A drill bit is initially at
rest. When the drill is activated, the
drill bit rotates 47.5 times per second. The drill bit reaches this speed in
a time of 175 milliseconds. What is the angular acceleration of
the drill bit?
Okay, let’s say that this is an
end-on view of our drill bit. So we can say that the bit is
pointed out of the screen at us. Initially, the bit is at rest. But then, when the drill is turned
on, the bit starts to rotate, 47 and a half times every second. Knowing that the drill bit reaches
this rotation speed from rest over a time of 175 milliseconds, we want to know what
is the angular acceleration of the bit. As we get started, we can recall
that angular acceleration, 𝛼, is equal to a change in angular speed, Δ𝜔, divided
by a change in time, Δ𝑡.
One important thing to realize
about this equation is that this change in angular speed, Δ𝜔, assumes that this
angular speed is expressed in units of radians per second. That way, the angular acceleration
we calculate is in units of radians per second per second or radians per second
squared. Now we bring this up because in our
problem statement, we’re told that our drill bit rotates 47 and a half times every
second. That is, it goes through one
complete revolution 47.5 times every second. But that does not mean that our
angular speed is 47.5 radians per second. This is because one complete
rotation, one time around the circle we could say — that’s one revolution — is
equal to two 𝜋 radians.
So that means the real angular
speed in units of radians per second is 47.5 times two 𝜋. It’s this value that we’ll use in
our equation for angular acceleration. That acceleration is equal to the
change in angular speed, Δ𝜔. But since our drill bit started out
at rest, that means that this value here is equal to that change divided by the time
over which that change occurs. And that’s 175 milliseconds. Before we calculate this fraction,
we want to convert this time from milliseconds into units of seconds. That’s so that we can have a common
unit of time in both numerator and denominator.
We can recall that one millisecond
is equal to one thousandth of a second. And therefore, 175 milliseconds is
equal to 0.175 seconds. Now we’re ready to calculate
𝛼. And we’ll indeed get units of
radians per second squared when we do. Rounding our answer to three
significant figures, we get a result of 1710 radians per second squared. That’s the angular acceleration of
the drill bit.