Q2.6 A consolidation exercise (S)
Establish from first principles the acceleration of a particle moving at constant speed v in a circle of radius r.Hint
RevealSolution
RevealSolution
Consider two points on the circular path, P and Q say (above, left) It helps to take them to be not too far apart (since we are going to consider the limit in which they effectively coincide).
Now draw a triangle (right)
in which two sides are formed by vectors giving
(the velocity at P)
and
(the velocity at Q), tail-to-tail.
Then the third side of the triangle automatically gives the change in the
velocity as the particle moves from P to Q:
The average acceleration over this interval (of duration Δt, say) is then (recall Equation 1.4, written for the case of 1D motion)
By inspection we can see that the direction of
is that of
, and thus of
itself.
It is towards the centre of the circle (it happens to be downwards, because of
our choice of P and Q).
To find magnitude of
we use a little trigonometry on the
vector triangle, to give
