A tetherball is attached to a swivel in the ceiling by a light cord of length L,

Question

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, mass m, and length L of the rope are known, derive algebraic expressions for the tension FT in the cord and the angle .

What is the period T of the ball’s motion if L=4.0m and the cord makes an angle of =40 with the vertical?

Explanation / Answer

in horizontal,

FT sin(beta) = mv^2 /R

and 2 pi R / v = T

v/R = 2 pi / T

and v^2 / R^2 = (2pi /T)^2

v^2 / R = R (2pi /T)^2

So, FTsin(beta) = m R (2pi /T)^2

R = L sin(beta)

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

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

and in vertical,

FT cos(beta) = mg

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

beta = cos^-1 [ g T^2 / 4 pi^2 mL] .....................Ans

cos40 = 9.81 / [ 4 pi^2 4 / T^2 ]

T ^2 = 12.33

T =3.51 ..............Ans

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