Angular Speed

# Circular Motion for A-Level

Let's imagine a small body on a circular plane. The body is initially at rest but then begins to move with uniform acceleration. The tangential speed of the body increases with the acceleration, which is directed in the direction of the tangent to the circle at every moment. We call it tangential acceleration.

Tangential acceleration can be directed in the direction of the body's motion. Then the speed of the body increases (Figure 1, left):

If it is directed in the opposite direction to the body motion of the body, its speed decreases (Figure 1, right):

In addition to tangential acceleration, we also have radial (centripetal) acceleration , similar to uniform circular motion. This is directed towards the centre of rotation and forces the body into a circular motion track.

Figure 1: Uniformly accelerated circular motion

In uniformly accelerated rotation, the body moves in a circle with tangential acceleration given as:

## Angular acceleration

In a uniformly accelerated circular motion, the angular velocity is no longer a constant. It varies with angular acceleration :

Figure 2: Tangential and angular acceleration

Let's use the equation from uniform circular motion, which connects the angular and tangential velocities :

 We note that since the tangential velocity is not constant, it follows that the angular velocity is also not constant. Therefore, let's consider the changes in the tangential and angular velocities: We note that: We cancel out on both sides of the equation:

The angular acceleration is the change in the angular velocity with time:

The tangential and angular accelerations are related by the formula:

## Angular displacement

From the material, Uniform circular motion, we recall the connection between the angle described by the radius as the body travels in a circle and the angular velocity :

Analogously, in the material, Uniform motion for A-Level, we have a connection between the distance , the speed and time :

What if the angular velocity changes uniformly with time?

We have a similar case in the material, Uniformly Accelerated Motion for A-Level. In the material, we discussed the situation in which the speed changed uniformly with time and we calculated the distance as the area under the graph of the speed against the time . We can use all the equations we learned in the material. We only change the notations:

### Angular velocity increases uniformly with time

Let's consider the case where the angular velocity increases uniformly from 0 to the final value .

Figure 3: Angular velocity increases uniformly with time

The angular velocity changes with time according to the equation:

It attains the final value in time :

The area under the graph above is therefore given as:

 We note that:

The angular velocity is initially zero. At time , it begins to increase uniformly with the angular acceleration . The angle described by the line between the body and the centre of the circle is therefore given as:

Example

The example is available to registered users free of charge.

### Angular velocity decreases uniformly with time

A body on the circumference of a circle has an initial angular velocity . At time , its angular velocity begins to decrease uniformly with an angular acceleration .

Figure 4: Angular velocity decreases uniformly with time

When does the body stop?

At time , the angular velocity is:

 Let's note that when the body stops, its final angular velocity is zero: We make the subject of the formula:

Let's also check the angle at which the body stops.

The angle described by the body at time while stopping is given by the area under the graph (see Figure 4):

The body stops at an angle which is given as:

 We note that:

The body rotates around the circumference of the circle at an angular velocity when it begins to stop at the time . The angle described in time while stopping is given as:

The body stops at the time:

The total angle described by the body during stopping is given as:

Example

The example is available to registered users free of charge.

material editor: OpenProf website