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Lab-01 - To determine the radius of gyration of compound
pendulum
fluid mechanic (Khawaja Fareed University of Engineering and Information Technology)
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Experiment 01
Objective
To determine the radius of gyration of compound pendulum
Apparatus
The apparatus shown in Figure 1, consists of a thin rod of length 1000 mm with adjustable
additional bob of mass 0.492 kg suspended from a knife-edge bearing. The point of suspension of
the rod can be adjusted at the knife-edge bearing.
Figure. 1 Compound Pendulum
Theory
Vibration
Any motion that repeats itself after an interval of time is called vibration or oscillation. The
swinging of a pendulum and the motion of a plucked string are typical examples of vibration. The
theory of vibration deals with the study of oscillatory motions of bodies and the forces associated
with them.
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Elementary Parts of Vibrating Systems
A vibratory system, in general, includes a means for storing potential energy (spring or elasticity),
a means for storing kinetic energy (mass or inertia), and a means by which energy is gradually lost
(damper).
The vibration of a system involves the transfer of its potential energy to kinetic energy and of
kinetic energy to potential energy, alternately. If the system is damped, some energy is dissipated
in each cycle of vibration and must be replaced by an external source if a state of steady vibration
is to be maintained.
Classification of Vibration
Vibration can be classified in several ways. Some of the important classifications are as follows.
Free Vibration
If a system, after an initial disturbance, is left to vibrate on its own, the ensuing vibration is known
as free vibration. No external force acts on the system. The oscillation of a simple pendulum is an
example of free vibration.
Forced Vibration
If a system is subjected to an external force (often, a repeating type of force), the resulting
vibration is known as forced vibration. The oscillation that arises in machines such as diesel
engines is an example of forced vibration.
Compound Pendulum
Any swinging rigid body free to rotate about a fixed horizontal axis is called a compound
pendulum. It always pivoted other than its center of mass and oscillate with its own weight.
Radius of Gyration
Radius of gyration or gyradius of a body about an axis of rotation is defined as the radial distance
of a point from the axis of rotation at which, if the whole mass of the body is assumed to be
concentrated, its moment of inertia about the given axis would be the same as with its actual
distribution of mass.
Let G be the center of gravity of a compound pendulum of mass m that oscillates about a point O
with OG = h if the pendulum is moved so that the line OG is displaced through an angle θ.
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Figure 2- Compound Pendulum
The restoring couple is:
τ = mghsinθ = mghθ
Iα = mghθ
α
= mg h /I
θ
As parallel axis theorem is given as:
∵θ≈0
∵ τ= Iα
(1)
I = IG + mh2
From equation (1),
I = mK G 2 + mh2
Time Period of a compound pendulum:
α
gh
=
2
θ
K G + h2
(2)
Displacement
T = 2π�
Acceleration
K G 2 + h2
T = 2π�
gh
(3)
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Where, K G is radius of gyration.
K G = �(
T2 g h
) − h2
4π2
(4)
Time period of a simple pendulum is given as;
T = 2π�
L
g
(5)
From equation (3) & (5)
h2 − Lh + K G 2 = 0
(6)
Let l1 & l2 are the roots of quadratic equation (6)
K G = �l1 × l2
l1 & l2 can be calculated from T-h graph.
Figure 3- T-h Graph
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Experimental Procedure
1)
Support the rod on knife edge. Measure the length of the pendulum and point out
the center of gravity of the rod
2)
Allow the bar to oscillate with small amplitude
3)
Note the time for 20 oscillations by a precision stop watch. Make this observation
three times and find mean time ‘t’ for 20 oscillations
4)
Measure the distance ‘h’ between the point of suspension and center of gravity of
the pendulum with the help of meter rod
5)
Calculate the radius of gyration of the compound pendulum from the following
relation
6)
Draw the graph with the distance ‘h’ as abscissa and time period as ordinate and
calculate the value of radius of gyration graphically.
Observations and Calculations
Time for 20 Oscillation
(t)
t1
t2
t3
tavg
Time for one
Oscillation
(T)
(KG)th
(m)
(KG)exp
(m)
31.97
1.598
0.288
0.285
30.87
30.88
1.544
0.277
0.285
30.78
30.74
30.75
1.537
0.288
0.285
30.5
30.40
30.45
30.45
1.522
0.287
0.285
0.25
29.87
29.60
29.72
29.73
1.486
0.273
0.285
6
0.2
31.63
31.44
31.54
31.53
1.576
0.288
0.285
7
0.15
33.72
33.72
33.71
33.71
1.685
0.288
0.285
8
0.1
39.74
39.62
39.53
39.53
1.976
0.294
0.285
9
0.05
53.32
53.47
53.40
53.39
2.669
0.293
0.285
Sr.
No.
h (m)
1
0.45
31.85
321.16
31.9
2
0.4
30.8
30.91
3
0.35
30.75
4
0.3
5
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Graph
3
2.5
2
T (s)
l1
l2
1.5
1
0.5
0
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
h (m)
Specimen Calculations
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Discussion
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CLO-1
Measure different mechanical properties like moment of inertia, radius of gyration of different
systems.
Rubric 01
Marks
CLO-1 – Level P4 mapped to PLO-2 (Problem Analysis)
9-10
Draws the time period graph from the measured data with confidence and
proficiency and draws correct conclusions. (no mistakes and single attempt)
7-8
Draws the time period graph from the measured data without mistakes however
he does it with hesitancy after few repeated attempts and draws correct
conclusions
3-6
Draws the time period graph from the measured data with some mistakes and
hesitancy after few repeated attempts and draws incorrect conclusions
0
Unable draw the time period graph from the measured data
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