Loading....
Coupon Accepted Successfully!

 

Slipping, Spinning, and Rolling

Slipping When the body slides on a surface without rotation, then its motion is called slipping motion. In this condition, friction between the body and surface F = 0 (Fig. 7).
 
43370.png
Fig. 7
 
Body possess only translatory kinetic energy, 43364.png.
 
Example

Motion of a ball on a frictionless surface.

 

 
Spinning When the body rotates in such a manner that its axis of rotation does not move, then its motion is called spinning motion.
 
In this condition, axis of rotation of a body is fixed.
 
43358.png
Fig. 8
 
Example

Motion of blades of a fan.

 

 
In spinning, body possess only rotatory kinetic energy,
43352.png
 
43346.png
 
i.e., rotatory kinetic energy = (K2/R2) times translatory kinetic energy.
 
Here, K2/R2 is a constant for different bodies. Value of K2/R2 = 1 (ring), K2/R2 = 1/2 (disc), and K2/R2 = 1/2 (solid sphere).
 
Rolling If in case of rotational motion of a body about a fixed axis, the axis of rotation also moves, the motion is called combined translatory and rotatory.
 
43308.png
Fig. 9
 
Example
  1. Motion of a wheel of cycle on a road.
  2. Motion of football rolling on a surface.
 
In this condition, friction between the body and surface, F = 0.
 
Body possesses both translational and rotational kinetic energy.
 
Net kinetic energy = (Translatory + Rotatory) kinetic energy
 
43302.png
46962.png43295.png
 
∴ 43289.png

Rolling without slipping

In case of combined translatory and rotatory motion, if the object rolls across a surface in such a way that there is no relative motion of object and surface at the point of contact, the motion is called rolling without slipping.
 
Friction is responsible for this type of motion but work done or dissipation of energy against friction is zero as there is no relative motion between body and surface at the point of contact.
 
The rolling motion of a body may be treated as a pure rotation about an axis through the point of contact with same angular velocity ω (Fig. 10).
 
44260.png
Fig. 10
 
By the law of conservation of energy,
KN43257.png    [as v = ]
43251.png
43245.png
43239.png [as IP = I + mR2]
 
By theorem of parallel axis, where I = moment of inertia of rolling body about its center O and IP = moment of inertia of rolling body about point of contact P.

Rolling on an inclined plane

When a body of mass m and radius R rolls down on inclined plane of height ‘h’ and angle of inclination θ, it loses potential energy. However it acquires both linear and angular speeds and hence, gain kinetic energy of translation and that of rotation.
 
44254.png
Fig. 11
 
By conservation of mechanical energy,
43206.png
 
Velocity at the lowest point
 
43200.png
 
Acceleration in motion From equation v2 = u2 + 2aS
 
By substituting u = 0, 43194.png and
43187.png we get
43181.png
 
Time of descent Consider equation v = u + at.
 
By substituting u = 0 and value of v and a from above expressions,
 
43174.png
 
From the above expressions, it is clear that
 
43168.png

 

Note
  • Factor (k2 / R2) is a measure of moment of inertia of a body. Its value is constant for given shape of the body and it does not depend on the mass and radius of a body.
  • Velocity, acceleration, and time of descent (for a given inclined plane) all depend on k2 / R2. Lesser the moment of inertia of the rolling body lesser will be the value of k2 / R2. So greater will be its velocity and acceleration and lesser will be the time of descent.
  • If a solid and hollow body of same shape are allowed to roll down on inclined plane, then as k2 / R2S < k2 /R2H, solid body will reach the bottom first with greater velocity.
  • If a ring, cylinder, disc, and sphere run a race by rolling on an inclined plane,then as (k2 / R2)sphere = minimum, while(k2 / R2)ring = maximum, the sphere will reach the bottom first with greatest velocity while ring at last with least velocity.
  • The angle of inclination has no effect on velocity, but time of descent and acceleration depend on it. Velocity ∝ θ°, time of decent ∝ θ –1, and acceleration ∝ θ.




Test Your Skills Now!
Take a Quiz now
Reviewer Name