compound pendulum movement analysis

2. 1- to determine the value of “g “ at a place by the
help of a bar pendulum
2- to determine of “K” ( radius of gyration of the
body )

3. 1- bar pendulum
2- stop watch
3- knife edge fixed to a rigid support
4- a meter scale

4. As we see .. We have body oscillate freely around its
center of gravity
m represent the mass of the body moving
L represent the length of rod
θ represent the angel between the rod and Y axis
S represent the restoring force

5. Hence equation if motion for the whole body can be represented
by: I =
𝑑2
θ
𝑑𝑡2 = - mgLsin θ
Where I is the moment of inertia of the body about the horizontal axis
passing through S and negative sign indicates that the force is directed in
a direction opposite to the motion
𝑑2
θ
𝑑𝑡2 +
𝑚𝑔𝐿
𝐼
sin θ = 0
OR
𝑑2
θ
𝑑𝑡2 +
𝑚𝑔𝐿
𝐼
θ = 0 if θ is small
OR
𝑑2
θ
𝑑𝑡2 + P2
θ =0 where P2
=
𝑚𝑔𝐿
𝐼
This equation represents a S.H.M of time period :-
T =
2π
𝑃
= 2π
1
𝑚𝑔𝐿

6. If I is the moment of inertia of the body about an axis passing
through the c. g. these
I = I’ + mL2
= mk2 + mL2
= m(k2 + L2
) (where k is the radius of gyration of the body)
Hence T= 2π
m(k2 + L2
)
𝑚𝑔𝐿
= 2π
k2 + L2
𝑔𝐿
= 2π
𝑔2
+𝐿
𝑔
= 2π
𝑘2
2𝐿(𝐿
2
+2𝐿𝑘)
𝐿𝑔
This time period will be minimum when L – k = 0 i.e. L = k

7. s.No. distance of hole time for 20 oscillations time T period
from the e.g. “x” 1 2 3 Mean
1
2
3
4
Observations :-

8. Now plot the graph between LT on Y axis and L2 on X axis