This document describes an experiment to determine the effective length of a simple pendulum using a L-T square graph. The experiment involves measuring the time period of oscillations for a pendulum with varying lengths. The time periods are then plotted on a graph of L vs T^2 to determine the effective length. Key steps included measuring the pendulum bob diameter, varying the pendulum length, measuring oscillation times, calculating time periods, plotting an L-T^2 graph, and determining the effective length. Sources of error and precautions taken are also outlined.

1. INDEX
Contents
AIM....................................................................................................................2
INTRODUCTION..................................................................................................3
THEORY..............................................................................................................5
DIAGRAM............................................................................................................6
APPARATUS...........................................................................................................8
PROCEDURE.......................................................................................................9
OBSERVATION....................................................................................................11
CALCULATIONS...................................................................................................14
GRAPH...............................................................................................................17
RESULT….............................................................................................................18
PRECAUTIONS....................................................................................................19
SOURCES OF ERROR...........................................................................................20
BIBLIOGRAPHY .................................................................................................21

2. AIM :-
Using a simple pendulum, plot a L-T
square graph, find effective length of
seconds pendulum.

3. INTRODUCTION :-
• A pendulum is an object, hung from a fixed point,
that swings freely back and forth under the action
of gravity. A playground swing is an example of a
pendulum. The swing is supported by chains that
are attached to fixed points at the top of the swing
set. When the swing is raised and released, it will
move freely back and forth. These back-and-forth
movements are called oscillations. The swing is
moving due to the force of gravity on the swing.
The swing continues moving back and forth until

4. • We see pendulums in other areas of our lives as
well, such as in long-case clocks, commonly
known as grandfather clocks. But pendulums can
do more than entertain and help us tell time.
Among other applications, they can show that the
Earth is rotating! This was done in the mid-1800s
C.E. using perhaps the most famous pendulum,
Foucault's pendulum. However, pendulums were
being used for centuries before this. One of the
first known pendulum uses was around 100 C.E.,
when a Chinese scientist, Zhang Heng, used it to
detect distant earthquakes in a device called a

5. THEORY :-
• A simple pendulum consists of a heavy metallic (brass)
sphere with a hook (bob) suspended from a rigid stand, with
clamp by a weightless inextensible and perfectly flexible
thread through a slit cork, capable of oscillating in a single
plane, without any friction, with a small amplitude (less than
150) as shown in figure. There is no ideal simple pendulum.
In practice, we make a simple pendulum by tying a metallic
spherical bob to a fine cotton stitching thread.
• The spherical bob may be regarded by as a point
mass at its center G. The distance between the point of
suspension S and the center G of the spherical bob is to be
regarded as the effective length of the pendulum as shown
in figure. The effective length of a simple pendulum, L = l +
h + r. Where l is the length of the thread, h is length of hook,

7. • The simple pendulum produces Simple Harmonic Motion
(SHM) as the acceleration of the pendulum bob is directly
proportional to its displacement from the mean position
and is always directed towards it. The time period (T) of a
simple pendulum for oscillations of small amplitude, is
given by the relation,
• T = 2 π √ (L/g)
• Where, g = value of acceleration due to gravity and L is
the effective length of the pendulum.
• T2 = (4π2/g) X L or T2 = KL (K= constant)
and, g = 4π2(L/T2)

8. APPARATUS :-
•A Clamp With Stand
•Bob with Hook
•Split Cork
•Stop Clock/Stopwatch
•Vernier Calipers
•Cotton Thread
•Half Meter Scale

9. PROCEDURE :-
• Find the vernier constant and zero error of the vernier callipers same as
experiment 1.
• Measure the radius (r) of the bob using a vernier callipers same as experiment
1.
• Measure the length of hook (h) and note it on the table 6.1.
• Since h and r is already known, adjust the length of the thread l to make L = l
+ h + r an integer (say L = 80cm) and mark it as M1 with ink. Making L an
integer will make the drawing easier. (You can measure the distance between
the point of suspension (ink mark) and the point of contact between the hook
and the bob directly. Hence you get l + h directly).
• Similarly mark M2, M3, M4 , M5, and M6 on the thread as distance (L) of 90
cm, 100 cm, 110cm, 120cm and 130 cm respectively.
• Pass the thread through the two half-pieces of a split cork coming out just from

10. • Mark a point A on the table (use a chalk) just below the position of
bob at rest and draw a straight-line BC of 10 cm having a point A at its
center. Over this line bob will oscillate.
• Find the least count and the zero error of the stop clock/watch. Bring
its hands at zero position
• Move the bob by hand to over position B on the right of A and leave.
See that the bob returns over line BC. Make sure that bob is not
spinning.
• Now counting oscillations, from the instant bob passes through its
mean position L, where its velocity is maximum. So, starting from L it
traverses LL2, L2L, LL1, L1L hence, one oscillation is completed. We
must find time for 20 such oscillations.
• Now start the stopwatch at the instant the bob passes through the
mean position A. Go on counting the number of oscillations it
completes. As soon as it completes 20 oscillations, stop the watch.
Note the time t for 20 oscillations in the table 6.1.

11. OBSERVATION :-
•Vernier constant = 0.01 cm
•Zero error of Vernier capiller = (i) 0cm,
(ii) 0cm, (iii) 0cm
•Mean error = 0 cm
•Mean zero corrected error, ±e = 0 cm

12. Diameter of the bob and length of
hook
• Observe diameter of the bob:= (i) 2.2 cm, (ii) 2.3 cm, (iii)
2.2 cm
• Mean diameter of bob, d0 = 2.2 cm
• Mean corrected diameter of bob, d = d0 ±e = 2.2 cm
• Radius of the bob, r = d/2= 1.1 cm
• Length of the hook, h= 1.7 cm
• Standard value acceleration due to gravity, g1 : 980 cm s-2
• Least count of stop clock = 0.01 s
• Zero error of stop clock = 0.00 s

13. Table 6.1 Determination of time-periods for different lengths of the pendulum.
78.2
88.2
98.2
108.2
118.2
80
90
100
110
120
36.04
37.93
40.63
41.87
44.21
35.88
37.97
40.34
41.85
43.85
35.78
40.38
38.00
41.75
43.44
37.96
35.90
40.45
41.82
43.82
1.79
1.89
35.88
2.02
2.09
2.19
3.3
3.6
4.0
4.4
4.8

14. Calculation:
• For each length, wire mean of 20 vibrations:
• T=(t1+t2+t3)/3
• T=(36.04+35.88+35.78)/3=35.90
=>t/20=35.90/20=1.795
• T=(37.93+37.97+38.00)/3=37.96
=>t/20=37.96/20=1.895
• T=(40.63+40.34+40.38)/3=40.45
=>t/20=40.45/20=2.025
• T=(41.87+41.85+41.75)/3=41. 82

15. •T²=(1.79) ²=3.26²
•T²=(1.89) ²=3.65²
•T²=(2.02) ²=4.05²
•T²=(2.09) ²=4.45²
•T²=(2.19) ²=4.85²

16. • Graph:
• L vs T2 Graph
•Plot the graph between L and T2 from the
observations recorded in the table 6.1. Take
L along X-axis and T2 along Y-axis. The L-T
curve is a straight line passing through the
(0, 0) point. So, the origin of the graph
should be chosen (0, 0). As shown in the
figure 6.3.

18. RESULT :-
•Experimental length = 100 cm
•Actual length = 99.4 cm
•Percentage error = 0.6/99.4 X
100 = 0.6 %

19. PREACAUTIONS :-
• The thread should be very light and strong.
• The point of suspension should be reasonably rigid.
• The pendulum should oscillate in the vertical plane without any
spin motion.
• The floor of the laboratory should not have vibration, which may
cause a deviation from the regular oscillation of the pendulum.
• The amplitude of vibration should be small (less than 15) .
• The length of the pendulum should be as large as possible in
the given situation.’
• Determination of time for 20 or more oscillations should be
carefully taken and repeated for at least three times.

20. SOURCES OF ERROR :-
•The string may not be weightless and
inextensible point of suspension may not
be rigid. The amplitude may not be
small. The bob may spin.
•The air currents may disturb vibrations.
There may be an error in counting. The
stopwatch may be inaccurate. There

21. BIBLIOGRAPHY :-
•www.google.com
•www.wikipedia.org
•www.physicsprojects.com
•Comprehensive practical book