Topic 1  Circular Motion

Motion in a Circle

The rules of circular motion help us to describe:

• movement of a car going round a corner,

• a tethered model aeroplane;

• the planets in their orbits.

We have so far applied Newton’s Laws to motion in a straight line; the forces are pushing or pulling the object along in the direction of its travel.  What happens if we put a force perpendicular to the direction of motion?

The path is a parabola.

Now let us suppose that the force applied at 90^o to the object was always at 90^o.  This time the path would be circular:

In this case, the linear speed remains constant, but the direction is always changing.  This means that there is a change in velocity, hence acceleration.  Along with this come concepts that are important in circular motion, such as angular velocity.

Consider an object going round in a circle of radius r.

We know that anything going round in a circle has a constant speed but a changing velocity.  This is because the direction is constantly changing.  If we alter the radius, the linear speed also changes, even though we have not changed the rate of turning.  So we have to think up another quantity that we can use to describe the rate of turning.  We use angular velocity, physics code w (omega, a Greek letter long ‘ō’), how big an angle is turned in one second.  We could use degrees per second, but instead we use another kind of angular measurement, the radian.

·       One radian is the angle that subtends an arc whose length is the same as the radius. 

·        q rad = s/r. 

·        We can easily work out that 1 rad » 57 ^o. 

·        1 revolution is 2p radians.

·        For small angles in radians, q » sin q » tan q.  This is another reason why radians are so useful.  It does not work for large angles in radians, nor does it work for degrees. 

·        In dimensional analysis the radian is a dimensionless unit.  In some texts, you may see it missed out altogether, although here we will always include it.

·        w rad s-1 is the angular velocity. w = 2pf Þ linear speed v = wr = 2pfr m/s.

·        The direction of the velocity is tangential.

A common bear-trap is to fail to convert revolutions per minute to radians per second.  Divide the rpm by 60, then multiply the answer by 2p.

Centripetal Acceleration

We need to distinguish between an object spinning on its axis, and an object moving in a circular path.  We will consider the latter only.  The former situation is part of rotational dynamics.  You can read about the derivation of the relationship in any textbook, so we will not cover it here.

·        Acceleration is always towards the centre of the circle and is given by a = w²r.

• We can also express this in terms of frequency.  a = (2pf)²r = 4p²f²r.  A very useful dodge here is that p² is approximately 10.

·        We can write this as a = v²/r

·        Where there is acceleration, there is a force.   We call the force centripetal force (NOT centrifugal force!), which is described by the formula:

F = mv²

    r

·        The force acts towards the centre of the circle.

This is the circular motion version of Newton II, since F = ma and a = v²/r.

Suppose we were to whirl a stone around on a string, the forces would be governed by the relationship above.  However, if the string were to snap, the stone would fly off in a straight line at a tangent to the circle (NOT straight out from the centre).  The following are other examples that obey this relationship:

• Satellite orbiting the Earth (gravity provides the centripetal force)

• Vehicle going a bend (friction)

• Electron orbiting the nucleus (electrostatic attraction).

Problem solving strategy

1.      State clearly what object is being considered.

2.      Draw a free body sketch of the object.

3.      Mark the weight of the object

4.      Mark in any points where the object touches anything else and the forces involved.

5.      Decide which direction you will call positive.

6.      Apply Newton II (F = ma).

Worked example