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Rolling Without Slipping - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • 8 Questions around this concept.

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 A roller is made by joining together two cones at their vertices O.  It is kept on two rails AB and CD which are placed asymmetrically (see figure), with its axis perpendicular to CD and its centre O at the centre of line joining AB and CD (see figure).  It is given a light push so that it starts rolling with its centre O moving parallel to CD in the direction shown.  As it moves, the roller will tend to :

Concepts Covered - 1

Rolling Without Slipping

 

  • The linear velocity of different points 

In pure Translation-

In pure Rotation-

And in Rolling all points of a rigid body have same angular speed (\omega) but different linear speed.

      I.e

  • During Rolling motion

If V_{cm}> Rw\rightarrow slipping \: motion

If V_{cm}= Rw\rightarrow pure \: rolling

If V_{cm}< Rw\rightarrow skidding \: motion

 

When the object rolls across a surface such that there is no relative motion of object and surface at the point of contact, the motion is called rolling without slipping.

Here the point of contact is P.

Friction force is available between object and surface but work done by it is zero because there is no relative motion between body and surface at the point of contact.

Or we can say No dissipation of energy is there due to friction.

I.e., Energy is conserved.

Which is K_{net} =K_T +K_R = \frac{1}{2}mV^2+\frac{1}{2}I\omega^2

Now  using V = \omega.R

And using  K_{net}= \frac{1}{2}mV^2+\frac{1}{2}I\omega ^2=\frac{1}{2}(I+mR^2)\omega ^2

Where I = moment of inertia of the rolling body about its centre ‘O’

And using Parallel axis theorem 

We can write I_p=I+mR^2

So we can write K_{net}=\frac{1}{2}I_p\omega ^2

 

Where I_p=moment of inertia of rolling body about point of contact ‘P’.

 

So this Rolling motion of a body is equivalent to a pure rotation about an axis passing through the point of contact  (here through P) with the same angular velocity \omega.

Here axis passing through the point of contact P is also known as Instantaneous axis of rotation.

(Instantaneous axis of rotation-Motion of an object may look as pure rotation about a point that has zero velocity.)

  •  Net Kinetic Energy for  different rolling bodies

As K_{net} = K_T+K_R=\frac{1}{2}mV^2(1+\frac{K^2}{R^2})

So the quantity \frac{K^2}{R^2} will have different values for different bodies.

 

Rolling body

\frac{K^2}{R^2}

K_{net}

Ring

Or Cylindrical shell

1

mV^2

Disc

Or solid cylinder

\frac{1}{2}

\frac{3}{4}mV^2

Solid sphere

\frac{2}{5}

\frac{7}{10}mV^2

Hollow sphere

\frac{2}{3}

\frac{5}{6}mV^2

 

  • The direction of friction-

Kinetic friction will always oppose the rolling motion. While Static friction on the other hand only opposes the tendency of an object to move.

  1. When an external  force is in the upward diametric part

  • If K^{2}= Rx then no friction will act

  • If K^{2}> Rx then Friction will act in the backward direction

  • If K^{2}< Rx then Friction will act in a forward  direction 

  1. If an external force is in the lower diametric part,

Then friction always act backwards

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Rolling Without Slipping

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