Short Answer:

Rolling motion without slipping is a type of motion in which a body, such as a wheel or a cylinder, rolls over a surface in such a way that there is no relative motion between the point of contact and the surface. In this case, the body rolls smoothly without sliding or skidding.

In simple words, when a wheel or sphere moves on a surface and the point of contact with the ground is momentarily at rest, it is called rolling without slipping. This motion is a combination of pure rotation and translation, where the linear velocity of the center of the body equals the angular velocity multiplied by its radius.

Detailed Explanation :

Rolling Motion Without Slipping

The concept of rolling motion without slipping is an important topic in mechanics, particularly in the study of dynamics of rigid bodies. It describes the motion of a circular object, like a wheel or a ball, moving on a surface where the motion is a perfect combination of rotation and translation.

When an object rolls without slipping, every point on the object has a unique motion. The point of contact with the surface remains momentarily stationary, while other points move in a circular path around the center. This condition ensures that no energy is lost due to frictional sliding, making the motion smooth and efficient.

Definition

Rolling motion without slipping can be defined as:

“The motion of a rigid body on a surface in which the point of contact between the body and the surface has zero relative velocity with respect to the surface.”

Mathematically, this condition is expressed as:

where,

= Linear velocity of the center of mass
= Angular velocity of the body
= Radius of the body

This equation ensures that the tangential velocity due to rotation equals the translational velocity of the center of mass, thus preventing slipping.

Nature of Rolling Motion

The motion of a rolling object can be considered as a combination of two types of motion:

Translational Motion:
The entire body moves forward such that every point on the body has the same linear velocity equal to that of the center of mass.
Rotational Motion:
The body rotates about its own axis with an angular velocity .

When both these motions occur simultaneously and satisfy the condition , the motion is called pure rolling motion or rolling without slipping.

Condition for Rolling Without Slipping

The condition that must be satisfied for rolling without slipping is:

Where,

= Linear velocity of the center of mass of the body
= Radius of the rolling body
= Angular velocity

This condition ensures that the instantaneous point of contact between the rolling object and the ground has zero velocity with respect to the surface.

If the velocity of the point of contact is greater than zero, the body is slipping forward; and if it is less than zero, it is slipping backward.

Velocity and Acceleration in Rolling Motion

For a rolling body, the velocity of any point on the rim is determined by combining the translational and rotational components of motion.

Velocity of the Topmost Point (A):
The topmost point has both translational and rotational velocities in the same direction.

Velocity of the Center (C):
The center moves with translational velocity .
Velocity of the Bottommost Point (B):
The bottommost point, in contact with the ground, has rotational velocity opposite to the translational velocity, making its resultant velocity zero:

This shows that the point of contact does not slip, fulfilling the rolling condition.

Similarly, for acceleration:

where,

= Linear acceleration of the center
= Angular acceleration

Examples of Rolling Motion Without Slipping

A wheel moving smoothly on a road.
A cylinder rolling on an inclined plane without skidding.
A ball rolling on the ground.
A pulley rotating with a belt moving smoothly.
A rolling drum used in conveyors or transport systems.

In all these cases, the frictional force ensures that the point of contact does not slide.

Role of Friction in Rolling Motion

Friction plays a crucial role in enabling rolling without slipping. Although the rolling object does not experience kinetic friction (since there is no relative motion at the point of contact), static friction is necessary to prevent slipping.

Static Friction: Acts at the point of contact to prevent slipping. It adjusts its magnitude as needed, up to a maximum limit , where is the normal reaction.
Kinetic Friction: Comes into play only when slipping occurs, reducing efficiency and causing wear and tear.

In ideal rolling motion, only static friction acts, and the energy loss is minimal.

Energy in Rolling Motion

In rolling motion without slipping, the total kinetic energy is the sum of translational and rotational kinetic energies.

Substituting :

Where,

= Mass of the rolling body
= Moment of inertia of the body about its axis of rotation

For example:

For a solid sphere, ,
so .
For a solid cylinder, ,
so .

These relationships show that the total energy is distributed between linear and rotational forms.

Advantages of Rolling Without Slipping

Smooth Motion: The object moves without jerks or vibrations.
Energy Efficiency: Minimal energy is lost to friction and heat.
Reduced Wear: Surfaces last longer because there is no sliding friction.
Better Control: Ensures steady and predictable movement of wheels.
Higher Performance: Used in vehicles, rollers, and machinery for smooth power transmission.

Difference Between Rolling With and Without Slipping

Aspect	Rolling Without Slipping	Rolling With Slipping
Relative motion at contact	No relative motion	Relative motion exists
Friction type	Static friction	Kinetic friction
Energy loss	Very small	High due to sliding
Condition
Example	Smooth car tire motion	Car tire skidding on road

Applications

Automobile tires rolling on roads
Conveyor rollers in industries
Gear and pulley mechanisms
Railway wheels on tracks
Rolling bearings in machines

These applications depend on the principle of rolling without slipping for smooth and efficient operation.

Conclusion

The rolling motion without slipping is a combination of pure rotation and translation in which the point of contact between a rolling body and the surface remains momentarily at rest. The condition ensures that the body rolls smoothly without sliding. Static friction plays a crucial role in maintaining this condition. This type of motion is vital in the design of wheels, gears, and rollers, as it ensures efficiency, stability, and minimal energy loss in mechanical systems.