A ball of mass m is attached to a string of length L and secured at a point O. I

Question

A ball of mass m is attached to a string of length L and secured at a point O. It is swung in a circular orbit at constant angular velocity theta. and constant apex angle beta as shown in the diagram. The string is under tension T. In cylindrical polar coordinates, using the origin as the fixed point O at the top of the string, write down the acceleration of the ball in terms of r, theta, r., theta, and. Sketch a free body diagram showing all the forces on the ball Resolve forces in the radial (r) direction to show that T sin beta = mr theta ^2 Resolve forces in the vertical direction to show that T cos beta = mg Therefore show that h = g/theta ^2.

Explanation / Answer

a) a = v^2/r = [(theta)']^2*r
b) Forces on ball : mg (downwards)
T (tension due to string)
c) Along radial direction : F = Tsin(beta) = ma = m[(theta)']^2 *r
d) Along vertical direction : Tcos(beta) = mg
e) Tcos(beta) = T*h/L = mg, but T sin(beta) = ma
ma*h/L*sin(beta) = mg
ahL/Lr = g
ah/r = g
[(theta)']^2 * r * h / r = g
h = g/[(theta)']^2

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