Pan It

Classical Mechanics Level pending

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 the figure below. 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 6 m/s, find the speed with which the system moves just after the collision.


The answer is 2.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Let F = magnitude of the contact force between the particle and the pan

Let T = tension in the string

Let V = the speed after collision

Consider the impulse imparted to the particle. The force is F in upward direction and the impulse is ∫ F d t \int F dt . This should be equal to the change in its momentum. The striking speed of 6 m/s has been denoted by v for convenience.

Thus, ∫ F d t \int F dt = mv - mV ... (i)

Similarly, considering the impulse imparted to the pan,

∫ ( F − T ) d t \int (F-T) dt = mV ... (ii)

And that to the block,

∫ T d t \int T dt = mV ... (iii)

Adding (ii) and (iii),

∫ F d t \int F dt = 2 mV

Comparing with (i),

mv - mV = 2 mV

or, m v mv = 3 m V 3 mV

or, V = v 3 \frac{v}{3} = 6 3 \frac{6}{3} = 2 m/s

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