Question

A small mass is suspended from a light string of length r. What should be the least horizontal velocity that must be given to mass in its lowest position so that tension at the highest point is equal to the weight of the mass attached?

Solution

Let u be the velocity of mass at the lowest point and v be at the highest point.

Tension at highest point is equal to weight.
∴

space space fraction numerator m v squared over denominator r end fraction minus m g equals T equals m g

rightwards double arrow

fraction numerator m v squared over denominator r end fraction equals 2 m g

v squared equals 2 r g

The mass rises by height 2r in going from the lowest to the highest point,
∴

v squared minus u squared equals negative 2 g left parenthesis 2 r right parenthesis

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rightwards double arrow

u squared equals v squared plus 4 g r equals 2 r g plus 4 r g

rightwards double arrow

u squared equals 6 r g

u equals square root of 6 r g end root

, is the least horizontal velocity that is given to the mass.

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Laws Of Motion

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