9.2 Uniform Circular Motion: Direction of the Acceleration

9.2 Uniform Circular Motion: Direction of the Acceleration


For a particle that’s
moving in a circle, we found that when it’s moving
at a constant rate of d theta dt— and let’s recall what
we meant by theta of t– and here’s our particle,
and we introduced our polar coordinates r hat and theta hat,
then we found that the velocity was r d theta dt theta hat. And so let’s assume that
this quantity is positive, in which case the velocity is
pointing in the positive theta direction. And that means that
everywhere in the circle, the velocity is
tangential to the circle, and the magnitude is a constant. So for this case of
uniform circular motion, we calculated that
the acceleration was equal to minus r d theta dt
quantity squared r hat, which means that at every point, the
acceleration vector is pointing towards the center. Now we can write that
acceleration vector as a component a of r– r hat–
where this component is given by r times d theta dt squared. It it’s always negative, because
when you square this quantity, it’s always a positive quantity. The minus sign, just
to remember– that means that the acceleration
is pointing inward. Now how can we think about that? Well, if we look at the velocity
vector, what’s happening here is the velocity is
not changing magnitude but changing direction. And if you compare
two points– and let’s just pick two arbitrary points. So let’s remove this
acceleration for a moment and consider two arbitrary
points– say, a time t1 and t2. So our velocity
vectors are tangent. The length of these
vectors are the same. And if we move them tail
to tail– [? Vt2– ?] and take the difference,
delta v, where delta v is equal to v
of t2 minus V of t1, then we can get an understanding
why the acceleration is pointing inward,
because recall that acceleration
by definition is a limit as delta t goes to 0. That means as this
point approaches that point of the change
in velocity over time. And so when we
look at this limit as we shrink down our time
interval between t2 and t1, then this vector will
point towards the center of the circle. And that’s why
the direction of a is in the minus r hat direction. Again, let’s just recall
that this is the case for we called uniform
circular motion, which is defined by the condition
that d theta dt is a constant.

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