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8 Questions around this concept.
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 :
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 () but different linear speed.
I.e
During Rolling motion
If
If
If
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
Now using
And using
Where I = moment of inertia of the rolling body about its centre ‘O’
And using Parallel axis theorem
We can write
So we can write
Where =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 .
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
So the quantity will have different values for different bodies.
Rolling body |
||
Ring Or Cylindrical shell |
1 |
|
Disc Or solid cylinder |
||
Solid sphere |
||
Hollow sphere |
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.
When an external force is in the upward diametric part
If then no friction will act
If then Friction will act in the backward direction
If then Friction will act in a forward direction
If an external force is in the lower diametric part,
Then friction always act backwards
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