Video Transcript
The wheel of a car has a moment of
inertia of 2.25 kilograms meters squared about its axis of rotation. As the car is driven along a road,
the wheel has an angular momentum of 70.7 kilograms meter squared per second. With what angular speed is the
wheel rotating?
Let’s say that this is the wheel
we’re considering and that as the car drives along, the wheel rotates about an axis
through its center. This rotation happens at some
angular speed, we’ll call it 𝜔, and it’s this value we want to solve for. We can do this by taking into
account that we’re given the moment of inertia of the wheel as well as its angular
momentum. We can recall that an object’s
angular momentum 𝐿 is equal to its moment of inertia multiplied by its angular
speed. And so we can equivalently write
that angular speed is equal to angular momentum divided by moment of inertia.
In our scenario, the wheel’s
angular momentum is 70.7 kilograms meter squared per second and its moment of
inertia is 2.25 kilograms meters squared. Looking at the units in this
expression, notice that kilograms and meters squared cancel out from top and bottom
and that we’ll be left with one over seconds in our final answer. When we calculate our result, which
to three significant figures has a numerical value of 31.4, we can recall that 𝜔,
an angular speed, has units of radians per second. Now, the reason that radians didn’t
appear in our fraction is that the radian is a dimensionless quantity. Nonetheless, it’s a unit that we
know to include when we’re talking about angular motion, as in this case with
angular speed. So the angular speed with which
this car wheel is rotating is 31.4 radians per second.
