Question

A body starts to slide from A, down the fixed smooth inclined plane AO having inclination $\theta$ with the horizontal and then ascends another fixed smooth plane OB up to B. If impact at O is neglected, then:

A. $t_{AO}$>$t_{OB}$
B. $t_{AO}$<$t_{OB}$
C. $t_{AO}$=$t_{OB}$
D. $h^{\prime} = h$

Answer 1

The principle of conservation of mechanical energy states that the work done by a conservative force on a body depends only on the initial and final state of the body, and the total mechanical energy of the system remains constant throughout. To this end, determine the influencing forces in the system and their type. In other words, a smooth plane implies a frictionless surface which only leaves the gravitational force, which we know to be a conservative force. Use these statements to arrive at the appropriate relation.
Formula Used:
Gravitational potential energy $U=mgh$

Complete step by step answer:
We are given that both the inclined planes are smooth, which means that the contribution of frictional force between the body and the plane is negligible. Therefore, the only influencing force acting on this system is the gravitational force.
Now, gravitational force is a conservative force. This means that the work done by this force in moving any object from one position to the other is independent of the path taken and depends only on the initial and final position of the object.
This brings us to the principle of conservation of mechanical energy which states that the total mechanical energy of the system remains constant as long as the only forces acting on the system are conservative forces.
Now, let us put all of these definitive deductions into the context of our question.
We have already established that the entire system is under the influence of only conservative (gravitational) force, and there are no non-conservative (frictional) forces coming into the picture, owing to the smooth inclined planes and neglection of impact at O.
Let the mass of the body be m. And let the acceleration due to gravity acting on the body be g.
Now, the body is at a height h on the smooth inclined plane AO. By virtue of its state of rest at a height h the body possesses a potential energy that is gravitationally influenced and is given as:
$U_{A} = mgh$
Similarly, when the body reaches the height h’ on another smooth inclined plane OB, it possesses a gravitational potential energy of
$U_{B} = mgh^{\prime}$
Now, according to our definition of a conservative force and the principle of conservation of mechanical energy we see that the work done by this conservative force depends only on the initial A and final B position of the object, irrespective of the path taken, and that the mechanical energy of the system is conserved. This means that the initial and final energy of the system is the same.
This means that $U_A = U_B \Rightarrow mgh = mgh^{\prime} \Rightarrow h=h^{\prime}$
Therefore, the correct choice would be: D. $h^{\prime}=h$

Note:
It is important to remember the distinction between conservative and non-conservative forces. A conservative force “conserves” the energy of the system by not spending it on collateral processes, whereas a non-conservative force does not shy away from energy expenditures on such processes via heat, sound, light, etc. The work done by a conservative force is independent of the path taken and depends only on the initial and final position of the body whereas the work done by a non-conservative force is dependent on the path taken. Gravitational and electrostatic forces are examples of conservative forces, whereas, air resistance and friction are examples of non-conservative forces. Note that if the path taken by an object is closed, then the work done by a conservative force is zero.