KATER’S PENDULUM
© Institute of Lifelong Learning, University of Delhi
1
KATER’S PENDULUM
2
PHYSICS (LAB MANUAL)
© Institute of Lifelong Learning, University of Delhi
PHYSICS (LAB MANUAL)
KATER’S PENDULUM
Introduction
Kater’s pendulum (Figure1) consists of a long metallic rod R of circular cross
section weighted at one end so that the center of mass is much nearer to one
end. To do this, a heavy metallic cylinder W1 and an identical wooden cylinder W 2
are placed at the two ends of the rod. It has two movable knife edges K 1 and K2.
A much smaller metallic cylinder W is kept at the middle of the rod. All the five
can be moved along the rod and fixed at any position using screws. The labeled
diagram of Kater’s pendulum is shown in Figure 1. This configuration ensures that
the center of gravity lies near one of the knife edges.
W2
K2
R
l2
W
G
l1
K1
W1
Figure 1:

•
•
•
•
•
•
Kater’s Pendulum
G - Center of gravity
R - Metallic rod
K1, K2 - Movable knife edges
W1 - Metallic cylinder
W2 - Wooden cylinder
W1, W2 are to be placed beyond K1and K2 and can be moved along R
W - metallic weight to be kept near the center of rod and can be moved
for adjustments
© Institute of Lifelong Learning, University of Delhi
3
PHYSICS (LAB MANUAL)
KATER’S PENDULUM
Apparatus






Kater’s Pendulum
Telescope
Stop watch
Meter scale
Sharp wedge
Rigid support
Theory
Kater’s pendulum is a compound pendulum based on the principle that the
center of suspension and center of oscillation are interchangeable. The
movable cylinders, knife edges and the metallic weight are so adjusted such that
the time periods of the pendulum about the two knife edges situated
asymmetrically with respect to the center of gravity are exactly equal. Then, the
distance between the knife edges is equal to the length of equivalent simple
pendulum whose time period is given by (refer to Equation (5), Bar pendulum)
T  2
L
g
and
g
4 2 L
T2
Hence, g may be calculated.
We resort to Bessel’s approximation where we do not require making the two
time periods to be exactly equal because it is quite difficult and time-consuming
to set the Kater’s pendulum for this configuration.
If T1 and T2 represent two nearly equal time periods (in sec) for positions of K 1
and K2 distant l1 and l2 (in cm) from C.G., then we can write
T1  2
l1  k 2 and
gl1
2
T2  2.
2
Hence,
l2  k 2
gl2
2
2
gl1T1
2
 l1  k 2
2
4
gl2T2
2
 l2  k 2
2
4
and
Subtracting and rearranging we obtain
T  T2
8 2 T1  T2

 1
l1  l 2  l1  l 2 
g
2
2
2
2
Since T1~T2 and positions of K1 and K2 are asymmetrical about C.G, l1-l2 is fairly
large. Hence, the second term in the denominator is negligibly small and thus, an
approximate value of l1-l2 is sufficient.
Therefore,
g
4
8 2
T1  T2
T  T2
 1
l1  l 2  l1  l 2 
2
2
2
2
© Institute of Lifelong Learning, University of Delhi
(1)
PHYSICS (LAB MANUAL)
KATER’S PENDULUM
where
g = Acceleration due to gravity in cm/s2
T1 = Time period about K1 in seconds
T2 = Time period about K2 in seconds
l1 = Distance of K1 from C.G. in cm
l2 = Distance of K2 from C.G. in cm
Learning Outcomes
This experiment will enable you:
1. To determine the acceleration due to gravity (g) using Kater’s pendulum.
2. To verify that there are two pivot points on either side of the centre of
gravity (C.G.) about which the time period is the same.
3. To determine the length of the equivalent simple pendulum.
Pre-lab Assessment
Now to know whether you are ready to carry out the experiment in the
lab, pick the correct answer from the following. If you score at least
80%, you are ready, otherwise read the preceding text again. (Answers
are given at the end of this experiment.)
(1) Example/s of compound pendulum is/are
a) Bar pendulum
b) Kater’s pendulum
c) both a and b
d) none of the above
(2) The oscillatory motion is
a) always periodic
b) periodic as well as bounded
c) only bounded
d) none of the above
(3) About how many points, the time period of a compound pendulum is same?
a) 2
b) 3
c) 4
d) 5
(4) The factors that cause damping is/are
a) friction at the point of suspension
b) viscosity of air
c) none of the above
d) both (a) and (b)
(5) The value of g
a) changes randomly
b) changes definitely according to some relation
c) is constant everywhere
d) can not be predicted
© Institute of Lifelong Learning, University of Delhi
5
KATER’S PENDULUM
PHYSICS (LAB MANUAL)
Procedure
Method 1 (Equalization of time periods)
1. Determine the middle point of the rod and fix the smaller metal weight W
there. Fix the brass weight W1 near one end of the Kater’s pendulum (5 cm
from end 1) and the knife edge K1 just below it (at a distance of about 2 cm).
2. Similarly, adjust the wooden weight W2 and the knife edge K2 at the other end
(end 2) of the pendulum with the same symmetry. The metallic and wooden
cylinders are placed at different ends to eliminate viscous drag of air and to
make the C.G. asymmetrical about the knife edges .Screw all the five tightly.
Knife edges must be sharp, horizontal and parallel to each other so that the
oscillations are confined to a vertical plane
3. Suspend the pendulum vertically about K1 and focus the telescope at the tip
of its lower end. Set it oscillating with amplitude of about 4-5 degrees for the
motion to remain simple harmonic. Note the time for 30 oscillations using a
stop watch.
4. Now suspend the pendulum vertically about K2 and repeat step 3.This time
will be quite different from that about K1.
5. Keep moving K1 and K2 towards W by small distance (approx. 1 cm) and
repeat steps 3 and 4 till the difference in time about K 1 and K2 is less than one
second. If at any stage the time difference increases, then K 1 and K2 should
be moved towards W.
6. Now, move the weight W and repeat step 5 to reduce the time difference to
about 0.5 second.
7. The apparatus is ready to record the measurements. Suspend the pendulum
about K1 and K2 vertically and record the time taken for 100 oscillations.
Repeat this 5 times each.
8. Remove the pendulum from support and place it horizontally on a wedge.
Balance it and find the C.G. of the system.
9. Measure the distances l1 and l2 from C.G. to the knife edges K1 and K2.
Method 2 (Graphical method)
10. Here, after the initial adjustments (steps1-2), measure the time for 30
oscillations about K1 and K2. Balance the pendulum on the wedge to mark the
C.G. Then measure l1 and l2.
11. After this, move K1 and K2 by 2 cm each towards the center of the rod.
12. Again measure the time for 30 oscillations about K 1 and K2, Mark the C.G. and
measure l1 and l2.
13. Repeat the process 8-9 times by moving K1 and K2 towards each other in
steps. Repeat step 12.
Observations
Least count of meter scale = ------ mm
Least count of stop watch = ------ sec
6
© Institute of Lifelong Learning, University of Delhi
PHYSICS (LAB MANUAL)
KATER’S PENDULUM
Method 1 (Equalization of time periods)
Table 1
S.No
1.
2.
3.
4.
5.
6.
7.
No. of
oscillations
(n)
30 (t1-t2~0.3
s)
50 (t1-t2~0.5
s)
100 (t1t2~1s)
100
100
100
100
Method 2
Time (t1)
about K1
(s)
Time (t2)
about K2
(s)
T1=t1/n
(s)
T2=t2/n
(s)
(T1-T2)
(s)
(Graphical method)
Table 2
S.No.
No. of
oscillations
(n)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
30
30
30
30
Time
(t1)
about
K1
(s)
Time
(t2)
about
K2
(s)
T1=t1/n
(s)
T2=t2/n
(s)
Distance
of K1from
CG
(l1)
Distance
of
K2from
CG
(l2)
Calculations
Method 1 (Equalization of time periods) (when T1 ~ T2)
T1 = ------ sec
T2 = ------ sec
l1 = ------ cm
l2 = ------ cm
Substitute in the Equation (1) and obtain the value of g.
© Institute of Lifelong Learning, University of Delhi
7
PHYSICS (LAB MANUAL)
KATER’S PENDULUM
Method 2 (Graphical method)
Draw a graph (Figure 2) of l1 v/s T1 and l2 v/s T2 on the same graph sheet.
Graph of l1 v/sT1 and l2 v/sT2
Time period (about k 1, k2 in sec.)
1.85
1.8
Line 1x
T1x
1.75
Line 1y
T1y
25
l2y
l1y
l1x
1.7
30
35
40
45
50
Distance from CG (l1, l2 in cm)
l2x
55
60
Figure 2: Graph between time period and distance from C.G.
Draw two horizontal lines (line 1x and line 1y, as shown in the graph in Figure 2)
on the graph intersecting the two experimental lines.
To find g1, use T1 = T2 = T1x, l1 = l1x and l2 = l2x in Equation (1) and obtain g1.
To find g2, use T1 = T2 = T1y, l1 = l1y and l2 = l2y in Equation (1) and obtain g2.
Find mean g= (g1 + g2) / 2
Estimation of error
Maximum log error
g
8 2
2
2
2
2
T1  T2 T1  T2

l1  l2  l1  l2 
Differentiating logarithmically, we have
g
g

X  Y 
(2)
X Y
where
T1  T2
l1  l2 
2
X
8
2
© Institute of Lifelong Learning, University of Delhi
PHYSICS (LAB MANUAL)
KATER’S PENDULUM
and
T1  T2
l1  l2 
2
Y

X
X


 T12  T2 2
T1  T2
2
2
2
   l  l   2T 
1
2
l1  l2
T
2l
l1  l2
Since T1 ~T2. Here δT corresponds to the smallest division of the stop watch
and δl to the smallest division of the meter scale. Thus, δX can be evaluated.
Also,
Y
Y
Using above equation
Y
may then be used to find

4T1T
2l

2
2
l1  l2
T1  T2
can also be calculated. The values of X, Y, δX and
g
g
Y
using Equation (2) and hence, the log error.
Percentage error
The percentage error can be calculated as
Standard value – calculated value
Percentage error = ----------------------------------------------ⅹ100
standard value
where
Standard value = 981 cm/s2
Calculated value = g
Result
The value of acceleration due to gravity g as calculated in the lab is (---------±
max. log error) cm/s2
Glossary
Acceleration: The rate of change of velocity with time.
Acceleration due to gravity: The acceleration imparted to freely moving bodies
towards the center of earth by the attractive gravitational force of the earth; its
value varies with latitude and elevation.
Center of gravity: The point in a material body through which the resultant
force of gravitational attraction acts.
Center of mass: The point in a material body or system of bodies which moves
as though the system’s total mass was concentrated at that point and all external
forces were applied at the point. It is also known as the center of inertia.
Center of oscillation: The point in a physical pendulum, on the line through the
point of suspension and the center of mass, which moves as if all the mass of the
pendulum were concentrated there.
© Institute of Lifelong Learning, University of Delhi
9
KATER’S PENDULUM
PHYSICS (LAB MANUAL)
Center of suspension: The intersection of the axis of rotation of a pendulum
with a plane perpendicular to the axis that passes through the center of mass.
Compound pendulum: A rigid body of any shape capable of oscillating about a
horizontal axis passing through it.
Oscillation: Any effect that varies periodically back and forth between two
values.
Simple pendulum: A device consisting of a small massive body suspended by an
inextensible object of negligible mass from a fixed horizontal axis about which the
body and suspension are free to rotate.
Simple harmonic motion: A periodic motion about an equilibrium position for
which the displacement is a sinusoidal function of time. The acceleration of the
object is always directed towards the equilibrium position and is proportional to
the displacement from that point.
Telescope: An assemblage of lenses or mirrors or both that enhances the ability
of the eye to see distant objects with greater resolution.
Time period: Time taken to complete one oscillation of a periodic motion.
Post-lab Assessment
Choose the correct answer
(1) When the distance of the point of suspension from the centre of gravity
increases, the time period of a compound pendulum
a) increases
b) decreases
c) remains constant
d) first decreases then increases
(2) As we go down below the surface of earth, the value of g
a) increases
b) decreases
c) first increases then decreases
d) does not change
(3) The difference in the value of g at equator and poles is equal to
a) radius of earth (R)
b) angular velocity of earth (ω)
c) Rω
d) Rω2
(4) If we interchange the center of suspension and center of oscillation, the time
period of compound pendulum
a) increases
b) decreases
c) remains same
d) cannot be predicted
(5) The length of an equivalent simple pendulum is equal to
a) the distance of the point of suspension from C.G.
b) radius of gyration (k)
c) (k2+l2)/l
d) infinite
(6) Which instrument is expected to give more accurate value of g?
a) simple pendulum
b) Kater’s pendulum
c) bar pendulum
d) mass spring system
10
© Institute of Lifelong Learning, University of Delhi
PHYSICS (LAB MANUAL)
KATER’S PENDULUM
(7) The time period of a compound pendulum about its C.G. is
a) zero
b) infinite
c) finite
d) unpredictable
(8) The time period of compound pendulum will be minimum when
a) l = k
b) l = k2
c) l > k
d) none of the above
(9) With altitude the acceleration due to gravity ‘g’
a) increases
b) decreases
c) remains constant
d) cannot be predicted
Answers to Pre-lab Assessment
1.
2.
3.
4.
5.
c
b
c
d
b
Answers to Post-lab Assessment
1.
2.
3.
4.
5.
6.
7.
8.
9.
d
b
d
c
c
b
b
a
b
© Institute of Lifelong Learning, University of Delhi
11