Further Mechanics: Circular Motion

Basics of Circular Motion

Circular motion involves an object moving along the circumference of a circular path, or in a circle.
The speed of an object moving in a circle (its scalar magnitude) may be constant, but its velocity (which considers direction) is not, as the direction is constantly changing.
A change in velocity, even if just directional, implies an acceleration. This is known as centripetal acceleration.

Centripetal Force

For an object to move in a circle, there must be a force acting towards the centre of the circle. This is the centripetal force.
Without a centripetal force, an object would continue to move in a straight line, as per Newton’s first law of motion.
Centripetal force is always perpendicular to the velocity of the object.
The formula for centripetal force is F=mv²/r, where m is the mass, v is the velocity, and r is the radius of the circle.

Centripetal Acceleration

The centripetal acceleration is the rate of change of tangential velocity and always points towards the centre of the circle.
The formula for centripetal acceleration is a=v²/r or a=ω²r, where ω is the angular velocity.

Angular Velocity and Frequency

Angular velocity represents how quickly an object moves around the centre of the circular path. It is often represented by the Greek letter ω (omega).
Angular velocity ω is equal to 2πf or 2π/T, where f is the frequency and T is the period of the motion.
Frequency is the number of rotations per unit of time, measured in hertz (Hz). Period is the time taken to complete one full rotation.

Radial and Tangential Velocity

In circular motion, radial velocity refers to the speed of an object along the radius of the circle, which is always zero.
Tangential velocity refers to the speed of an object along the tangent to the circular path.

Uniform and Non-uniform Circular Motion

Uniform circular motion describes an object moving in a circular path at a constant speed.
Non-uniform circular motion, also known as angular motion, involves an object moving in a circular path with changing speed.

Free Body Diagrams in Circular Motion

Free body diagrams can be useful to identify the forces acting on an object in circular motion, such as gravitational force, tension, normal force, or friction.

Banked Curves and Conical Pendulums

Banked curves and conical pendulums involve the concepts of vertical circular motion.
Banked curves help to balance the forces acting on an object travelling in a circular path, reducing the dependence on friction.

Satellite Motion

Satellite motion is a form of circular motion. A satellite orbits around the Earth due to the gravitational pull acting as a centripetal force.
For the satellite, the gravitational force FG = mv²/r, where m is the mass of the satellite, v is its velocity, r is the distance from the centre of the Earth, and G is the gravitational constant.

Kepler’s Laws of Planetary Motion

Kepler’s laws describe the motion of planets around the sun, which also involves circular motion.
This includes the principle that planets move in elliptical orbits, and that the line between the sun and the planet sweeps out equal areas in equal times.