Question

A block of mass m moves with a speed v towards the right block which is in equilibrium with a spring attached to a rigid wall. If the surface is frictionless and collisions are elastic, the frequency of collisions between the masses will be:

(A) $\dfrac{v}{2 L}+\dfrac{1}{\pi} \sqrt{\dfrac{K}{m}}$
(B) $2\times [\dfrac{v}{2L}+\dfrac{1}{\pi }\sqrt{\dfrac{K}{m}]}$
(C) $\dfrac{2}{[\dfrac{2L}{v}+\pi \sqrt{\dfrac{m}{K}]}}$
(D) $\dfrac{v}{2L}+\dfrac{1}{\pi }\sqrt{\dfrac{m}{K}}$

Answer 1

Hint: We know that friction is a force between two surfaces that are sliding, or trying to slide, across each other. For example, when you try to push a book along the floor, friction makes this difficult. Friction always works in the direction opposite to the direction in which the object is moving, or trying to move. It is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: Dry friction is a force that opposes the relative lateral motion of two solid surfaces in contact. Based on this concept we have to solve this question.

Complete step by step answer
We know that the time taken to collide on left wall and get back to the mass attached with spring is $t_{1}=\dfrac{2 L}{v} .$
Hence, the time to get the spring compressed once and to come back is, ${{t}_{2}}=\dfrac{T}{2}=\dfrac{2\pi }{2}\sqrt{\dfrac{m}{K}}=\pi \sqrt{\dfrac{m}{K}}$
$t_{2}=\dfrac{T}{2}=\dfrac{2 \pi}{2} \sqrt{\dfrac{m}{K}}=\pi \sqrt{\dfrac{m}{K}}$
$\therefore$Average time between two successive collisions, $t=\dfrac{t_{1}+t_{2}}{2}$
collision frequency $=\dfrac{1}{t}$
Therefore,
For Block B, $T=2\pi \sqrt{\dfrac{m}{k}}$
For block A, $T=\dfrac{2L}{V}+\dfrac{2\pi \sqrt{m}/k}{2}$
$\Rightarrow T=\dfrac{2L}{V}+\pi \sqrt{\dfrac{m}{R}}$

Therefore, the correct answer is Option C.

Note: We know that static friction is a force that keeps an object at rest. Static friction definition can be written as the friction experienced when individuals try to move a stationary object on a surface, without actually triggering any relative motion between the body and the surface which it is on. It hinders the movement of an object moving along the path. When two fabrics slide over each other, this friction occurs. There's friction all around us. When we walk, for instance, our feet are in touch with the floor. Static friction is caused by adhesion, light chemical attraction between two surfaces. And friction, in general, is caused by the imperfections in every surface gripping together and overlapping.
It should also be known to us that kinetic friction (also referred to as dynamic friction) is the force that resists the relative movement of the surfaces once they're in motion.