Question

A uniform rod of length L and mass M is free to swing about an axis that is perpendicular to the rod. The axis is a distance x from the rod's center of mass.

a) Find the period of oscillations for small angles as a function of L and x with appropriate constants.

b) make a sketch of the period as a function of x. If you use a spread sheet you may assume that L=1.0 m, then your graph should go from x = .05m to .50m

c) Find the value of x for which the period in part a) is a minimum. Give your answer as a function of L.

Answer 1

given uniform rod of length L

Mass M

free to swing about an axis perpendicular to the rod

distance of axis form the center of mass = x

a. time period of osscilation = T

T = 2*pi*sqrt(Is/mgLcm)

Is = mL^2/12 + mx^2

Lcm = x

hence

T = 2*pi*sqrt((L^2 + 12x^2)/12*gx)

b. following is a graph of T vs x

L = 1 m

c. for T to be minimum

T^2 is minimum

dT^2/dx = 0

hence

4*pi^2*(-L^2/12gx^2 + 1/g) = 0

L^2/12 = x^2

x = L/sqrt(12) = 0.28867513459 L