Circular motion is everywhere around us, from the spinning of the wheels on your bicycle to the fans in your living room. Studying the way that objects undergo motion in a circle thus has several real-world applications.
For example, have you ever wondered how a car speedometer calculates the speed of the car? What it is measuring is not the amount of distance travelled, but rather the number of rotations that a wheel is making in a fixed amount of time. Using concepts of angular velocity, and centripetal acceleration & force, we can then calculate how fast the wheel is going in a circle, and how fast the car is moving.
Before going into the concepts proper, there are a few terms that you need to understand.
- Radians (rad): This is typically the unit of choice for measuring angular displacement, with respect to motion in a circle. In a full circle, there are 2π rad. When converting degrees to radians, remember that π rad = 180°.
- Period of rotation (T): The period is the time taken for an object to complete a full rotation around a point of reference. It is measured in terms of seconds.
- Frequency (f): The frequency is the number of full rotations that an object completes in 1 second. It is measured in terms of Hertz (Hz). It is related to the period by the formula f = 1 / T.
When an object moves in a circular path, we measure its velocity in terms of angles, rather than the linear displacement. The angular velocity of an object, denoted by the symbol ω, is the rate of change of angle. It is a measurement of how fast an object’s angular position changes with time, and by extension, how fast an object rotates around a point.
Angular velocity is mathematically defined as the angle that an object has travelled in radians, divided by the amount of time that has passed in seconds (ω = θ/s). But given that an object travels 2π rad in one full rotation, the angular velocity can be calculated using the following formulas:
- Formula: Angular velocity = 2π / Period = 2π x Frequency
- Simplified formula: ω = 2π / T = 2πf
- SI Unit: Radians per second (rads-1)
To find the linear velocity, i.e. the displacement of the object per unit time, you can simply multiply the angular velocity by the radius of the circle. This relationship can be summarised by the following formula:
- Formula: Linear velocity = 2π x Radius / Time taken = Angular velocity x Radius
- Simplified formula: v = 2πr / t = ωr
- SI Unit: Metres per second (ms-1)
Centripetal acceleration & force
According to Newton’s first law of motion, an object will continue at the same speed in the same direction unless an unbalanced force acts upon it. Since an object undergoing motion in a circle is constantly changing direction and hence changing velocity, there must be some force acting on it.
This force is called centripetal force. Centripetal force is defined as the net force acting perpendicular to the velocity’s direction. This force is always directed towards the centre of the circle. It can also come from various types of forces, like gravitational force or friction. As such, if there are multiple forces acting on an object that is in circular motion, the resultant force should be used.
Furthermore, recall that the rate of change of an object’s velocity is also referred to as its acceleration. In this case, the acceleration provided by the centripetal force is called the centripetal acceleration.
The formulae of centripetal acceleration & force are shown below:
- Formula: Centripetal force = Mass x (Velocity)2 / Radius = Mass x (Angular velocity)2 x Radius
- Simplified formula: F = mv2 / r = mω2r
- SI Unit: Newtons (N)
- Formula: Centripetal acceleration = (Velocity)2 / Radius = (Angular velocity)2 x Radius
- Simplified formula: a = v2 / r = ω2r
- SI Unit: Metres per second squared (ms-2)
From planetary orbits to turning cars, circular motion drives the world as we know it. By learning the concepts of centripetal acceleration & force, you come one step closer to understanding how motion in a circle works, and how to apply it for practical purposes.
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