6.1.3 Centripetal Acceleration

Centripetal Acceleration

• For an object moving in a circle
• the acceleration is towards the centre of the circle
• The magnitude of the acceleration is v2/r

What Causes Centripetal Acceleration?

• Velocity and acceleration  are both vector  quantities
• An object in uniform circular motion is continuously changing direction, and therefore is constantly changing velocity
• The object must therefore be accelerating
• This is called the centripetal acceleration

The Direction of the Centripetal Acceleration

• The centripetal acceleration is perpendicular to the direction of the linear speed
• Centripetal means it acts towards the centre of the circular path

Checking the direction of the centripetal acceleration.

• The object moves from A to B during some time Δt
• The change in velocity during this time is Δv
• The centripetal acceleration is Δv (a vector) divided by Δt (a scalar)
• The centripetal acceleration points in the same direction as the change in velocity Δv
• Slide a ruler parallel to Δv towards the circle in the diagram above
• Midway between A and B, Δv points towards the centre of the circle
• The centripetal acceleration is caused by a centripetal force of constant magnitude that also acts perpendicular to the direction of motion (towards the centre)
• There is no component of the centripetal force in the direction of the velocity so there is no acceleration in the direction of the velocity
• Hence the uniform motion at constant speed
• The centripetal acceleration and force act in the same direction

The Magnitude of the Centripetal Acceleration

• In the diagram above notice how the angle Δθ is defined in terms of the arc length vΔt and the radius r
• v is the magnitude of v1 and v2.

Deriving the equation for the magnitude of the centripetal acceleration

• The vector triangle of the figure above so that Δv is horizontal
• The velocity vectors v are of the same length hence
• the vertical line bisects the angle Δθ and the vector Δv
• use trigonometry for one of the small triangles.
• The small-angle approximation requires that the angles are in radians.
• Two equations for Δθ lead to the magnitude of the centripetal acceleration
• Using v = r ω leads to equivalent expressions for the centripetal acceleration

• Where
• a = centripetal acceleration in m.s-2
• v = in m.s-1
• r = radius of circle in m
• ω = angular speed or velociy in rad.s-1
• It can be defined using the radius r and linear speed v:
• These equations can be combined to give another form of the centripetal acceleration equation:

Centripetal acceleration is always directed toward the centre of the circle, and is perpendicular to the object’s velocity

• Where:
• a = centripetal acceleration (m s-2)
• v = linear speed (m s-1)
• ⍵ = angular speed (rad s-1)
• r = radius of the orbit (m)v

Worked Example

A domestic washing machine has a spin cycle of 1200 rpm (revolutions per minute) and a diameter of 50cm. What is the centripetal acceleration experienced by the washing during the spin cycle?

Solution

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