Kinematics of circular motion (5.2.1)
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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.
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: