A small mass can slide without friction in a hemispherical bowl of radius R. The

Question

A small mass can slide without friction in a hemispherical bowl of radius R. The mass is set in motion from the lowest point in the bowl with a speed v at t = 0 seconds. Use the small angle approximation (sin Theta similar to theta, cos Theta similar to 1 - 1/2 theta^2) to show that the mass undergoes simple harmonic motion, by writing down the differential equation and determining the angular frequency and phase constant of the oscillation. Determine the angular amplitude theta of the oscillation. Determine the instant after t = 0 at which the mass reaches maximum angular displacement for the third time.

Explanation / Answer

a) Restoring force, F = -mg(sin)

For small ,

F = -mg

=> mR = -mg

=> d2/dt2 = -(g/R)

=> d2/dt2 = -2

where = angular frequency of oscillation = (g/R)1/2

This is the differential equation for SHM. Hence, the mass undergoes SHM.

The solution of the above differential equation is given by,

= osin(t + )

where o is the amplitude and is the phase constant.

At t = 0, x = 0

=> 0 = Asin

=> = 0

b) For the particle, conservation of energy gives:

mgh = mv2/2

=> h = v2/2g

Now,

coso = (R - h)/R = 1 - h/R

For small angles, coso = 1 - o2/2

So,

1 - o2/2 = 1 - v2/2gR

=> o = v/(gR)1/2

So, angular amplitude is given by:

= [v/(gR)1/2] * sin[(g/R)1/2t]

c) Time period of oscillation, T = 2/ = 2/(g/R)1/2 = 2(R/g)1/2

The mass will reach maximum angular displacement for the third time after t = T + T/4 = 5T/4 = 2.5(R/g)1/2

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