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