A mass m resting on a smooth horizontal slot is connected to...

A mass $\mathrm{m}$ resting on a smooth horizontal slot is connected to another equal mass over a smooth pulley. The system is at rest when $\theta=\theta_{0}\left(\Delta \mathrm{y}<<\ell=\ell^{\prime}+\Delta \mathrm{y}\right)$. The velocity of the mass on the floor at $\theta=90^{\circ}$ is (a) $\sqrt{2 \mathrm{gh}}$ (b) $\sqrt{2 \operatorname{gh}\left(1-\cos \theta_{0}\right)}$ (c) $\sqrt{2 \operatorname{gh}\left(1-\sin \theta_{0}\right)}$ (d) $\sqrt{2 \mathrm{gh}\left(\operatorname{cosec} \theta_{0}-1\right)}$

A mass $\mathrm{m}$ resting on a smooth horizontal slot is connected to another equal mass over a smooth pulley. The system is at rest when $\theta=\theta_{0}\left(\Delta \mathrm{y}<<\ell=\ell^{\prime}+\Delta \mathrm{y}\right)$. The velocity of the mass on the floor at $\theta=90^{\circ}$ is
(a) $\sqrt{2 \mathrm{gh}}$
(b) $\sqrt{2 \operatorname{gh}\left(1-\cos \theta_{0}\right)}$
(c) $\sqrt{2 \operatorname{gh}\left(1-\sin \theta_{0}\right)}$
(d) $\sqrt{2 \mathrm{gh}\left(\operatorname{cosec} \theta_{0}-1\right)}$

Key Concepts

Pulley System Dynamics

In ideal pulley systems, a massless, inextensible string transmits force between objects. The constraints imposed by the string and the frictionless pulley ensure that the movement of one mass is directly related to the movement of the other, a key concept when applying energy conservation to the system.

Kinetic Energy

Kinetic energy is the energy associated with the motion of an object. In frictionless systems, the increase in kinetic energy is directly related to the decrease in gravitational potential energy, allowing one to determine the speed of moving objects through energy conversion.

Gravitational Potential Energy

This is the energy an object possesses due to its position in a gravitational field. In many mechanics problems, a change in an object’s height leads to a change in its potential energy, which can be converted into kinetic energy, thereby driving its motion.

Conservation of Energy

This principle states that in the absence of non-conservative forces, the total mechanical energy (kinetic plus potential) of a system remains constant. It is used to equate the loss in potential energy with the gain in kinetic energy, providing a route to calculate velocities in problems involving motion under gravity.

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