Circular motion (Topic 6B)
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For objects in circular motion, it is useful to work with angles measured in radians rather than degrees.
In the diagram below, the angle represented is equal to one radian when the arc length is the same length as the radius of the circle.
The angle in radians is found by:
Half the circumference is equal to the radius multiplied by Using the formula for the angle in radians gives:
Therefore, radians is equivalent to The whole circumference is equal to two times the radius multiplied by Again using the formula above gives:
Therefore, radians is equivalent to
To convert from radians to degrees, use:
To convert from degrees to radians, use:
These equations should be memorised. Expressing angles in radians as fractions of pi when possible is useful for maintaining precision. The diagrams below show some common conversions.
Calculators are normally set by default to measure angles in degrees when calculating trigonometric functions, but they will have an option to change from ‘degrees mode’ to ‘radians mode’.
The time period of an object in circular motion is the time it takes to make one complete rotation Time period is measured in seconds ().
The frequency of an object in circular motion is how many revolutions it completes in one second. Frequency is measured in hertz () or revolutions per second ().
The frequency and time period of an object in circular motion are inversely proportional:
As the time taken for a complete revolution decreases, the number of revolutions per second increases, and vice versa.
Question walkthrough
Finding time period after tripling frequency
Uses the inverse relationship f=1/T to find the new time period for a full rotation after the frequency of rotation triples.
Similar to linear speed, which is the distance in metres travelled per second, angular speed is the angle turned through per second by an object in circular motion.
Angular speed, can be found using the equation:
Where:
- is the angle turned through in radians (rad)
- is how many seconds (s) the object has turned for
- is the angular speed and is measured in radians per second
The diagram above shows two points, A and B, on a rotating bike wheel. Both points will have the same angular speed as they will take the same amount of time to complete one revolution.
However, points A and B will have different linear speeds as the point furthest from the centre has a greater distance to travel to return to its starting point after one revolution.
In physics, quantities can either be scalar or vector. A scalar quantity has just magnitude, whereas a vector has both magnitude and direction.
- Angular speed is an example of a scalar quantity. The amount of radians per second an object is turning through can be measured, but the direction is not relevant.
- Angular velocity is a measurement of both the number of radians per second an object is turning through and the direction this rotation is occurring in. The direction will often be defined as being positive or negative, similar to what would happen when measuring linear velocity.
When calculating angular speed or angular velocity, the same formulae can be applied to both.
For a complete revolution, an object turns through an angle of in one time period, Therefore, the equation:
becomes:
Frequency is related to the time period by:
Thus, angular speed or angular velocity is also given by the equation:
The linear speed of an object can be found by dividing the distance travelled by the time taken.
In the case of an object moving with circular motion, the speed can be found by dividing the circumference of the circle representing the object’s trajectory by the time taken to complete one full rotation (the period):
Where:
- is linear speed, measured in metres per second (),
- is the radius of the circle, measured in metres (),
- is the time period, measured in seconds ().
The angular speed of an object moving with circular motion is given by:
Where is the angular speed, measured in radians per second ().
Combining these two equations using substitution leads to the relationship between linear speed and angular speed :
The equation implies that the linear speed of an object moving with circular motion is proportional to the radius, if the angular speed is constant.
An example of this is two points at different distances from the hub on a bike wheel. They will both have the same angular speed because they will both take the same amount of time to complete one full revolution (the same time period).
However, their linear speeds will be different, as they have to travel different distances to complete one rotation.
In the diagram above, point A is further from the centre than point B and will need a greater linear speed in order to complete one full rotation in the same amount of time as point B.
Question walkthrough
Linear Speed of Objects on a Rotating Disc
Use the shared angular speed of two children on a merry-go-round to find one's linear speed from the other's.
As an object travels in a circular path, its direction is constantly changing. Therefore, its velocity must be constantly changing as velocity is a vector. If the velocity of an object is changing, the object is accelerating.
This acceleration is known as a centripetal acceleration and is directed towards the centre of the circle and perpendicular to the velocity.
The centripetal acceleration of an object moving with circular motion is given by:
Where:
- is the centripetal acceleration, measured in metres per second squared (),
- is the linear speed, measured in metres per second (),
- is the radius, measured in metres ().
Combining the equation above with the equation that links linear speed to angular speed produces an alternative way to find the centripetal acceleration :
combined with returns
Where is the angular speed, measured in radians per second ().
Question walkthrough
Comparing Centripetal Acceleration on a Turntable
Use the shared angular speed of two objects on a turntable to find one's centripetal acceleration from the other's.
Newton’s first law states that an object will continue travelling at a constant speed in a straight line unless acted upon by a net force.
An object following a circular path changes direction, and therefore, a force must act on it. This force is directed towards the centre of the circle, perpendicular to the object’s velocity.
A centripetal (centre-seeking) force keeps an object moving at constant speed in a circle, but causes the direction of the object’s motion to change.
While the speed of an object undergoing uniform circular motion remains constant, the constantly changing direction means the object has a changing velocity. It is important to note that speed is a scalar quantity and velocity is a vector.
A change in velocity means an object is accelerating. The centripetal force provides this acceleration. Newton’s second law states that force and acceleration are proportional to each other; you cannot have one without the other.
The centripetal force acts towards the centre of an object’s circular motion. Examples of centripetal forces are shown below:
If an object is moving in a circular path and the centripetal force is removed, then the object will fly off at a tangent.
An example of this is cutting the string attached to a ball being swung in a circle parallel to the ground.
Once the centripetal force is removed, Newton’s first law applies again – the object will travel in a straight line at a constant speed unless another resultant force acts upon it.
The centrifugal force is the name given to the fictitious outward pseudoforce you experience when you are turning.
For example, if you were sitting on the back seat of a car as it went round a corner you might feel like you are being pushed away from the centre of the turn and slide away from the centre of the turn.
However, this is just your inertia wanting you to continue on a straight path. When the car turns, the side of the bus pushes you towards the centre of the turn. This would be a real centripetal force caused by the reaction force between you and the side of the car.
Be careful not to confuse centripetal force with centrifugal force in questions. Exam questions will only require you to perform calculations involving centripetal force.
In circular motion, the centripetal force is directed towards the centre of the circle and is also perpendicular to the velocity of the object. Newton’s second law states that force is proportional to acceleration:
Therefore, the equations for centripetal acceleration can be converted into equations for centripetal force by multiplying the acceleration by the mass of the object moving with circular motion. Equations for centripetal acceleration are:
So the equations for centripetal force are:
Where:
- is the mass of the object moving with circular motion, measured in kilograms (),
- is the linear speed, measured in metres per second (),
- is the radius, measured in metres (), and
- is the angular speed, measured in radians per second ().
Centripetal force it is not a force itself. It is a measure of the pull of another force towards the centre of a circle.
In circular motion, the centripetal force is provided by a force acting towards the centre, such as friction, gravity, tension, or a combination of forces.
An example of a combination of forces constituting centripetal force is a ball spun in a vertical circle on a string. The centripetal force () is a combination of the object’s weight () and the tension in the string ().
The weight of the object acts downwards and is constant (the mass does not change).
The centripetal force is also constant if the radius and speed stay constant (true for both linear and angular speed).
The tension in the string is the force that can change in size and direction:
- When the ball is at the bottom, the tension supports the weight and provides the centripetal force.
- When the ball is at the top, its weight points to the centre, providing some centripetal force; the tension provides the rest.
When calculating tension or weight, remember that the resultant centripetal force is always directed towards the centre.
Circular motion can be investigated using a whirling bung experiment:
- A rubber bung is attached to a thin piece of string, which is passed through a hollow glass tube.
- A weight is suspended from the other end of the string.
- The student holds the glass tube and whirls the rubber bung above their head horizontally with circular motion.
The suspended weight creates tension in the string as the bung is whirled, providing the centripetal force to keep it moving in a circular path.
In the whirling bung experiment, the suspended weight causes a tension in the string as the bung is whirled around. This provides the centripetal force to keep it moving in a circular path.
The relationship between the force of the suspended weight and the centripetal force determines how the weight will move.
Where:
- is the mass of the bung and is the mass of the weight, measured in kilograms (),
- is the linear speed, measured in metres per second (),
- is the radius, measured in metres (), and
- = 9.81 is the gravitational acceleration.
You can also determine the linear speed of the bung by measuring the time taken for one complete rotation:
Or find the angular velocity of the bung:
Then use the equation that links linear speed to angular speed:
Question walkthrough
Frictional Force for Cars on Different Radii
Calculate the frictional force needed for two cars at the same speed to take corners of different radii.
Question walkthrough
String Tension in Vertical Circular Motion
Calculate the tension in a string at the top and bottom of a vertical circle for a ball undergoing circular motion.