Consider a circular-motion problem in which tension not only must provide the pr

Question

Consider a circular-motion problem in which tension not only must provide the proper radial acceleration but also must balance gravity. As we shall see, it can do both. A tetherball is attached to a swivel in the ceiling by a light cord of length L, as measured from the ceiling to the center of the tetherball. When the ball is hit by a paddle, it swings in a horizontal circle with constant speed v, and the cord makes a constant angle with the vertical direction. The ball goes through one revolution in time T. Assuming that T, massm, and length L of the rope are known, derive algebraic expressions for the tension FTin the cord and the angle .

What is the period T of the ball’s motion if L=3.0m and the cord makes an angle of =40 with the vertical?

Express your answer in seconds to two significant figures.

Explanation / Answer

In horizontal,

FT sin(beta) = m v^2 / R   

v = 2 pi R / T and T = L sin(beta)

v = 2 pi L sin(beta) /T

FT sin(beta) = m (2 pi L sin(beta) /T )^2 / (L sin(beta))
FT sin(beta) = 4 pi^2 m L sin(beta) / T^2

FT = 4 pi^2 m L / T^2 .............Ans

In vertical,

FT cos(beta) = mg   

4 pi^2 m L cos(beta) / T^2 = m g

cos(beta) = g T^2 / 4 pi^2 L

beta = cos^=1 [ g T^2 / 4 pi^2 L ]

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L = 3m and beta = 40 deg

then

cos40 = (9.81) T^2 / 12pi^2

0.766 = 0.0828 T^2

T = 3.04 sec

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