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A bolt of mass 0.3 kg falls from the ceiling of an elevator moving down with a uniform speed of...

Question:

A bolt of mass 0.3 kg falls from the ceiling of an elevator moving down with a uniform speed of 7m/s. It hits the floor of the elevator (length of the elevator =3m) and does not rebound. What is the heat produced by the impact? Would the answer be different if the elevator were stationary?

The Elevator Frame of Reference

Newton's laws give the correct dynamics in any inertial frame of reference. However, if the frame of reference is accelerating or non- inertial then even in the absence of applied forces objects are seen to accelerate. For instance, if you are inside a train moving with a constant velocity and then if it so happens that the brakes are applied then you will find yourself hurtling forward even though no visible force has been applied on you. The Newtonian scheme may be resurrected in a non-inertial frame by invoking a fictitious force. If the acceleration of the frame is {eq}\displaystyle { a} {/eq} simply assign a force {eq}\displaystyle {- ma} {/eq} to a particle of mass {eq}\displaystyle {m} {/eq}. Now take into account all the real forces and apply Newton's law.

Answer and Explanation:

Since the elevator is moving with a constant speed of 7 m/s, the elevator frame of reference is an inertial frame. Hence Newton's laws apply directly and all physical phenomena are identical with that in a stationary frame. The bolt of mass 0.3 kg falls from a height of 3 m, hits the floor, and does not rebound. So its entire energy is thermalized. The initial energy at the instant of falling is entirely potential. It is,

{eq}\displaystyle {E=mgh=0.3\times 9.8\times 3=8.82\ J} {/eq}.

As it falls this gets converted to kinetic energy and at the bottom just before hitting the floor the entire energy is kinetic. On hitting the floor the kinetic energy is thermalized.

Thus the heat generated would be {eq}\displaystyle {8.82\ J} {/eq}.

The same result obtains in a stationary elevator.


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Inertial Frame of Reference: Definition & Example

from General Studies Science: Help & Review

Chapter 4 / Lesson 12
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