Understanding Circular Motion (3.6.1)

Welcome to one of the most exciting topics in Mechanics: Circular Motion!
While moving in a straight line is simple, moving in a curve—especially a perfect circle—requires a constant push or pull. This chapter explains why objects moving at a constant speed in a circle are actually accelerating, and how we calculate the forces involved. This knowledge is crucial for understanding everything from satellites orbiting Earth to how rollercoasters manage their loops.

Don't worry if the idea of constant speed but changing velocity seems strange at first—we'll break down why this happens using vectors!

1. Uniform Circular Motion (UCM)

Uniform Circular Motion describes an object moving in a circular path at a constant speed.

Speed vs. Velocity: The Key Difference
  • Speed is a scalar quantity (just magnitude). If you drive around a perfect circular roundabout at 30 km/h, your speed is constant.
  • Velocity is a vector quantity (magnitude AND direction). Even if your speed is 30 km/h, your direction is constantly changing as you turn the steering wheel.

Because velocity is continuously changing direction, the object must be accelerating. This acceleration always points inwards, towards the centre of the circle.

Key Takeaway: Motion in a circle, even at constant speed, requires acceleration because the direction of the velocity vector is continuously changing.

2. Describing Circular Motion: Angular Speed (\(\omega\))

When dealing with circular motion, it's often more convenient to measure how quickly the angle changes rather than the linear distance covered. This measure is called angular speed (\(\omega\), pronounced omega).

Defining Angular Speed

Angular speed is the rate at which an object rotates or revolves, measured in radians per second (\(\text{rad s}^{-1}\)).

The syllabus provides three ways to calculate angular speed:

Formula 1: Relating Angular Speed to Linear Speed (\(v\))

Angular speed (\(\omega\)) is related to the linear (tangential) speed (\(v\)) and the radius (\(r\)) by the formula:

$$ \omega = \frac{v}{r} $$

This means that for a rigid body rotating, points further from the centre (larger \(r\)) must have a greater linear speed \(v\) to maintain the same angular speed \(\omega\).

Formula 2: Relating Angular Speed to Frequency (\(f\))

If an object completes \(f\) revolutions per second (frequency), it covers \(2\pi\) radians per revolution.

$$ \omega = 2\pi f $$

Remember, frequency \(f\) is the reciprocal of the Period \(T\) (time for one full rotation), so \(\omega\) is also \(\frac{2\pi}{T}\).

Quick Review Box: Units

  • Linear Speed (\(v\)): \(\text{m s}^{-1}\)
  • Angular Speed (\(\omega\)): \(\text{rad s}^{-1}\) (Note: We use radians for angular calculations in Physics A-Level!)
  • Frequency (\(f\)): \(\text{Hz}\) or \(\text{s}^{-1}\)

Key Takeaway: Angular speed (\(\omega\)) tells us how many radians are covered per second. It links the linear motion variables (\(v\), \(r\)) to the periodic variables (\(f\)).

3. Centripetal Acceleration (\(a\))

Since the velocity vector is constantly changing direction, there must be an acceleration. This acceleration is called Centripetal Acceleration.

The Direction is Everything

The word Centripetal means ‘centre-seeking’.
Crucially, the centripetal acceleration always acts perpendicular to the instantaneous velocity, pointing directly towards the centre of the circle.

  • If the acceleration pointed backwards (tangentially), the object would slow down.
  • If the acceleration pointed forwards (tangentially), the object would speed up.
  • Since the speed is constant (UCM), the acceleration must only change the direction, hence why it points to the centre.
Centripetal Acceleration Formulas

Centripetal acceleration (\(a\)) can be expressed using linear speed (\(v\)) or angular speed (\(\omega\)):

$$ a = \frac{v^2}{r} $$

OR, substituting \(v = \omega r\) into the first equation:

$$ a = \omega^2 r $$

(Phew! The syllabus states that the derivation of these formulas will not be examined, so focus on knowing how and when to use them.)

Memory Aid: Look at the formulas. The acceleration is proportional to \(v^2\). This tells you that doubling the speed increases the required acceleration by a factor of four!

Key Takeaway: Centripetal acceleration ensures the object changes direction without changing speed. It always points towards the centre of the circular path.

4. Centripetal Force (\(F\))

According to Newton's Second Law (\(F = ma\)), if there is acceleration, there must be a resultant force causing it. This resultant force is called the Centripetal Force.

Defining Centripetal Force

The centripetal force is the resultant force that acts towards the centre of the circle, necessary to maintain circular motion.

Centripetal Force Formulas

By substituting the acceleration formulas (\(a = \frac{v^2}{r}\) or \(a = \omega^2 r\)) into \(F = ma\):

$$ F = \frac{mv^2}{r} $$

OR:

$$ F = m\omega^2 r $$

A Crucial Distinction: Centripetal Force is the Resultant Force

A common pitfall is treating the centripetal force as a separate type of force, like gravity or friction. It is not.

The centripetal force is simply the name given to the resultant force (the net force) acting towards the centre. This force must be provided by one or more physical forces present in the situation:

  • Example 1: Ball on a string. The centripetal force is provided by the Tension in the string.
  • Example 2: Satellite orbiting Earth. The centripetal force is provided by Gravitational Attraction.
  • Example 3: Car turning a corner. The centripetal force is provided by Friction between the tyres and the road.

What happens if the centripetal force disappears? If you cut the string (Example 1), the tension is gone, the centripetal force is zero, and the object flies off in a straight line, tangential to the circle, obeying Newton's First Law!

Solving Problems in UCM

When solving problems, remember to identify which real physical force (or combination of forces) is providing the required centripetal force.

Step-by-Step Problem Strategy:

  1. Identify the forces acting on the object (Tension, Gravity, Normal Reaction, Friction, etc.).
  2. Determine the direction of the centre of the circle.
  3. The resultant force acting towards the centre must equal the required centripetal force \(F = \frac{mv^2}{r}\).
  4. Set up the equation: \(\sum F_{\text{towards centre}} = \frac{mv^2}{r}\).

Did you know? The feeling of being 'pushed outwards' when a car turns is often called centrifugal force. However, this is a fictitious force felt inside the accelerating frame of reference (the car). In Physics (9630), we only deal with the real, inward-acting centripetal force.

Key Takeaway: Centripetal force is the resultant inward force \(F = \frac{mv^2}{r}\) required by Newton’s Second Law to maintain circular motion. It must be provided by a real physical force.