In the set up shown, the floor is smooth, m1=1 kg, m2=10 kg and the coefficient of friction b

step 1 we have problem number 149 in which this system is being shown okay so we have to find in this c

In the set up shown, the floor is smooth, m1=1 kg, m2=10...

In the set up shown, the floor is smooth, $m_{1}=1 \mathrm{~kg}, \mathrm{~m}_{2}=10 \mathrm{~kg}$ and the coefficient of friction between their surfaces is $\mu=0.6$. Assume the surface of $\mathrm{m}$, is large enough so that when $\mathrm{m}_{2}$ is given a velocity $\mathrm{v}$ as shown, it slips on $\mathrm{m}_{1}$ till the spring is compressed to maximum i.e., till $m_{1}$ stops. The spring constant $\mathrm{k}=200 \mathrm{~N} \mathrm{~m}^{-1}$. The maximum velocity achieved by $\mathrm{m}_{1}$ is (a) $3 \sqrt{2} \mathrm{~m} \mathrm{~s}^{-1}$ (b) $5 \sqrt{2} \mathrm{~m} \mathrm{~s}^{-1}$ (c) $3 \sqrt{3} \mathrm{~m} \mathrm{~s}^{-1}$ (d) $\sqrt{7} \mathrm{~m} \mathrm{~s}^{-1}$

In the set up shown, the floor is smooth, $m_{1}=1 \mathrm{~kg}, \mathrm{~m}_{2}=10 \mathrm{~kg}$ and the coefficient of friction between their surfaces is $\mu=0.6$. Assume the surface of $\mathrm{m}$, is large enough so that when $\mathrm{m}_{2}$ is given a velocity $\mathrm{v}$ as shown, it slips on $\mathrm{m}_{1}$ till
the spring is compressed to maximum i.e., till $m_{1}$ stops. The spring constant $\mathrm{k}=200 \mathrm{~N} \mathrm{~m}^{-1}$. The maximum velocity achieved by $\mathrm{m}_{1}$ is
(a) $3 \sqrt{2} \mathrm{~m} \mathrm{~s}^{-1}$
(b) $5 \sqrt{2} \mathrm{~m} \mathrm{~s}^{-1}$
(c) $3 \sqrt{3} \mathrm{~m} \mathrm{~s}^{-1}$
(d) $\sqrt{7} \mathrm{~m} \mathrm{~s}^{-1}$

Key Concepts

Energy Transfer in Systems with Dissipative Forces

While energy conservation is a powerful tool, in systems where friction is present, some mechanical energy is dissipated as heat. Analyzing the energy transfer, including both the work done by friction and the energy stored in a compressed spring, is critical for understanding the complete dynamics of the system, particularly when determining extreme values like maximum velocities during interaction.

Conservation of Momentum

In an isolated system with no external forces, the total momentum remains constant despite internal forces such as friction or spring forces. The conservation of momentum principle is key to relating the velocities of the different masses during their interaction, allowing for the determination of how momentum is distributed through events like the maximum compression of a spring.

Spring Mechanics and Hooke's Law

A spring stores potential energy when compressed or stretched, a relationship described by Hooke's law, which states that the restoring force is proportional to the displacement from equilibrium with the constant of proportionality given by the spring constant. This concept is fundamental when analyzing how energy is stored and released within the system, influencing the dynamics and maximum velocities achieved by the masses.

Frictional Force and Relative Motion

When two objects in contact move relative to each other, friction acts at the interface, opposing this motion. The magnitude of the friction force depends on the coefficient of friction and the normal reaction force, and it mediates the momentum exchange between the objects. Understanding friction is crucial in analyzing how one mass influences the other during their interaction, especially when they slide against each other.

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