Circular Motion
The radian
● The SI unit for angle is the radian
● A radian is the angle subtended (connected) by a circular arc with a length equal
to the radius of the circle
○ The angle is around 57.3° for any circle
● The angle in radians subtended by any arc is defined
as follows:
𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ
○ Angle in radians = 𝑟𝑎𝑑𝑖𝑢𝑠
● For a complete circle, the arc length is equal to the
circumference of the circle
○ Angle in radians = = 2𝛑 radians
● Therefore, 360° is equal to 2𝛑 radians, or about 6.3 radians
● To convert from degrees into radians, divide the angle in degrees by
Angular velocity
● To describe the motion of moving objects fully, we
need to describe their linear motion and also how
objects rotate as they move - circular motion
● Any object moving in a circle or circular path moves
through an angle 𝛉 in a certain time t
● For example, all points on the London Eye rotate
through the same angle in the same period of time
● This gives a method of describing movement in
terms of angular motion
○ The wheel has an average angular velocity of
0.20°s-1 or 3.5 x 10-3 rad s-1
● The angular velocity 𝛚 of an object moving in a circular path is defined as the rate
of change of angle
○ Therefore, 𝛚 =
● In a time t equal to 1 period T, the object will move through an angle 𝛉 equal to 2𝛑
radians
○ Therefore, 𝛚 =
● The angular velocity is measured in radians per second
, 1
● As frequency f is the reciprocal of the period T, f = 𝑇 , we can also express angular
velocity 𝛚, as:
○ 𝛚 = 2𝛑f
● Angular velocity can be expressed in several units, including degrees per second
(° s-1), revolutions per second (rev s-1), and revolutions per minute (rpm)
○ You can just use rad s-1
Angular acceleration
● As a result of moving in a circular path, your direction continuously changes
● Therefore, the velocity is changing even if your speed is constant
● This change in velocity means that objects following a circular path must be
accelerating
● Any force that keeps a body moving with a uniform speed along a circular path is
the centripetal force
○ Newton’s 1st Law
○ Center-seeking force
● Centripetal force is always perpendicular to the velocity of the object
○ This force has no component in the direction of motion so no work is done
on the object
○ Therefore, the speed remains constant
● Centripetal force might be:
○ A gravitational attraction for a satellite in orbit
around a planet
○ Friction for a car going round a bend
○ Tension in the string when a yo-yo is swung
around in a vertical circle
● At any point on a circular path, the linear velocity is always at a tangent to the
circular path
○ 90° to the radius
● For an object moving in a circle at constant speed, we can calculate its speed using
the equation:
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑
○ Speed = 𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛
● In 1 complete rotation, the distance travelled is the circumference of the circle,
and the time is the period T
○ Therefore, v =
● Since angular velocity 𝛚 = , we can express the
speed as
○ v = r𝛚
● For objects with the same angular velocity, the linear
velocity at any instant is directly proportional to the
radius
○ Double the radius, double the linear velocity
The radian
● The SI unit for angle is the radian
● A radian is the angle subtended (connected) by a circular arc with a length equal
to the radius of the circle
○ The angle is around 57.3° for any circle
● The angle in radians subtended by any arc is defined
as follows:
𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ
○ Angle in radians = 𝑟𝑎𝑑𝑖𝑢𝑠
● For a complete circle, the arc length is equal to the
circumference of the circle
○ Angle in radians = = 2𝛑 radians
● Therefore, 360° is equal to 2𝛑 radians, or about 6.3 radians
● To convert from degrees into radians, divide the angle in degrees by
Angular velocity
● To describe the motion of moving objects fully, we
need to describe their linear motion and also how
objects rotate as they move - circular motion
● Any object moving in a circle or circular path moves
through an angle 𝛉 in a certain time t
● For example, all points on the London Eye rotate
through the same angle in the same period of time
● This gives a method of describing movement in
terms of angular motion
○ The wheel has an average angular velocity of
0.20°s-1 or 3.5 x 10-3 rad s-1
● The angular velocity 𝛚 of an object moving in a circular path is defined as the rate
of change of angle
○ Therefore, 𝛚 =
● In a time t equal to 1 period T, the object will move through an angle 𝛉 equal to 2𝛑
radians
○ Therefore, 𝛚 =
● The angular velocity is measured in radians per second
, 1
● As frequency f is the reciprocal of the period T, f = 𝑇 , we can also express angular
velocity 𝛚, as:
○ 𝛚 = 2𝛑f
● Angular velocity can be expressed in several units, including degrees per second
(° s-1), revolutions per second (rev s-1), and revolutions per minute (rpm)
○ You can just use rad s-1
Angular acceleration
● As a result of moving in a circular path, your direction continuously changes
● Therefore, the velocity is changing even if your speed is constant
● This change in velocity means that objects following a circular path must be
accelerating
● Any force that keeps a body moving with a uniform speed along a circular path is
the centripetal force
○ Newton’s 1st Law
○ Center-seeking force
● Centripetal force is always perpendicular to the velocity of the object
○ This force has no component in the direction of motion so no work is done
on the object
○ Therefore, the speed remains constant
● Centripetal force might be:
○ A gravitational attraction for a satellite in orbit
around a planet
○ Friction for a car going round a bend
○ Tension in the string when a yo-yo is swung
around in a vertical circle
● At any point on a circular path, the linear velocity is always at a tangent to the
circular path
○ 90° to the radius
● For an object moving in a circle at constant speed, we can calculate its speed using
the equation:
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑
○ Speed = 𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛
● In 1 complete rotation, the distance travelled is the circumference of the circle,
and the time is the period T
○ Therefore, v =
● Since angular velocity 𝛚 = , we can express the
speed as
○ v = r𝛚
● For objects with the same angular velocity, the linear
velocity at any instant is directly proportional to the
radius
○ Double the radius, double the linear velocity
