Circular Motion, why’s it so important? Wherever we look in the man made world we tend to find circles in motion.
The most obvious of example of circular motion is the wheel but there are many others all around us:
Fairground rides carry their passengers through circular paths.
Internal combustion and electrical motors have rotating parts.
Spinning turbine blades are used in aircraft jet engines and in electrical power stations.
Centrifuges in hospitals spin blood samples in circles to separate red blood cells from plasma.
The computer that you are probably reading this text on will have a rapidly rotating hard disc.
Not all circular motion is created by humans though, looking into the natural world:
Bacteria have evolved a circular motor to power the flagellum that many species use for motion.
Charged particles move in circular paths in the presence of magnetic fields, this leads to the trapping of electrons and protons in the Van Allen Belts around the Earth.
The elliptical orbits of the planets and of stars within galaxies are a more general form of circular motion.
Circular motion really is all around us and so we have to study it if we wish to understand much of the natural as well as the designed world.
The basic rules of mechanics are set down in Newton’s three laws of motion. The first law states:
"Any object will continue in a state of rest or of uniform motion unless acted upon by an external force."
In other words the natural state of any object is to move in a straight line and at a constant speed (a stationary object is just the extreme case where the speed happens to be zero). It follows that if an object is moving in a circular path then it must have an external force acting on it!
We can work out the size and the direction of that force as follows:
Consider an object moving in a circle of radius r and taking a time T to complete one revolution (2π radians). In one complete revolution the object travels around the circumference once, a distance of 2πr. We can use this to define the speed, v and the ‘angular velocity’, ω.
Where f is the frequency of the motion (rotations per second)
It follows that in a period of time Δt the path sweeps out an angle Δθ:
The diagram below shows how the velocity (a vector) changes in the time Δt. The change in velocity, Δv is the difference between the velocity at time zero and at time Δt.
If we want to know about the instantaneous acceleration then we have to look at what happens when Δt becomes very small. This case is shown by the following diagram:
As δθ is such a small angle we can write:
Rearranging:
This is the rate of change of velocity, i.e. the acceleration. Using v=rω it can also be rearranged as follows:
It should also be clear that the acceleration is at right angles to the velocity and directed towards the centre of the circle. This is the ‘centripetal’ force which means ‘centre aiming’. It should not be confused with ‘centrifugal’ (or centre fleeing) force that does not really exist… so more of that later!
There is another intuitive reason to expect the acceleration to be at right angles to the motion; if the acceleration were at any general angle to the motion there would always be a component of acceleration in the direction of the motion and so we would expect the speed as well as the velocity to change!
We can combine the equation for centripetal acceleration with the equation relating force mass and acceleration to give:
How fast (linear speed) would the rim of each craft need to be traveling at?
Is there a minimum strength for the thread in order to achieve a vertical circle?