Circular Motion

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.

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.

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.

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.

Examples of objects in circular motion

There are many real-life examples of circular motion. Consider the following:

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.