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"Angular Speed"

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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.

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

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

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

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

The tangential and angular accelerations are related by the formula:

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:

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

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:

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:

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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 .

When does the body stop?

At time , the angular velocity is:

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:

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:

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