A ball rolls down an inclined planc and acquires a velocity...

A ball rolls down an inclined planc and acquires a velocity $v_{r}$ when it reaches the bottom of the planc. If the same ball slides without friction and acquires rolling from the same hcight down and cqually inclined smooth planc and aequires a velocity $v$ then which of the following statements are nol correct? (a) $v_{r}<v_{x^{\prime}}$ becausc a work is done by the rolling ball against the frictional force (b) $v_{r}>v_{s^{2}}$, because the angular velocity acquired makes the rolling ball to travel faster (c) $v_{r}=v_{s^{\prime}}^{\prime}$ bocause kinctic cnergy of two balls is samc at bottom of plancs (d) $v_{r}<v_{x^{\prime}}$, because the rolling ball acquires rotational as well as translational kinctic encrgy

A ball rolls down an inclined planc and acquires a velocity $v_{r}$ when it reaches the bottom of the planc. If the same ball slides without friction and acquires rolling from the same hcight down and cqually inclined smooth planc and aequires a velocity $v$ then which of the following statements are nol correct?
(a) $v_{r}<v_{x^{\prime}}$ becausc a work is done by the rolling ball against the frictional force
(b) $v_{r}>v_{s^{2}}$, because the angular velocity acquired makes the rolling ball to travel faster
(c) $v_{r}=v_{s^{\prime}}^{\prime}$ bocause kinctic cnergy of two balls is samc at bottom of plancs
(d) $v_{r}<v_{x^{\prime}}$, because the rolling ball acquires rotational as well as translational kinctic encrgy

Key Concepts

Rotational Kinetic Energy

Rotational kinetic energy is the energy associated with an object's rotation about its own axis. For a ball rolling down an incline without slipping, not all the potential energy is converted into translational motion; some is used to generate rotational motion. This division of energy means that the translational speed of a rolling ball is less than that of a sliding ball, given the same drop in height.

Friction and Work in Rolling Motion

In rolling without slipping, static friction is necessary to prevent slipping and to initiate rotation; however, static friction does not do work because there is no displacement at the point of contact. Therefore, even though friction is essential for rolling, it does not dissipate energy. The energy difference between a rolling ball and a sliding ball arises because the rolling ball’s energy is partitioned into both rotational and translational forms, not due to energy lost in overcoming friction.

Conservation of Energy

This concept involves the idea that in the absence of non-conservative forces doing work (or when work done is accounted for), the total mechanical energy (potential plus kinetic) remains constant. In problems involving objects moving down an incline, the gravitational potential energy lost is converted into kinetic energy; for sliding objects all the potential energy converts to translational kinetic energy, whereas for rolling objects the energy is shared between translational and rotational kinetic energies.

Translational Kinetic Energy

Translational kinetic energy is the energy due to the motion of the center of mass of an object. In the context of a sliding object on an incline, it is the only form of kinetic energy acquired when there is no rotation. This is why a frictionless sliding ball converts all its potential energy into translational kinetic energy, resulting in a higher speed compared to a rolling ball.

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