### Video Transcript

A felled tree trunk rolls down a slope in a time of 7.2 seconds. The trunk is initially at rest at the top of the slope and has an angular velocity of 12 radians per second at the base of the slope. How many complete rotations does the trunk make as it rolls down the slope?

We can begin by drawing a diagram. Here is the trunk on the slope. We know that it starts at rest at the top of the slope. And once it rolls down and reaches the bottom, it’s rotating at an angular velocity of 12 radians per second. Let’s call this final angular velocity 𝜔 subscript f. And we can call the initial angular velocity 𝜔 subscript i, which, we know, equals zero radians per second because the tree trunk was initially at rest. Then, as the constant force of gravity caused the trunk to roll down the slope, its angular velocity increased at a constant rate, meaning it had a constant angular acceleration.

Under this condition, we know that the equations of angular motion apply. These equations are written in terms of angular displacement 𝜃, angular velocity 𝜔, angular acceleration 𝛼, and time 𝑡. Notice though that in the equations, angular velocity is always given as either 𝜔 sub i or 𝜔 sub f, not just plain 𝜔. And although all of these equations would correctly model this situation, it’s our job to pick which equation will best help us solve this question in particular.

Thus far, we’ve written down values for the trunk’s initial and final angular velocities. And because we’ve been told that the trunk rolls down the slope in 7.2 seconds, we also have a value for time 𝑡. Now, this whole question is asking us to find the number of complete rotations the trunk makes as it’s rolling. So we want to know its angular displacement 𝜃. Also, notice that even though we know the trunk’s angular acceleration is constant, we don’t actually know what it is, nor will we need its value in order to answer this question. We can use this equation at the bottom to solve for angular displacement in terms of only values that we know. So let’s clear some room on screen and get started.

Substituting in our values for 𝜔 sub f, 𝜔 sub i, and 𝑡, the formula tells us that 𝜃 equals one-half times 12 radians per second plus zero radians per second all times 7.2 seconds. Before we calculate though, it’s always a good idea to check out the units. We have seconds, the SI unit of time, so that’s good. And we also have radians. Recall that even though there are several different angular units that we can use, such as degrees or revolutions, it is best to use radians in calculations like this. So we’re all set.

Canceling out the zero term and combining the factors of one-half and 12, we have six radians per second times 7.2 seconds. Notice that units of seconds cancel out of this expression entirely, leaving the result expressed in just radians. Now, multiplying six by 7.2, we have that 𝜃 equals 43.2 radians.

Remember though, we were asked about how many rotations the trunk makes. So we need to convert from radians to rotations or revolutions. Recall that one revolution or one rotation refers to a full turn around a circle, which measures two 𝜋 radians. So we can write this conversion factor, which itself is just equal to one, to cancel out radians, leaving only revolutions and the factor of one over two 𝜋 to adjust the magnitude accordingly. Now, calculating, we have that 𝜃 equals 6.88 and so on revolutions.

But this isn’t our final answer, since we were asked how many complete rotations the trunk makes. It didn’t quite make that seventh rotation, so our answer is six. As the trunk rolls down the slope, it makes six complete rotations.