Question

A sphere cannot roll on
(A) A smooth horizontal surface
(B) A smooth inclined surface
(C) A rough horizontal surface
(D) A rough inclined surface

Answer 1

Hint: When a body continues to roll on a surface and the body has acceleration a then It must have an angular acceleration $\alpha $and if it has an angular acceleration $\alpha $, it must have an acceleration a such that the relation between them is $a = \alpha R$, where R denotes the radius of the body. In all the above cases given in the options, we should look for such relation, and if such a relation is not possible, the body cannot roll.

Complete step by step answer:
Discussing the possibility of option (a)
Consider a ball having radius R and is placed on a smooth horizontal surface. The forces acting on the ball will be
Weight perpendicular to the surface (in the downward direction) and Normal reaction perpendicular to the surface (in the upward direction), this means that the horizontal force on the sphere will be absent which can accelerate the sphere. Hence $a = 0$.
Both these forces (Normal reaction and weight) are passing through the centre of the ball hence the torque on the ball about the centre will also be zero, that means $\alpha = 0$.
Since the values are such that $a = \alpha R$, hence rolling motion with uniform speed and angular speed is possible on such a surface.

Discussing the possibility of option (b)
Consider a ball having radius R and is placed on a smooth inclined surface. The forces acting on the ball will be
Weight in the downward direction will have two components, one is along the surface and one will be perpendicular to the surface and Normal reaction force will be perpendicular to the surface, this means that the component of weight along the plane will be a force on the sphere which can accelerate the sphere. Hence $a \ne 0$.
Both these forces (Normal reaction and weight) are passing through the centre of the ball hence the torque on the ball about the centre will also be zero, that means $\alpha = 0$.
Since the values are such that $a \ne \alpha R$, hence rolling motion is not possible on such a surface.

Option (c) and (d) do not need mathematical calculations since they can be seen in everyday life. And the possibility of rolling motion can be easily discussed.
For option (c), answer this question: Can you roll a ball on the floor of your home? The answer is Yes, hence you can roll a ball on a rough horizontal surface.
For option (d), answer this question: Have you seen a pen rolling off the edge of a table by itself? The answer is Yes, hence you can roll a ball on a rough inclined surface.

Therefore, the correct answer to the question is option : B

Note:A common mistake that students make while solving such questions is not relating physics with everyday phenomena but at the same time, mathematical equations are required to judge events not possible in everyday life, like a horizontal smooth surface. Secondly, the condition $a = \alpha R$ gives us the equation for rolling motion to continue, but if there is slipping initially and we want the rolling motion to start then rolling motion always starts when the condition $v = \omega R$ is valid, where v denotes the speed of the object and $\omega $ denotes the angular speed. Once rolling motion starts the condition $a = \alpha R$ is used to decide whether the ball will continue to roll.