Angular velocity and the relationship between linear and angular speed, centripetal acceleration, and centripetal force in horizontal and vertical circular motion.

What this dot point is asking

Edexcel wants you to define angular velocity, relate linear and angular speed, calculate centripetal acceleration, and apply centripetal force ideas to horizontal and vertical circular motion, identifying which real force provides the centripetal force in each case.

The answer

Angular velocity and linear speed

Every point on a rotating rigid body has the same angular velocity, but points farther from the axis move faster in linear terms because $v = r\omega$. One full revolution is $2\pi$ radians.

Centripetal acceleration

Speed can be constant while velocity changes, since velocity is a vector. The acceleration is perpendicular to the velocity, so it changes direction, not magnitude, of the motion.

Centripetal force

For a car cornering, friction between tyres and road supplies it; for a satellite, gravity supplies it; for a ball on a string, tension supplies it. If the required force is not available (for example, not enough friction), the object cannot follow the circle and flies off tangentially.

Horizontal and vertical circles

In a horizontal circle (such as a conical pendulum or a banked corner) the vertical forces balance while a horizontal component provides the centripetal force. In a vertical circle (such as a ball on a string or a loop-the-loop) the required centripetal force is constant in size for constant speed, but gravity helps at the top and opposes at the bottom, so the tension or normal force is least at the top and greatest at the bottom. At the very top, the minimum speed to maintain the circle is when gravity alone provides the centripetal force, $v_{\min} = \sqrt{gr}$.

Examples in context

Banked race tracks and bends in roads are angled so that a component of the normal force supplies the centripetal force, reducing reliance on friction. A centrifuge spins samples so that the large $\omega^2 r$ acceleration separates components by density. Satellites orbit because gravity provides exactly the centripetal force for their speed and radius. Fairground rides, from the rotor to the loop-the-loop, are direct applications of vertical and horizontal circular motion.

Try this

Q1. State the direction of the centripetal acceleration. [1 mark]

• Cue. Towards the centre of the circle.

Q2. A wheel rotates at $5.0$ revolutions per second. Find its angular velocity. [2 marks]

• Cue. $\omega = 2\pi f = 2\pi \times 5.0 = 31$ rad per second.

Q3. A $2.0$ kg mass moves in a circle of radius $0.50$ m at $4.0$ m per second. Find the centripetal force. [2 marks]

• Cue. $F = \frac{mv^2}{r} = \frac{2.0 \times 4.0^2}{0.50} = 64$ N.