1. What is Rolling Motion?
Rolling motion occurs when an object, such as a wheel or ball, moves forward while simultaneously rotating. A rolling object combines two types of motion:
- Translational motion — the entire body moves forward.
- Rotational motion — the body spins about its axis.
When both motions happen in such a way that the point of contact is momentarily at rest relative to the ground, the object is said to be rolling without slipping.
1.1. Everyday Examples
You see rolling motion everywhere:
- Wheels of bicycles, cars, and carts
- A ball rolling down a slope
- A roller moving on a table
- A coin spinning and rolling on a surface
1.1.1. Intuitive Idea
During rolling, the bottom point of the object touches the ground without sliding. This special condition links rotation and translation together.
2. Rolling Without Slipping
Rolling without slipping is a condition where the object rolls smoothly without sliding at the point of contact. This happens when the linear velocity of the centre of mass matches the rotational motion.
The key relation is:
\( v = \omega R \)
Where:
- \( v \) is the linear speed of the centre of mass
- \( \omega \) is the angular velocity
- \( R \) is the radius of the rolling object
2.1. Why No Slipping Occurs
The point of contact must have zero velocity relative to the ground. This makes the object grip the surface, allowing pure rolling.
2.2. What Happens During Slipping?
If \( v \neq \omega R \), slipping occurs. This happens in cases like:
- A car accelerating too quickly
- A ball skidding on a smooth floor
- A wheel braking suddenly
3. Velocity Distribution in Rolling Motion
Different points on a rolling object have different velocities. The centre of mass has velocity \( v \), while other points have velocities formed by combining translational and rotational components.
3.1. Top and Bottom Points of a Rolling Body
- Bottom point: Momentarily at rest (velocity = 0).
- Centre of mass: Moves forward with velocity \( v \).
- Top point: Has velocity \( 2v \) because translational and rotational velocities add up.
3.2. General Velocity Expression
For any point on the object:
\( \vec{v} = \vec{v}_{cm} + \vec{v}_{rot} \)
4. Kinetic Energy in Rolling Motion
A rolling object has two kinds of kinetic energy:
- Translational KE: \( K_t = \dfrac{1}{2} M v^2 \)
- Rotational KE: \( K_r = \dfrac{1}{2} I \omega^2 \)
Total kinetic energy is the sum of both:
\( K_{total} = \dfrac{1}{2} M v^2 + \dfrac{1}{2} I \omega^2 \)
4.1. Using the No-Slip Condition
Using \( v = \omega R \), the total energy can be written in terms of either \( v \) or \( \omega \). Different shapes (disc, sphere, ring) have different rotational inertias, which changes how their energy is shared between rotation and translation.
4.2. Examples
- A hollow ring has more rotational KE than a solid disc of the same mass.
- A sphere reaches the bottom of a slope faster than a cylinder because it has smaller moment of inertia.
5. Rolling Down an Inclined Plane
When an object rolls down a slope without slipping, gravitational potential energy is converted into both translational and rotational kinetic energies. The acceleration depends on the moment of inertia of the object.
5.1. Acceleration Formula
For an object of moment of inertia \( I \), the acceleration down an incline of angle \( \theta \) is:
\( a = \dfrac{g \sin \theta}{1 + \dfrac{I}{M R^2}} \)
5.2. Which Object Reaches First?
Objects with smaller moment of inertia (solid sphere, solid disc) accelerate faster and reach the bottom sooner than objects with larger MI (ring, hollow cylinder).
6. Why Rolling Motion Matters
Rolling motion combines ideas from translation, rotation, energy, and friction. It is essential in understanding vehicle motion, machinery, sports physics, wheel dynamics, and many natural processes.
In the next topic, we explore the concept of rotational kinetic energy in more detail.