Rotational Motion

Rotational kinematics: angular displacement, velocity, and acceleration in rigid bodies.

General idea of the topic

This topic describes the rotation of a rigid body about an axis without yet considering the forces that produce it. It carries over to angles the quantities you already know from translation: position, velocity and acceleration, now angular. It connects circular motion with the rotation of wheels, discs and celestial bodies.

What you will truly understand

Why angular position, angular velocity and angular acceleration are derivatives of one another, just as in translation. How to move from angular to linear quantities through the radius, and why a point can accelerate even though the body rotates at a constant angular velocity. Which geometric condition links the rotation of a wheel with its advance over the ground.

Recommended learning path

1. Angular quantities: angular position, displacement, velocity and acceleration, measured in radians and radians per second.
2. Rotational kinematics: uniform and uniformly accelerated circular motion, with equations analogous to those of translation.
3. Linear–angular relationship: arc length travelled, tangential velocity and tangential and centripetal accelerations.
4. Rolling condition: the centre of mass advances at the rate set by the rotation, with no slipping at the contact.

Physical criterion to acquire

An angle is measured in radians and is dimensionless, so angular velocity is expressed in radians per second. Centripetal acceleration always points towards the centre even when the magnitude of the velocity does not change. In rolling without slipping, the contact point has zero velocity relative to the ground.

Connection with real problems

• Wheels and tyres: relationship between the engine's revolutions and the vehicle's speed.
• Hard disks and electric motors: start-up and braking with constant angular acceleration.
• Bicycle wheels: the forward speed is obtained from the angular velocity and the radius.
• Satellites and planets: rotation of celestial bodies from their period.

Final objective of the topic

To describe the rotation of any rigid body with the same tools used for translation, to convert between linear and angular quantities, and to apply the rolling condition to wheels and cylinders.