Circular Motion
General Notes
Circular motion is based around angle rather than displacement, as after one full rotation the
displacement resets to zero
Angles of a circle can be measured in radian. A radian is the angle subtended from the
centre of a circle intersecting an arc that is equal in length to the radius. 2 pi radians is equal
to 360 degrees.
Angular frequency is the number of rotations, or revolutions, per second, over multiple
rotations (Hz). An angular speed is how fast an object rotates, in radians per second, at any
point in the cycle. A period of rotation is the time taken for one complete cycle.
An alternate measurement of speed is the linear speed, visualised as if the object were
travelling in a straight line tangential from the circle, and measured in metres per second.
We only work with constant linear speeds/velocities, but the object will still accelerate as it is
constantly changing direction. This acceleration acts centripetal, meaning acting towards the
centre. While the angular speed is constant, the linear speed increases proportionally with
the distance from centre, or radius.
Centripetal acceleration implies a centripetal force (F = ma), pushing objects moving in a
circle inwards. A centripetal force can be any resultant force acting towards the centre. As
the force acts inwards and the linear velocity acts at 90 degrees from that, there can be no
work done by the force and some satellites can orbit without any energy loss.
Equations
𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ
θ in radians : 𝑟𝑎𝑑𝑖𝑢𝑠
Angular speed ω (full cycle): number of radians in one rotation times time period 2π𝑇
θ
Angular speed ω(generic): angle divided my time 𝑡
Linear speed V : angular speed times radius ω𝑟
Resultant force : rate of change of momentum ξ𝐹 = 𝑚𝑎
2
𝑣
Centripetal acceleration a: velocity squared divided by radius 𝑟
2
Centripetal acceleration a: radius times angular speed squared 𝑟ω
2
𝑚𝑣
Centripetal force: 𝑟
2
Centripetal force: 𝑚ω 𝑟
General Notes
Circular motion is based around angle rather than displacement, as after one full rotation the
displacement resets to zero
Angles of a circle can be measured in radian. A radian is the angle subtended from the
centre of a circle intersecting an arc that is equal in length to the radius. 2 pi radians is equal
to 360 degrees.
Angular frequency is the number of rotations, or revolutions, per second, over multiple
rotations (Hz). An angular speed is how fast an object rotates, in radians per second, at any
point in the cycle. A period of rotation is the time taken for one complete cycle.
An alternate measurement of speed is the linear speed, visualised as if the object were
travelling in a straight line tangential from the circle, and measured in metres per second.
We only work with constant linear speeds/velocities, but the object will still accelerate as it is
constantly changing direction. This acceleration acts centripetal, meaning acting towards the
centre. While the angular speed is constant, the linear speed increases proportionally with
the distance from centre, or radius.
Centripetal acceleration implies a centripetal force (F = ma), pushing objects moving in a
circle inwards. A centripetal force can be any resultant force acting towards the centre. As
the force acts inwards and the linear velocity acts at 90 degrees from that, there can be no
work done by the force and some satellites can orbit without any energy loss.
Equations
𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ
θ in radians : 𝑟𝑎𝑑𝑖𝑢𝑠
Angular speed ω (full cycle): number of radians in one rotation times time period 2π𝑇
θ
Angular speed ω(generic): angle divided my time 𝑡
Linear speed V : angular speed times radius ω𝑟
Resultant force : rate of change of momentum ξ𝐹 = 𝑚𝑎
2
𝑣
Centripetal acceleration a: velocity squared divided by radius 𝑟
2
Centripetal acceleration a: radius times angular speed squared 𝑟ω
2
𝑚𝑣
Centripetal force: 𝑟
2
Centripetal force: 𝑚ω 𝑟
