A truck with a 44-inch diameter wheel
is traveling at 50 miles per hour.
A, find the angular velocity in radians per minute,
and B, how many revolutions per minute do the wheels make?
The first thing to recognize here
is if the diameter is 44 inches,
then the radius, r, is 22 inches,
which is half of the diameter.
And now looking at the formulas below,
the linear velocity, v,
equals the radius, r, times the angular velocity, omega.
And because we're looking for the angular velocity,
we divide both sides by r to solve for omega,
which gives us omega, the angular velocity,
is equal to v, the linear velocity, divided by the radius.
Before we use this formula to answer part A, though,
let's better understand why the linear velocity, v,
is equal to the radius, r, times the angular velocity.
Remember, when the central angle of a circle
has a measure of one radian,
then the intercepted arc,
this arc here, has a length of one radius,
which means when the wheel rotates one radian,
the linear distance traveled is equal to this arc length,
which is equal to one radius,
which is why to find the linear velocity,
we find the product of the radius and the angular velocity.
As an example, if the angular velocity
is eight radians per minute,
again each radian of rotation the wheel would travel
a linear distance of one radius,
and therefore if the radius is 10 inches,
the linear velocity is equal to 10 times eight,
or 80 inches per minute.
So now going back to our example,
we are given the linear velocity of 50 miles per hour,
but before we find the angular velocity,
we will have to convert 50 miles per hour
into inches per minute.
So beginning with the given velocity of 50 miles
per one hour,
let's begin by converting miles to inches.
And then we'll convert hours to minutes.
We'll perform the conversion using unit fractions.
And because there are 5,280 feet in one mile,
and we want miles to simplify out, we'll begin
by multiplying by the unit fraction of 5,280 feet
divided by one mile.
The units of miles simplify out
and now to convert feet to inches,
because there are 12 inches in one foot,
we want feet to simplify out,
we multiply by the unit fraction of 12 inches divided by
one foot.
And again notice how feet simplifies out.
And now to convert hours to minutes,
there are 60 minutes in one hour,
we want hours to simplify out
and therefore we multiply by the unit fraction
of one hour divided by 60 minutes.
And the hours simplify out.
And now multiplying, in the numerator we have 50
times 5,280
times 12 inches
divided by, in the denominator, we have 60 minutes.
And now simplifying
50 times 5,280 times 12
divided by 60 is equal to 52,800.
And we now know the linear velocity, v,
is equal to 52,800 and the units are now inches per minute.
So now using our formula we know the angular velocity
is equal to the linear velocity, v, divided by the radius,
and this quotient gives us 52,800
divided by the radius of 22 inches
and this quotient is equal to 2,400.
And therefore the angular velocity
is 2,400 radians per minute.
And now for part B, we need to determine
how many revolutions per minute the wheels make.
Well the revolutions per minute is equal
to the angular velocity divided by two pi, because remember,
two pi radians is equal to one rotation or one revolution.
So the revolutions per minute are equal
to the angular velocity, omega, divided by two pi radians.
And we know the angular velocity
is equal to 2,400 radians per minute
and therefore the revolutions per minute are equal to 2,400
divided by two pi,
which is equal to 1,200 divided by pi,
and of course units here
are going to be revolutions per minute,
so this would be the exact revolutions per minute.
Let's also get a decimal approximation.
We have 1,200
divided by pi
which gives us approximately 381.9719
revolutions per minute.
So we have the exact RPMs
and the approximate RPMs.
I hope you found this helpful.