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A rod of mass m and length l is kept on a smooth wedge of mass M.if the system is released when the rod is at the top of the wedge ,find the speed of the wedge when the rod hits the ground level.(neglect friction between all surfaces in contact)

A rod of mass m and length l is kept on a smooth wedge of mass M.if the system is released when the rod is at the top of the wedge ,find the speed of the wedge when the rod hits the ground level.(neglect friction between all surfaces in contact)

Grade:12th pass

1 Answers

Arun
25750 Points
5 years ago
We can use conservation of energy to solve this... or by using forces (normal) between wedge M and the rod m.
Here on the system of rod m and wedge M, work is done only by the gravitational force on rod m.
The PE lost by the rod :   m g H
Wedge displaces horizontally by x. The rod displaces vertically by y.
=> y = x tanФ.        Differentiate wrt time, to get:
=> v1 = v2 tanФ
=> a1 = a2 tanФ
When the rod touches the ground, velocity of wedge = v2.
KE of Wedge = 1/2 M v2²
KE of the rod  = 1/2 m v1² = 1/2 m v2 tan²Ф
Using conservation of energy:  m g H = 1/2 v2² (M + m tan²Ф)
v2² = 2 m g H /(M + m tan²Ф)

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If we use the method of forces, then Normal reaction force N between the rod m and wedge M gives the acceleration to Wedge.
Draw the Free body diagram for wedge and the rod.
N SinФ = M a₂
m g - N CosФ = m a₁ = m a₂ tanФ
=>  m g - M a₂ CotФ = m a₂ TanФ
=>  a₂ = mg Tan Ф /(M+m tan²Ф )
Distance traveled by the wedge: H CotФ   (the base of wedge)
=>  v₂² = 2 a₂ H CotФ
 

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