A Level Physics
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Circular motion is simply the physics of things moving in circles. This happens all the time, for example you are moving in a circle around the earth as it rotates about its axis. If you are on the equator you would be moving through a larger circle per day than if you were in London, which is much further north. However, both of these places would have the same angular displacement at any point in the day. The angular displacement is measured in radians rather than degrees because it makes calculations much more straightforward. For example, the distance you move along the circle is given by the formula s = r θ. Where s is the arc length in metres, r is the radius of the circle in metres and θ is the angular displacement in radians.
There are 2π radians in a circle, which means that 2π = 360 degrees and using this fact we can convert between degrees and radians.
For example, an aeroplane performing a loop de loop of radius 60m turns through 135 degrees. To calculate the distance the plane has moved we have to first convert the degrees into radians. This is done by dividing 2π/360 to get the radians for 1 degree and multiplying it by 135, which gives ¾ π or 2.36 radians.
This can then be used to work out the distance the plane has travelled as being 141m.
When an object undergoes circular motion, the object will move at a constant speed but changing velocity. The velocity is changing because the direction is constantly changing and velocity is both speed and direction. This means that there is an acceleration, called the centripetal acceleration. The formula for this is a = v^2/r, where v is the velocity and r is the radius. Another relationship in circular motion is ω = v/r, where ω is the angular speed, which is a measrure of how many radians are being turned through per second and it is measured in rad/s. Another formula for ω is 2π/T where T is the time period of one complete rotation measured in seconds.
A centripetal force must be occuring here in order to produce this acceleration. The formula for centripetal force is F = mv^2/r. Where m is the mass of the body undergoing circular motion, v is the velocity and r is the radius of the circle.
For example, another aircraft display team perform a horizontal circle of radius 350m travelling at 100m/s. From this we can work out the angular speed as being 0.286 rad/s. If the mass of the pilot was 75kg, then we could work out the centripetal force as being 2140N.
It’s important to determine which forces are directed towards and away from the centre of the circle in order to determine the resultant force which is equal to the centripetal force. For example going back to the loop de loop performed by the aircraft, there are 2 points in the stunt that are of interest, at the top and at the bottom of the circle. At the top, the normal reaction force of the pilot from his seat and his weight are both acting downwards, towards the centre of the circle. At the bottom of the circle, the normal reaction force acts up towards the centre of the circle and the weight acts down. Given that the pilot had a mass of 80kg and the plane was travelling at 100m/s we can work out the normal reaction force on the pilot at the bottom of the circle to be 14100N and at the top the normal reaction force as 12500N which is a very strong force.
1. CGP AS & A2 Physics for OCR A, ISBN: 9781847624192