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