A pendulum is formed by pivoting a long, thin rod about a point on the rod. In a

Question

A pendulum is formed by pivoting a long, thin rod about a point on the rod. In a series of experiments, the period is measured as a function of the distance x between the pivot point and the rod’s center.

If the rod’s length is L = 2.20 m and its mass is m = 22.1 g, what is the minimum period the pendulum can have?

If x is chosen to minimize the period and then L is increased, does the period increase, decrease, or remain the same? Justify your answer.

If m is increased without L increasing, does the period increase, decrease, or remain the same? Justify your answer.

Explanation / Answer

L =2.2m , m =22.1 g

From parallel axis thorem I = (1/12)mL^2 +md^2

T = 2pi(I/mgx)^1/2

T = 2pi(m(L^2/12 +x^2)/mgx)^1/2

T = 2pi(L^2+12x^2/12gx)^1/2

To find T min , we use dT/dx =0

then x = L/(12)^1/2

Tmin = 2pi((L^2+12*L^2/12)/12g(L/12^1/2))^0.5

Tmin = 2*3.14((2.2^2 +2.2^2)/(12*9.8*2.2/12^0.5))^0.5

Tmin =2.26 s

(b) If x is chosen to minimize the period, then as L is increased the period will increases as well

(c) time period does not depend on mass. T remains same

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