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Objective questions, contained in this module with hidden solutions, help improve understanding of the topics covered under the module "Work and energy in rolling".

The questions have been selected to enhance understanding of the topics covered in the module titled " Work and energy in rolling ". All questions are multiple choice questions with one or more correct answers. There are two sets of the questions. The “understanding level” questions seek to unravel the fundamental concepts involved, whereas “application level” are relatively difficult, which may interlink concepts from other topics.

Each of the questions is provided with solution. However, it is recommended that solutions may be seen only when your answers do not match with the ones given at the end of this module.

Understanding level (work and energy in rolling)

A spherical ball rolls without sliding. Then, the fraction of its total mechanical energy associated with translation is :

(a) 1 2 (b) 2 3 (c) 3 4 (d) 5 7

The kinetic energy for translation is distributed between translation and rotation. The ratio of two kinetic energy is given by :

K R K T = I M R 2

Adding “1” to either side of the equation and solving, we have :

K K T = I + M R 2 M R 2

Inverting the ratio,

K T K = M R 2 I + M R 2

For solid sphere,

I = 2 M R 2 5

Putting MI’s expression in the ratio,

K T K = M R 2 2 M R 2 5 + M R 2 = 5 7

Hence, option (d) is correct.

Important to note here is that this distribution of kinetic energy between translation and rotation is independent of the path of rolling. It only depends on the MI of rolling body i.e. on mass and its distribution about the axis of rotation. As such, the distribution of kinetic energy between translation and rotation are different for different bodies. In the case of ring and hollow cylinder, the distribution is 50% - 50%.

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Two identical spheres start from top of two incline planes of same geometry. One slides without rolling and other rolls without sliding. If loss of energy in two cases are negligible, then which of the two spheres reachs the bottom first ?

(a) sphere sliding without rolling reaches the bottom first (b) sphere rolling without sliding reaches the bottom first (c) both spheres reaches the bottom at the same time (d) sphere sliding without rolling stops in the middle

Here, no loss of energy is involved, so we can employ law of conservation of energy for the two cases. B

Since both spheres move same vertical displacement, they have same gravitational potential energy. Further, there is no loss of energy. As such, spheres have same kinetic energy at the bottom.

In the case of sliding without rolling, the total kinetic energy is translational kinetic energy. On the other hand, kinetic energy is distributed between translation and rotation in the case of rolling without sliding. The sphere in rolling, therefore, has lesser translational kinetic energy.

Therefore, velocity of the sphere sliding without rolling reaches the ground with greater speed. Again, as geometry of the inclines are same, the sphere sliding without rolling reaches the bottom first. With respect to option (d), we note that once sliding starts, the sphere does not stop on an incline.

Hence, option (a) is correct.

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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