IB Physics SL

Revision Notes

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 diagram, downloadable AS & A Level Physics revision notes

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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