A block of mass 'm' and a pan of equal mass are connected by a string going over a smooth light pulley as shown in figure. Initially the system is at rest when a particle of mass m falls on the pan and sticks to it. If the particle strikes the pan with a speed 'v' find the speed with which the system moves just after the collision.
A block of mass 'm' and a pan of equal mass are connected by a string going over a smooth light pulley as shown in figure. Initially the system is at rest when a particle of mass m falls on the pan and sticks to it. If the particle strikes the pan with a speed 'v' find the speed with which the system moves just after the collision.
The correct option is B v3
Let the required speed be V.
As there is a sudden change in the speed of the block, the tension must change by a large amount during the collision.
Let N = magnitude of the contact force between the particle and the pan
T = tension in the string
Consider the impulse imparted to the particle. The force is N in upward direction and the impulse is ∫ N dt. This is equal to the change in its momentum.
Thus, ∫ N dt = mv − mV ......................(i)
Similarly for the pan: ∫ (N − T) dt = mV ................(ii)
And for the block: ∫ T dt = mV ................(iii)
Adding (ii) and (iii),
∫ N dt = 2mV
Comparing with (i),
mv mV = 2mV
Therefore, V = v3
v3
Let the required speed be V.
As there is a sudden change in the speed of the block, the tension must change by a large amount during the collision.
Let N = magnitude of the contact force between the particle and the pan
T = tension in the string
Consider the impulse imparted to the particle. The force is N in upward direction and the impulse is ∫ N dt. This is equal to the change in its momentum.
Thus, ∫ N dt = mv − mV ......................(i)
Similarly for the pan: ∫ (N − T) dt = mV ................(ii)
And for the block: ∫ T dt = mV ................(iii)
Adding (ii) and (iii),
∫ N dt = 2mV
Comparing with (i),
mv mV = 2mV
Therefore, V = v3
