I have mentioned a few times in previous blogs that an object moving in a circle at a constant speed does so because of a force acting towards the centre. We call this force the centripetal force. The force is given by the equation
where is the mass of the object moving in the circle, is its speed, and is the radius of the circle. More correctly, remembering that force is a vector, it should be written
where is the magnitude (size) of the radial vector , and is the unit vector in the direction of .
But, from where does this formula come?
The acceleration of an object moving in a circle
The acceleration of any object is defined at the change in its velocity divided by the change in time. Both acceleration and velocity are vectors, so mathematically we can write this as
We are going to consider an object moving at a constant speed in a circle, as illustrated in the diagram below. In a time the object has moved through an angle , and its initial velocity has changed to , where the only change in the velocity is its direction. Remember, the velocity is tangential to the radius, so makes a right angle with the radius vector . The direction of the radius vector is, by definition, radially outwards.
In my blog about vectors I mentioned that, to combine vectors which have different directions, we need to split the vectors into components, add the components and then recombine the resultant components. The components need to be at right-angles to each other, and usually (but not always) we choose the x and y-directions when the vectors are in two dimensions.
To find the acceleration of our object in this example, we want to find the change or difference in the velocity, that is . We start by splitting the two vectors into their x and y-components.
Looking at our diagram, we can write
where is the speed, the size of the vectors .
The change in the velocity in the x-direction, which we will call is then just
Similarly, the change in the velocity in the y-direction, which we will call is given by
To correctly calculate the acceleration, we need to find the change in velocity with time as the time interval tends to zero. This means the angle tends towards zero also, and when is very small (and expressed in radians) we can write
So we can then write
The overall change in velocity, is then just the change in velocity in the x-direction, . The direction of the change in velocity is in the positive x-direction, which as is along the radial vector towards the centre of the circle (that is, in the direction).
However, we can do a substation for the angle . Remember, the arc-length, which we shall call is related to the angle via our definition of the radian (see this blog here), , so we can write
The acceleration . But the speed and time are related via , so we can write that . Substituting this into equation 1 above gives
and so the acceleration is given by
where we have added the minus sign to remind us that the change in velocity, and hence the acceleration, is directed towards the centre of the circle.
If you prefer to think of vectors pictorially, then the direction of can be seen from this diagram:
The centripetal force is found by remembering that (Newton’s 2nd law), so finally we can write that the centripetal force is