Edexcel A Level (IAL) Physics-4.9 Centripetal Acceleration- Study Notes- New Syllabus

Centripetal Acceleration: Derivation Using Vector Diagrams and Applications

In uniform circular motion, a particle moves at constant speed but continuously changes direction. This change in velocity produces an acceleration directed towards the centre of the circle, called the centripetal acceleration.

 Why There Is Acceleration in Circular Motion

• Speed is constant, but velocity changes direction.
• Acceleration is the rate of change of velocity.
• Therefore, circular motion requires an acceleration.
• This acceleration always points towards the centre of the circle.

 Vector Diagram Derivation of \( a = \dfrac{v^2}{r} \)

Consider a particle moving in a circle of radius \( r \) with constant speed \( v \).

• At two nearby points, the velocity vectors have the same magnitude \( v \) but different directions.
• Place the velocity vectors tail-to-tail.
• The change in velocity \( \Delta \vec{v} \) is given by the vector joining their tips.
• For a small angular displacement \( \Delta \theta \), the velocity vectors form an isosceles triangle.

From the velocity vector triangle:

\( |\Delta v| \approx v\Delta\theta \)

Angular displacement in time \( \Delta t \) is:

\( \Delta\theta = \dfrac{v\Delta t}{r} \)

Substitute into the change in velocity:

\( \Delta v = v \cdot \dfrac{v\Delta t}{r} = \dfrac{v^2\Delta t}{r} \)

Acceleration is:

\( a = \dfrac{\Delta v}{\Delta t} = \dfrac{v^2}{r} \)

Direction: The change in velocity vector points towards the centre → acceleration is centripetal.

Deriving \( a = r\omega^2 \)

The linear speed is related to angular velocity by:

\( v = \omega r \)

Substitute into \( a = \dfrac{v^2}{r} \):

\( a = \dfrac{(\omega r)^2}{r} = r\omega^2 \)

Thus:

\( a = \dfrac{v^2}{r} = r\omega^2 \)

Key Features of Centripetal Acceleration

• Always directed towards the centre of the circular path.
• Perpendicular to the instantaneous velocity.
• Increases with speed.
• Decreases with increasing radius for the same speed.

When to Use Each Form

• Use \( a = \dfrac{v^2}{r} \) when linear speed is known.
• Use \( a = r\omega^2 \) when angular velocity is known.
• Both give the same physical acceleration.