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Centripetal Acceleration
Acceleration that acts towards the center of a circular path.
Centripetal Force
The resultant force towards the centre of the circle required to keep a body in uniform circular motion.
Constant Magnitude
Refers to a centripetal force that always acts perpendicular to the direction of motion.
Friction
Centripetal force acting on a car travelling around a roundabout.
Tension
Centripetal force acting on a ball attached to a rope moving in a circle.
Gravitational Attraction
Centripetal force acting on the Earth orbiting the Sun.
Linear Speed
Sometimes referred to as the 'tangential' speed.
Tangent
A straight line which touches a circle or curve at exactly one point.
Angular Speed (⍵)
The magnitude (size) component of angular velocity that stays constant.
Angular Velocity
The direction component of angular speed that is constantly changing.
Velocity
A vector quantity that includes both speed and direction.
Acceleration
Defined by a change in direction, not just a change in magnitude.
Uniform Circular Motion
Motion where an object travels in a circular path at a constant speed.
Perpendicular
Describes the direction of centripetal force and acceleration towards the centre of the circle.
Changing Linear Velocity
Occurs because an object in uniform circular motion is continuously changing direction.
Examiner Tip
Acceleration can occur without an increase in speed when an object travels in a circle.
Key Idea
Angular speed does not change with radius, but linear speed does.
Centripetal acceleration
The acceleration of an object towards the centre of a circle when an object is in motion (rotating) around a circle at a constant speed.
Centripetal acceleration formula
a = v² / r, where a = centripetal acceleration (m s⁻²), v = linear speed (m s⁻¹), r = radius of the circular orbit (m).
Angular speed relation
v = rω, where ω = angular speed (rad s⁻¹).
Centripetal acceleration in terms of angular speed
a = rω².
Direction of centripetal acceleration
Centripetal acceleration is always directed toward the centre of the circle and is perpendicular to the object's velocity.
Centripetal force formula (linear velocity)
F = mv² / r, where F = magnitude of the centripetal force (N), m = mass of object (kg), v = velocity (m s⁻¹), r = radius of orbit (m).
Centripetal force formula (angular velocity)
F = mvω, where ω = angular velocity (rad s⁻¹).
Centripetal force simplification
F = mrω².
Direction of centripetal force
Centripetal force is always perpendicular to the direction of travel.
Minimum speed for a bucket
The minimum speed the bucket must have at the top of the circle so no water spills out is calculated as v = √(gr).
Calculation of minimum speed
For a bucket of mass 8.0 kg and length 0.5 m, v = √(9.81 × 0.5) = 2.21 m s⁻¹.
Worked example of centripetal acceleration
If a ball tied to a string is rotating with a radius of 1.5 m and an angular speed of 3.5 rad s⁻¹, its centripetal acceleration will be 147 m s⁻² when the radius is doubled and angular speed is doubled.
Centripetal acceleration increase factor
The centripetal acceleration will be 8× bigger when the radius is doubled and angular speed is doubled.
Weight of the bucket
The weight of the bucket is calculated as mg, which is equal to the centripetal force at the top of the circle.
Centripetal force equivalence
The two equations for centripetal force are equivalent because of the equation linking linear and angular velocity.
Units of centripetal acceleration
Centripetal acceleration is measured in meters per second squared (m s⁻²).
Units of linear speed
Linear speed is measured in meters per second (m s⁻¹).
Units of angular speed
Angular speed is measured in radians per second (rad s⁻¹).
Units of radius
Radius of the circular orbit is measured in meters (m).
Units of mass
Mass of the object is measured in kilograms (kg).
Units of centripetal force
Magnitude of the centripetal force is measured in newtons (N).
