change the length of the string to a number of values like 40, 50, 60,70cm and measure the time required to complete 30 oscillations. Evaluate the time period by dividing the measured time by 30.
Evaluate g by using the equation.

OBSRVATIONS & CALCULATIONS:

String Length (l) in cm	Time for 30 oscillations (sec)	τ_exp (sec)	τ_theo (sec)	% error in τ	ω_n_,exp (rad/s)	ω_n_,theo (rad/s)	% error in ω_n
30
40
50
60
70

PRECAUTIONS:

Make sure the pendulum only vibrates in vertical plane.
Do not displace the bob with very large angle.
Switch off the fan to reduce the air resistance.
Include the hook and radius of the bob in length of the pendulum.
String should be light in weight.

COMMENTS:

Compare the measured values with actual value.
Discuss the sources of inaccuracies and state how errors can be reduced.

EXPERIMENT 2

OBJECTIVE:

To study un-damped free vibrations of a vertical spring mass system by determining the natural frequency and time period of oscillation and comparing the experimental results with theoretical expectations.

APPARATUS:

Simple Mass Spring System

THEORY:

Helical or coil springs are commonly used in wide variety of mechanical systems. Their basic work is to produce a force which is proportional to the deflection or vice versa. Figure-1 shows a typical force-deflection diagram for a helical spring. In the linear region of this diagram, the relation between force and deflection obeys Hook's Law:

∆F=k∆x

Where k is called stiffness of the spring (N/m). The reciprocal of k is called deflection coefficient which is the deflection introduced by a unit force. If a mass is attached to one end of a spring while the other end is fixed, the resulting system is called simple mass-spring which oscillates harmonically according the following equation (neglecting all types of damping forces):

PROCEDURE:

Part A; Static Deflection

1. Hang the spring with the hook of the frame.

2. Add weights in incremental fashion and record the corresponding deflection.

Part B; Oscillatory Motion

1. Pull down the mass and release it to introduce oscillatory motion.

2. Add masses incrementally and measure the time of 20 oscillations.

3. Record your readings as shown in the following table.

Mass (g)	Static Deflection (δ_st) (mm)	Time for 20 oscillations (sec)	Time Period			ω_n			% error inω
Mass (g)	Static Deflection (δ_st) (mm)	Time for 20 oscillations (sec)	τ_exp (sec)	τ_insp (sec)	τ_theo (sec)	ω_n,exp (rad/s)	ω_n,insp (rad/s)	ω_n,theo (rad/s)	% error inω

PRECAUTIONS:

Make sure the system only vibrates in vertical plane.
Do not produce very large deflections/oscillations in spring.
Include the mass of the hanger in total mass.
Spring should not be deformed.

COMMENTS:

Find reasons of inaccuracies for both parts of the experiment and state how we can reduce errors.

EXPERIMENT 3

OBJECTIVE:

To study the role of different parts of universal vibration system and develop an understanding of free and forced & damped and un-damped vibrations

APPARATUS:

Universal Vibration System

THEORY:

A vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium/mean position. It can be useful as well as harmful, so we need to understand this phenomenon quite clearly in order to avoid it or take advantage from it.

It can cause one or more of the following undesirable effects:

· Structural failure due to excessive displacement and stress

· Fatigue failure

· Noise

· Slippage and dislocation of joints

· Wear due to relative motion between components

· Discomfort while transportation

· Vibration transmission to connected structures

It can be exploited for positive purposes such as:

· Transporting powder

· Sieving and sizing products

· Consolidating poured concrete

· Applying massage

· Removing dental plaque

A vibratory system consists of three basic parts spring (stores potential energy), mass (stores kinetic energy) and a damper (dissipates energy).

Free vibration occurs when a mechanical system is set in motion with an initial input and allowed to vibrate freely. Examples of this type of vibration are pulling a child back on a swing and letting go, or hitting a tuning fork and letting it ring.

Forced vibration is when a time-varying disturbance (load, displacement or velocity) is applied to a mechanical system. The disturbance can be a periodic and steady-state input, a transient input, or a random input. Examples of these types of vibration include a washing machine shaking due to an imbalance, transportation vibration caused by an engine or uneven road, or the vibration of a building during an earthquake.

Damped vibrations occur when the energy of a vibrating system is gradually dissipated by friction and other resistances, the vibrations are said to be damped. The vibrations gradually reduce or change in frequency or intensity or cease and the system rests in its equilibrium position.

Undamped vibrations have no resistive force to act on the vibrating object. As the object oscillates, the energy in the object is continuously transformed from kinetic energy to potential energy and back again, and the sum of kinetic and potential energy remains a constant value.

Main Parts of Universal Vibration System:

1. Unbalance exciter

2. Beam

3. Damper

4. Exciter Control Unit

5. Drum for Recording Vibrations

6. Suspension and Oscillating Spring

7. Frame

Frame:

The experimentation set-ups are mounted in an aluminum sectional frame which is rigid but light weight. Rapid attachment and simple adjustment of components using T-slots, T-slot blocks and clamping levers can be done on this.

Imbalance Exciter:

It is a kind of exciter in which an eccentric mass is provided which rotates and provides a forcing factor to get the forced vibrations.

Beam:

A cantilever is a beam anchored at only one end. The beam carries the load to the support where it is forced against by a moment and shear stress.

Damper:

An oil filled damper is provided in the system for the dissipation of the energy produced during vibrations. Its damping severity can be changed screwing the needle valve in or out.

Exciter Control Unit:

The unit controls the frequency of excitation through potentiometer and a digital counter is also placed on it for displaying excitation frequency in Hertz. Power and recorder switches are also the part of this unit.

Drum Recorder:

Drum recorder is placed in front of the stylus. Graph paper rotates at its outer surface and records results.

Springs:

The helical suspension springs are available with universal vibration system to provide the required stiffness.

Spring Characteristics:

D = diameter of coil, d=diameter of spring wire, k = stiffness

(a) 0.75N/mm , D= 18.3mm , d =1.05mm

(b) 1.5N/mm , D= 17.6mm , d =1.6mm

EXPERIMENT 4

OBJECTIVE:

To determine the time period and frequency of free undamped vibrations of a spring-dashpot system and compare the experimental results with theoretical outcomes.

APPARATUS:

Universal Vibration System

THEORY:

A system is said to be a cantilever beam system if one end of the system is rigidly fixed to a support and the other end is free to move.

Vibration analysis of a cantilever beam system is important as it can explain and help us analyse a number of real life systems. Real systems can be simplified to a cantilever beam, thereby helping us make design changes accordingly for the most efficient systems.

When given an excitation and left to vibrate on its own, the frequency at which a cantilever beam will oscillate is its natural frequency. This condition is called Free vibration. The value of natural frequency depends only on system parameters of mass and stiffness. When a real system is approximated to a simple cantilever beam, some assumptions are made for modelling and analysis (Important assumptions for undamped system are given below):

v The mass (m) of the whole system is considered to be lumped at the free end of the beam

v No energy consuming element (damping) is present in the system i.e. undamped vibration

v The complex cross section and type of material of the real system has been simplified to equate to a cantilever beam

The governing equation for such a system (spring mass system without damping under free vibration) is as below:

mẍ + kx=0

ẍ + ω_n²x=0

k the stiffness of the system is a property which depends on the length (l), moment of inertia (I) and Young's Modulus (E) of the material of the beam and for a cantilever beam is given by:

OBSERVATIONS & CALCULATIONS:

Time period is the time needed for one complete cycle of vibration to pass a given point. Frequency and time Period are in a reciprocal relationship. Theoretical time period & frequency are calculated by following formulae:

Where;

m= mass of beam = 1.64kg L= length of the beam = 670mm

k = stiffness of the spring in N/m a = distance between beam’s fixed end and spring pivot point

Experimental time period & frequency are calculated by following formulae:

Where;

d = wave length of vibration (mm) v = velocity of recorder = 20 mm/s

PROCEDURE:

Measure the length of beam.
Set the distance ‘a’ from fix end of beam to spring pivot point and measure it.
Insert the stylus into the holder at the free end of the beam.
Wrap graph paper onto the recorder.
Dismount the unbalance exciter.
Switch on the apparatus.
Switch on the recorder from the control unit.
Now apply force by hand on the free of the beam in downward direction and release it.
The oscillations are started. Now wait for the beam to return to its initial position.
Switch off both the recorder and the apparatus.
Take out the graph paper from the recorder carefully.
Measure the distance between initial and end points a single wave of vibration.
Now calculate the time period and frequency using the above given formulae.
Repeat the experiment by varying the spring constant (k) and distance (a).

TABLE:

Sr. No.	Spring Stiffness (k) in N/mm	Distance (a) in mm	τ_theo in sec	τexp in sec	f_theo in Hz	f_exp in Hz	% error
1	1.5	400
2	1.5	500
3	1.5	600
4	3.0	400
5	3.0	500
6	3.0	600

PRECAUTIONS:

1. Do not disturb the beam during vibrations.

2. Secure the stylus in the holder tightly so that correct pattern can be recorded.

3. Control unit should be operated with care.

4. Beam should be aligned horizontally before starting the experiment.

5. Base of the apparatus should be stable while taking measurements.

COMMENTS:

……………………………………………………..……………………………………………………………………...……………………………………………………………………...………………………………………………………………………..……………………………………………………

EXPERIMENT 5

OBJECTIVE:

To determine the damping ratio(ζ), actual damping co-efficient (C) and critical damping co-efficient(C_c ) for free damped vibrations of spring-dashpot system.

APPARATUS

Universal Vibration System

THEORY:

In the absence of any form of friction the system will continue to oscillate with no decrease in amplitude. However, if there is some form of friction then the amplitude will decrease as a function of time. This frictional force provides damping to the system.

The damping can be of three major types’ i.e. viscous damping, coulomb damping & hysteresis damping.

Viscous damping is caused by such energy losses as occur in liquid lubrication between moving parts or in a fluid forced through a small opening by a piston, as in automobile shock absorbers. The viscous-damping force is directly proportional to the relative velocity between the two ends of the damping device.

Coulomb damping is a type of constant mechanical damping in which energy is absorbed via sliding friction. The friction generated by the relative motion of the two surfaces that press against each other is a source of energy dissipation.

Hysteresis damping is caused by the friction between the internal planes that slip or slide as the material deforms is called hysteresis damping.

The type of damping used in our system is viscous damping. When a viscous damper is added to the model this outputs a force that is proportional to the velocity of the mass. The damping is called viscous because it models the effects of a fluid within an object. The proportionality constant c is called the damping coefficient and has units of Force over velocity (N s/m).

F_d= -cv = -cẋ

PROCEDURE

1) Set the distance ‘a’ from fix end of beam to spring pivot point equals to 600mm.

2) Engage the spring having k = 1.5 N/mm.

3) Insert the stylus into the holder at the free end of the beam.

4) Wrap graph paper onto the recorder.

5) Engage the viscous damper with the beam.

6) Set the position of needle valve to five turns clockwise.

7) Switch on the apparatus.

8) Switch on the recorder from the control unit.

9) Now apply force by hand on the free of the beam in downward direction and release it.

10) The oscillations are started. Now wait for the beam to return to its initial position.

11) Switch off both the recorder and the apparatus.

12) Take out the graph paper from the recorder carefully.

13) Measure the peak amplitude of 1^st and 2^nd cycle.

14) Now calculate the ζ , C & C_c using the formulae given below.

15) Put the readings in the table.

16) Repeat the experiment for different positions of needle valve.

OBSERVATIONS & CALCULATIONS:

Find damping ratio using this formula:

Where; δ = logarithmic decrement =

Now find C_C using this relation: C_C= 2mω_n

Where;

Now; C = ζC_c

TABLE:

Sr. No.	Needle Valve Position (No. of Clockwise Turns)	C_c (Ns/m)	C (Ns/m)	ζ
(1)	5
(2)	10
(3)	15

PROCEDURE

1) Engage the damper in proper manner so that its screw does not hinder the motion of the beam.

2) Measure the amplitude from the graph very carefully.

3) Don’t touch the beam during vibrations.

COMMENTS

…………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

EXPERIMENT 6

OBJECTIVE:

To determine the amplitude and magnification factor of forced un-damped vibrations due to rotating unbalance and draw corresponding resonance curve.

APPARATUS:

Universal Vibration System

THEORY:

In order to define the resonance of the system we need to find the natural frequency of the system in free vibration state. By that, we may know theoretically value of the natural frequency. Next, the exciter will be used to give desired forced to the system. As we know exciter is capable to generate different type of forcing signal e.g. sine, swept sine, rectangular, triangular etc.

The effect of damping is to limit the maximum response amplitude and to reduce the sharpness of resonance, which can be defined as occurring when the drive frequency Ω equals the natural frequency of the system, ω.

Based on our learning of the resonance, this phenomenon only occurs if the frequency of the excitation coincides with the frequency of the system. As the reaction of the phenomenon happen in a short time, we may need to define a suitable frequency interval to record the amplitudes that will occur.

Free-Body Diagram:

PROCEDURE:

Tabulate a table that consist value of the desire frequency and responding values of amplitude. Plot a suitable frequency interval in order to keep a good record keeping afterward.
Set up mechanical drum recorder on the Spring-Dashpot system for plotting the graph.
Switch on the control unit; adjust the desire frequency on the ten-turn potentiometer.
Mount the rotating unbalance exciter.
Switch on rotating unbalance exciter with the frequency adjustable on the ten-turn potentiometer as the mechanical drum recorder record the graph plotted.
After the graph plotted, switch of the control unit and analyses the data on the graph to find the amplitude.
Repeat above steps with different forcing frequencies as mentioned in the table below.

OBSERVATIONS & CALCULATIONS:

At: a=600 k=3N/mm

Frequency (f) in Hz	7	7.5	8	8.5	9	8.5
Forcing Frequency (ω) in rpm	420	450	480	510	540	570
Frequency Ratio (r)
Peak Amplitude (X) in mm
Magnification Factor (M)

Where;

m= mass of beam = 1.64kg L= length of the beam = 670mm

k = stiffness of the spring in N/m a = distance between beam’s fixed end and spring pivot point

M = mass of the unbalance exciter = 0.4kg

L₁ = distance between beam’s fixed end & exciter location = 350mm

1 rad/s =

COMMENTS:

1. Draw the resonance curve between ‘M’ and ‘r’.

2. Discuss the effect of ‘r’ on ‘M’.

EXPERIMENT 7

OBJECTIVE:

To experimentally determine the range of frequency ratio for region of vibration amplification and vibration isolation for forced vibrations using base excitation in Universal Vibration System.

APPARATUS:

Universal Vibration System

THEORY:

The vibrational amplitude of a system depends upon the ratio between its natural frequency and the frequency with which it gets excitation. Up to certain range of frequency ratio the system exhibits vibration amplification and afterwards its vibrations start getting isolated.

This graph shows the phenomena described above.

PROCEDURE:

Tabulate a table that consist value of the desire frequency and responding values of amplitude. Plot a suitable frequency interval in order to keep a good record keeping afterward.
Set up mechanical drum recorder on the Spring-Dashpot system for plotting the graph.
Dismount the rotating unbalance exciter.
Switch on the control unit; adjust the desire frequency on the ten-turn potentiometer.
Mount the rotating unbalance exciter.
Switch on rotating unbalance exciter with the frequency adjustable on the ten-turn potentiometer as the mechanical drum recorder record the graph plotted.
After the graph plotted, switch of the control unit and analyses the data on the graph to find the amplitude.
Repeat above steps with different forcing frequencies as mentioned in the table below.

OBSERVATIONS & CALCULATIONS:

Forcing Frequency (ω) in rpm
Frequency Ratio (r)	0.6	0.7	0.8	0.9	1.0	1.1	1.2	1.3	1.4	1.5	1.6	1.7
Peak Amplitude (X) in mm
Magnification Factor (M)

Where;

m= mass of beam = 1.64kg L= length of the beam = 670mm

k = stiffness of the spring in N/m a = distance between beam’s fixed end and spring pivot point

1 rad/s =

COMMENTS:

……………………………………………………………………………………………………………………………………………………………………………………………………………………

…………………………………………………………………………………………………………

EXPERIMENT 8

OBJECTIVE:

To determine the moment of inertia of a hollow cylinder using Trifilar Suspension.

APPARATUS:

Bifilar-Trifilar Suspension Equipment

THEORY:

The model permits oscillations on pendulums with bifilar or Trifilar suspension to be investigated. For this purpose a hollow cylinder made of galvanized steel can be hung from a wall mounted carrier plate made of aluminum and placed in oscillation. The bodies used in the experiments have strong steel hooks for attachment to the suspension cords. The length of the cords can be rapidly changed and securely fixed using clamping wheels. The beam can oscillate, by translation, in the plane of suspension like an ideal mathematical pendulum. The cylinder and the circular ring work as rotary pendulums. Measurement of period of oscillation of torsion pendulums enables their mass moment of inertia to be calculated.

Moment of inertia is the property of a rigid body that defines the torque needed for a desired change in angular velocity about an axis of rotation. Moment of inertia depends on the shape of the body and may be different around different axis of rotation. A larger moment of inertia around a given axis requires more torque to increase the rotation, or to stop the rotation, of a body about that axis. Moment of inertia depends on the amount and distribution of its mass.

PROCEDURE:

1. Insert the rope loops into the hooks of test specimen and tie the other end of the ropes to the wall mounted frame.

2. Adjust the length of the ropes as desired and measure it using measuring tape.

3. Give the specimen a slight angular displacement and release so that it can have rotational oscillations about the vertical axis passing through its center of mass.

4. Let the specimen oscillate for 20 cycles.

5. Note the time required (T) to complete 20 cycles using stop watch and then calculate the time period by dividing that time by 20.

6. Compute the radius of gyration.

7. Compute the experimental mass moment of inertia and compare it with the theoretical value.

8. Repeat the experiment for various lengths of rope.

OBSERVATIONS & CALCULATIONS:

Where: m=mass of the test specimen=3.924kg

r₂ = perpendicular distance between axis of rotation & outer surface/outer radius (m)

r₁= perpendicular distance between axis of rotation & inner surface/inner radius (m)

I_exp= mk²

Where:

τ = time period of oscillation=

r = distance between the rope and the center of mass of specimen (m)

l = (m)

g= gravitational acceleration (m/s²)

Hollow Cylinder
Obs.	m	r	l	τ	𝑘	I_exp	I_th	e
	(kg)	(m)	(m)	(s)	(𝑚)	(kg∙m²)	(kg∙m²)

PRECAUTIONS:

1. Length of all the ropes must be equal.

2. All ropes should be equidistant from the center of mass of the specimen.

3. Avoid non-planar oscillations during the experiment.

4. Measure the time required for 20 oscillations thrice and take mean to reduce the human error.

COMMENTS:

……………………..……………………………………………………...………………………………………………………………………………………………………………………………….……………………………………………………………………………………………………………….

EXPERIMENT 9

OBJECTIVE:

To determine the moment of inertia of a solid cylinder using Trifilar Suspension.

APPARATUS :

Bifilar Trifilar Suspension Equipment

THEORY:

PROCEDURE:

1. Insert the rope loops into the hooks of test specimen and tie the other end of the ropes to the wall mounted frame.

2. Adjust the length of the ropes as desired and measure it using measuring tape.

3. Give the specimen a slight angular displacement and release so that it can have rotational oscillations about the vertical axis passing through its center of mass.

4. Let the specimen oscillate for 20 cycles.

5. Note the time required (T) to complete 20 cycles using stop watch and then calculate the time period by dividing that time by 20.

6. Compute the radius of gyration.

7. Compute the experimental mass moment of inertia and compare it with the theoretical value.

8. Repeat the experiment for various lengths of rope.

OBSERVATIONS & CALCULATIONS:

Where:
m = mass of the test specimen = 3.012kg

R = radius of the cylinder (m)

I_exp= mk²

Where:

τ = time period of oscillation=

r = distance between the rope and the center of mass of specimen (m)

l = (m)

g= gravitational acceleration (m/s²)

SOLID Cylinder
Obs.	m	r	l	τ	𝑘	I_exp	I_th	e
	(kg)	(m)	(m)	(s)	(𝑚)	(kg∙m²)	(kg∙m²)

PRECAUTIONS:

1. Length of all the ropes must be equal.

2. All ropes should be equidistant from the center of mass of the specimen.

3. Avoid non-planar oscillations during the experiment.

4. Measure the time required for 20 oscillations thrice and take mean to reduce the human error.

COMMENTS:

……………………..……………………………………………………...……………………………………………………………………………………………………………………………….………………………………………………………………………………………………………………….

EXPERIMENT 10

OBJECTIVE:

To determine the moment of inertia of a square beam by using Bifilar Suspension.

APPARATUS:

Bifilar-Trifilar Suspension Equipment

THEORY:

PROCEDURE:

1. Insert the rope loops into the hooks of test specimen and tie the other end of the ropes to the wall mounted frame.

2. Adjust the length of the ropes as desired and measure it using measuring tape.

3. Give the specimen a slight angular displacement and release so that it can have rotational oscillations about the vertical axis passing through its center of mass.

4. Let the specimen oscillate for 20 cycles.

5. Note the time required (T) to complete 20 cycles using stop watch and then calculate the time period by dividing that time by 20.

6. Compute the radius of gyration.

7. Compute the experimental mass moment of inertia and compare it with the theoretical value.

8. Repeat the experiment for various lengths of rope.

OBSERVATIONS & CALCULATIONS:

Where:
m = mass of the test specimen = 3.012kg

R = radius of the cylinder (m)

I_exp= mk²

Where:

τ = time period of oscillation=

r = distance between the rope and the center of mass of specimen (m)

l = (m)

g= gravitational acceleration (m/s²)

SQUARE BEAM
Obs.	m	r	l	τ	𝑘	I_exp	I_th	e
	(kg)	(m)	(m)	(s)	(𝑚)	(kg∙m²)	(kg∙m²)

PRECAUTIONS:

1. Length of all the ropes must be equal.

2. All ropes should be equidistant from the center of mass of the specimen.

3. Avoid non-planar oscillations during the experiment.

4. Measure the time required for 20 oscillations thrice and take mean to reduce the human error.

COMMENTS:

……………………..……………………………………………………...……………………………………………………………………………………………………………………………………….…………………………………………………………………………………………………………

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