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Problem 2.04PP
Write the equations of motion of a pendulum consisting of a thin, 2 kg stick of length / suspended 
from a pivot. How long should the rod be in order for the period to be exactly 1 sec? (The inertia I 
of a thin stick about an end point is Assume that 0 is small enough that sin Q ^ Q.)
Why do you think grandfather clocks are typically about 6 ft high?
Step-by-step solution
step 1 of 3
Consider the following equation for the sum of all external moments about the center of mass of
Consider the following equation for the sum of all external moments about the center of mass of 
a body.
M ^ I a ...... (1)
Where.
/ is the body’s mass moment of inertia about its center of mass, 
a is the angular acceleration of the body.
The coordinates and forces relationship is shown in Figure 1.
o
Figure 1
Step 2 of 3
From Figure 1, the equation of moment about point O is given below. 
M o = -m g n ^ s m 0
— ........(2)
Substitute for /p in equation (2).
-m g =
^ m /*tf+ mg X ̂ sin = 0
f l + ^ 8 i n f l = 0 ...... (3)
2i
Consider ^ is very small in equation (3).
g + ^ e = o (4)
Consider the following general equation.
^+24^a>e+ai‘ f f= 0 (5)
Compare equation (4)and equation (5).
Step 3 of 3 ^
Consider the following equation for the period for a swing.
r=— ....(7)
Substitute equation (6)in equation (7).
2/r - 2 > r / - = - ...... (8)
Substitute 2 for / and 9.81 for g in equation (8).
r = 2 ; r j
I 2x2 
’'V 3 x 9.8I
= 2.31 
= 2sec
Consider the following equation for the value of length.
/= -? § - (9)
Substitute 9.81 for g in equation (9).
3x9.81/ * -
8^*
* 0.3727 m
Thus, the vaiue of length and period for a swing are [0.3727m]a
Hence, the 1 second for a swing from one side to the other. This pendulum is shorter because 
the period is faster. But if the period had been 2 second, the pendulum length is 1.5 meters.