A car accelerates for 20 seconds,
during which time a torque of 32 newton meters is applied to each of the wheels. What angular impulse was given to
each wheel over the time it accelerates for?
Okay, so, in this exercise, we’re
considering a car that’s accelerating. But our focus is really on the
wheels of the car, which each have a torque of 32 newton meters applied to them. We can just represent this torque
as 𝜏 sub w. And we know it’s applied for a time
of 20 seconds. And we’ll call that time 𝑡. We want to know, then, what is the
angular impulse given to each wheel over this time of 20 seconds.
To start figuring this out, we can
recall that the angular impulse, 𝐻, delivered to an object is equal to that
object’s change in angular momentum. And that, by what we could call a
rotational form of Newton’s second law of motion, is equal to the torque experienced
by the object multiplied by the time over which that torque acts. In the case of our car wheels,
then, the angular impulse that each one experiences will equal the torque, 𝜏 sub w,
multiplied by the time, 𝑡. So then, that’s 32 newton meters
multiplied by 20 seconds.
Before we calculate this product
though, let’s recall that a newton is equal to a kilogram meter per second
squared. If we substitute that in for
newtons and then consider just the units in this expression for a moment, we see
that one factor of seconds cancels from numerator and denominator. And if we collect all our units off
to the right, we get kilograms meter squared per second. So, 32 times 20 kilograms meter
squared per second is the angular impulse delivered to each wheel. And this is 640 kilograms meter
squared per second. This is the angular impulse given
to each wheel over the time of the car’s acceleration.