Moving searchlight beam The figure shows a bo...

Question

Moving searchlight beam The figure shows a boat 1 km offshore, sweeping the shore with a searchlight. The light turns at a constant rate, dθ/dt = -0.6 rad/sec.

b. How many revolutions per minute is 0.6 rad/sec?

Accepted Answer

Welcome back, everyone. A ceiling fan rotates at a constant speed DC divided by the T equals 0.8 radiance per second. How many revolutions per minute does the fan make? We're given 4 answer choices A says 24 divided by pi, B 12 divided by pi, C 6 divided by pi, and D 3 divided by pi. So let's understand the context of the problem. We're given a speed given in a radiance per second, and we essentially want to calculate how many revolutions per minute the fan makes. So essentially we just want to convert that speed into revolutions per minute. So we're going from radiance per second into revolutions per minute. And we can perform this using a basic conversion. We're going to begin with 0.8 radiance per second, and we can write it as 0.8 radiance per 1 2nd, right? We're going to apply the technique called dimensional analysis, and as we understand, first of all, we can convert seconds to minutes. To do that, we want to include a conversion factor in which seconds must be in the numerator to cancel them out after multiplication, and under the denominator, we want to include the desired time unit which is minutes. We will be able to notice that those seconds can be canceled out and we will end up with radiance per minute first. And then our next step is to convert those radiants into revolutions because we have radiance at the top of the fraction. We're going to include radiance at the bottom of the next fraction, so those radiances can also be canceled out, and we wish to get revolutions, so that goes into the numerator of the conversion factor. Our next step is to include appropriate numbers for each conversion factor. Let's recall that 1 minute is equal to 60 seconds. So for the first conversion factor, we are going to include 1. In front of minutes, and then for seconds we're adding 60. So it shows us that 1 minute is equivalent to 60 seconds, right? And now for the relationship between the revolutions and radiance, we have to recall that 2 pi radiance. Which is 360 degrees, corresponds to one revolution. So one revolution is equivalent to 2 pi radiance, and now what we have to do is simply multiply these, which gives us 0.8. Multiplied by 60, multiplied by 1 divided by 1 multiplied by 1, multiplied by 2 pi and our units are revolutions per minute as we can see. Overall, we get 48 in the numerator divided by 2, that's 24, and in the denominator we're left with pi because we have already performed that division by 2, so we get 24 divided by pi revolutions per minute. Which corresponds to the answer choice A. Thank you for watching.

Moving searchlight beam The figure shows a boat 1 km offshore, sweeping the shore with a searchlight. The light turns at a constant rate, dθ/dt = -0.6 rad/sec.

b. How many revolutions per minute is 0.6 rad/sec?

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Key Concepts

Radians and Revolutions

Angular Velocity

Conversion of Units

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