A grandfather clock uses a physical pendulum to keep time. The pendulum consists
ID: 1298886 • Letter: A
Question
A grandfather clock uses a physical pendulum to keep time. The pendulum consists of a uniform thin rod of mass M = 1.01 kg and length L = 0.37 m that is pivoted freely about one end, with a solid disk of mass 3M and a radius of L/4 attached to the free end of the rod.
a) What the moment of inertia of the pendulum about its pivot point?
b) What is the period of the pendulum for small oscillations?
c) Determine the length L that gives a period of T = 2.69 s.
please show the formula that you used for the question
Explanation / Answer
a) Moment of inertia of rod about such an axis = ML^2/3
Moment of inertia of disk about such an axis = 3M(L/4)^2/2 + 3M(5L/4)^2 (using parallel axes theorem)
Moment of inertia of pendulum = Moment of inertia of rod + Moment of inertia of disk
= ML^2/3 + [3M(L/4)^2/2 + 3M(5L/4)^2]
= ML^2 [ 1/3 + 3/32 + 75/16 ]
= 1.01* 0.37^2* (491/96)
= 0.707 kg m^2
b) Distance of center of mass of assembly from pivot = (ML/2 + 5ML/4)/4M = 7L/16 =0.161875 m
Time period = 2pi*sqrt(l/g) = 2pi*sqrt(0.161875/9.81) = 0.807 s
c) If the time period is 2.69 s
2.69 = 2pi*sqrt(l/g)
(2.69/2pi)^2 = l/g
l = 0.1833*9.81 = 1.796 m
7L/16 = 1.796m
L = 4.105 m