Circular motion is motion at a constant speed in a circular path around a central point and in a constant radius. An example of circular motion would be a rock tied to a string being swung in a circle. Given the conditions mentioned earlier, the time period during which the object completes one full orbit is also constant.
Centripetal force is the name given to the force that maintains circular motion. It acts along the radius and is directed towards the centre of the orbit.
How does velocity work in circular motion?
In circular motion, the velocity is always tangential to the direction of the motion. If the centripetal force was removed, the object would continue to move in the direction of its velocity, breaking the circular motion and moving forward in a straight line.
Figure 1. The centripetal force keeps the object in circular motion.
Angular speed
Angular speed is the scalar version of angular velocity. This quantity can be useful for calculating the speed of a rotating body by means of the angle through which it moves with respect to time.
Angular speed is angular displacement per unit time, where angular displacement is the number of degrees by which an object has moved.
Here, the degrees are measured in radians. One radian is equivalent to the angle created at the centre of a circle when an arc of equal length to its radius is created on its circumference.
We can divide the total number of degrees by which an object has moved by the time it has taken for this angular displacement to occur to determine the angular speed.
\[\omega = \frac{\text{angular displacement}}{\text{time taken}} = \frac{\theta}{t}\]
- ω is the angular speed measured in rad/s.
- t is the time taken, which is different from the time period.
- θ is the total angular displacement in the given time period (this can exceed 360º or 2π).
Alternatively, we can measure the time it takes the object to complete a full orbit to derive the angular speed.
\[\omega = \frac{2 \pi}{T} = 2 \pi \cdot f\]
- ω represents the angular speed, measured in rad/s.
- T is the time period, i.e., the time it takes for the object to complete a full orbit. It is measured in seconds.
- f represents the frequency of the movement and is measured in Hertz.
- 2π is 360º in radians, representing the maximum angular displacement and a full circle.
Linear speed
Linear speed differs from angular speed. Linear speed is simply the distance that an object in motion has travelled divided by time. To calculate linear speed v, we need to calculate the circumference of a circle with radius r and divide it by the time T it takes to complete one cycle (one orbit).
\[v = \frac{2 \pi r}{T}\]
How does acceleration work in circular motion?
The constant change of direction of an object means that the velocity is constantly changing (remember that velocity is a vector, while speed is a scalar), which means that acceleration, too, is constantly changing. However, we can derive an equation for the acceleration of an object in circular motion, beginning with the equation that describes circular motion where:
\[F = \frac{m \cdot v^2}{r}\]
- F represents the centripetal force in Newtons.
- m is the mass of the orbiting object in kilograms.
- v is the velocity of the object in metres over seconds.
- r is the radius of the object’s orbit in metres.
Equating this to \(F = m \cdot a\), we get:
\[\begin{align} \frac{m \cdot v^2}{r} = m \cdot a \\ \frac{v^2}{r} = 1 \end{align}\]
This shows that angular acceleration in metres over square seconds is related to velocity and radius.
Given this relationship, we can determine the acceleration of an object in a circular motion if we know values, such as the centripetal force, the radius of the orbit, and the mass and velocity of the object. Let’s take a look at the following example.
A ball connected to a bar by a chord follows a circular motion around the bar at constant velocity. The ball has a mass of 300g and moves at a velocity of 3.2m/s. Calculate the centripetal force if the chord has a length of 1.5m. Then calculate the acceleration of the ball around the bar.
First, calculate the centripetal force with m = 300g, v = 3.2m/s, and r = 1.5m. Convert 300g to kg and then calculate F.
\(F = \frac{0.3 kg \cdot (3.2 m/s)^2}{1.5m} = 2.048 N\)
The result is 2,048 Newtons. We know from Newton’s laws that force is equal to mass multiplied by acceleration:
\(F = 2.048 \space N = m \cdot a\)
As we know the mass m of the object, we can divide F by m to determine a.
\(a = \frac{2.048 N}{0.3 kg} = 6.83 m/s^2\)
What forces act on an object in circular motion?
The centripetal force is a key concept in circular motion. It is not to be confused with the similarly named centrifugal force. The centripetal force maintains the angular acceleration, while the centrifugal force is a pseudo force. It is felt only by the object describing the circular path. The centrifugal force acts in the opposite direction to the centripetal force, which is to say in the outwards direction.
Figure 2. The centrifugal force (in blue) is contrary to the centripetal force (in green).
Examples of objects in circular motion
There are many real-life examples of circular motion. Consider the following:
A car turning a corner
In this case, the frictional force is the centripetal force:
\[F_{friction} = \frac{m \cdot v^2}{r}\]
Vehicles on banked surfaces
This may include velodromes for track cycling or oval NASCAR style speedways. This is a more complex example of circular motion, which allows vehicles to travel at higher speeds compared to paths with zero gradients. The weight of the car provides the centripetal force. It must be resolved as the normal reaction force.
\[\text{Normal force} \cdot \cos(\theta) = m \cdot g\]
The normal force is the component of the weight vector that is perpendicular to the slope. The normal force is equal to the mass per square velocity divided by the radius of the curve.
\[\text{Normal force} \cdot \sin(\theta) = \frac{m \cdot v^2}{r}\]
The system for these forces can be seen below in figure 3.
Figure 3. Force components for a car in a curved slope (pink), where r is the radius of the circle into which the curve fits.
Here, r is the radius of the imaginary circle into which the curve fits (see figure 4).
Figure 4. Imaginary circle that can fit the path of the curve the vehicle has taken.
A swing
A swing works like a pendulum. Applying Newton’s Second Law F=ma, the centripetal force is the sum of the forces in the axis along the string that supports the swing. The forces, in this case, are the gravity force F=mg and the tension force ‘T’.
\[F = (m \cdot g) - T = m \cdot a\]
When the swing passes through its lowest point, both forces are acting against each other (see figure 5).
Figure 5. Forces on a swing, with the gravity force pointing downwards and the tension force pointing upwards.
Understanding this, we can simply equate both as follows:
\[F = \frac{m \cdot v^2}{r}\]
Circular Motion - Key takeaways
- The defining characteristics of circular motion are uniform speed, radius, and time period.
- The equation for circular motion is \(F = \frac{m \cdot v^2}{r}\).
- There is a difference between linear vectors and their angular counterparts. It’s important to use angular scalars or the magnitudes of angular vectors.
- Circular motion can be applied to real-life situations.
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