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Circular Motion

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:

Not all circular motion is created by humans though, looking into the natural world:

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 Basics

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.

 Equation

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:

 Equation

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.

 Acceleration Daigram 1

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:

 Acceleration Daigram 2

As δθ is such a small angle we can write:

 Equation

Rearranging:

 Equation

This is the rate of change of velocity, i.e. the acceleration. Using v= it can also be rearranged as follows:

 Equation

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:

 Equation

How fast (linear speed) would the rim of each craft need to be traveling at?

In the film the astronaut, Dave Bowman, is seen jogging around the inside of the Discovery One cabin. Assuming that he can jog at 12 km/h will it have any effect on the way he feels? Will the direction of his jogging have any effect?

Assume that Discovery One and the space station had diameters of 17 m and 600 m respectively. Take g as 9.8 m s-1.

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