Circular Motion
As you know according to Newton's First law a
body remains at rest or travels in a straight line unless an unbalanced force acts on it. So to make
a body travel in a curve a force must act to pull the body away from its straight line. This curved
path could be a parabola, as in the case of a projectile moving in a gravitational field, a hyperbola,
as for an alpha particle near a nucleus, an ellipse, planets in motion round the Sun or a circle, as
in a stone whirled round your head on a string.
In a circle the radius is always at right
angles to the motion and since the string is the radius of the circle the force that acts on the stone
must be along the string - at right angles to the path of the stone.
Examples of "everyday"
circular motion are given below together with the force that makes the paths circular.
Centripetal force
The force that pulls an object out of a straight line path into a circular orbit is called the
centripetal force. (The word means centre seeking). It is the force on the orbiting object.
Thinking about the stone being whirled round on the end of a string. From Newton's Third Law states we know that if a force acts on the stone then there must be an equal and opposite force acting on another body - this will be a force on the hand holding the string and this force is known as the
centrifugal force (the word means fleeing from the centre). If you like to put it this way the centrifugal force is the reaction of the centripetal force on the pivot. As far as the orbingt object (in this case the stone) is concerned the centrifugal force is a fictitious force – it does
NOT act on the stone.
If you remove the centripetal force, in the example of the stone and string by cutting the
string, the stone will move off along a straight line in the direction it was moving at the instant the
string was cut along the tangent to the circle (ignoring gravity for a moment) and not along a
radius.
Angular displacement is expressed in
radians. A radian is defined as the angle subtended at the centre of
the circle by an arc of length equal to the radius of that circle. In one complete circle there are 2
p
radians and so one radian equals just over 57
o.
Angular
velocityWhen an object is travelling in a circle it has an instantaneous linear
velocity but it also has an angular velocity (
w). This is defined as the
rate of change of angle with time and is usually expressed in
radians per
second.
For a rotating body that is rigid such as a CD or a wheel the angular
velocity is the same at all points on the body while the linear gets greater as the distance from the
centre gets larger. However if the body is not rigid like soup in a bowl that is being spun round the
angular velocity changes. A fun experiment to demonstrate the forces in rotating objects and to
simulate what happens to a spinning car tyre is to rotate a jelly on a turntable and see when it
begins to break apart.
If we take T to be the time for one complete rotation - called the
period of the motion then
Since T is the period (the time to make one
complete rotation) the number of rotations per second (n) is 1/T and is the frequency of the
motion.
Linear velocity and angular velocity are therefore connected by the formula:
When an object
moves in a circle the linear velocity must be constantly changing as the direction of motion is
changing - there must therefore be an acceleration - the centripetal acceleration, and therefore a
force - the centripetal force. Both the centripetal acceleration and the centripetal force are directed
towards the centre of the circle.
The rotating jelly
A very good way to simulate the break up
of car tyres is to make a jelly and put it on a rotating table. As the rate of rotation of the table is
increased so the jelly spreads out and flattens. The forces within the jelly keep it together but a
point will be reached where these forces are no longer great enough, it can no longer stay in one
piece and flies apart, just like car tyres on a motorway.
