Video Transcript
𝐴𝐵𝐶𝐷𝐸 is a regular pentagon whose side length is 16 centimeters. Five forces, each of magnitude 11 newtons, are acting at 𝐴𝐵, 𝐵𝐶, 𝐶𝐷, 𝐷𝐸, and 𝐸𝐴, respectively. If the system is equivalent to a couple, find the magnitude of its moment, considering the positive direction is 𝐴𝐵𝐶𝐷𝐸, rounded to two decimal places.
All right, let’s say that this is a regular pentagon. The fact that this is a regular pentagon means that all its interior angles are equal and all its side lengths are the same. If we label the vertices of our pentagon like this, then we’re told that along all of these side lengths, acting in what we could call the clockwise direction, are forces of 11 newtons. This system of forces, we’re told, is equivalent to a couple. That means the net force on this pentagon is zero. But the net moment is not. Here, we want to solve for the magnitude of that moment. And we’ll consider a moment positive if it tends to create a rotation in the direction of 𝐴, 𝐵, 𝐶, 𝐷, and 𝐸. That is a clockwise rotation as we’ve drawn it.
This overall moment will be about a point at the geometric center of our pentagon. And to solve for this moment, the critical question is, what is this distance here? That is, what is the perpendicular distance between the side length of our pentagon and the center point. We’ll call it 𝑑. The reason this question is so important is because 𝑑 is the same for all five sides. And we can recall that to calculate the moment created by a given force, we multiply that force by the perpendicular distance between where the force is applied and an axis of rotation. In our scenario, we know all the forces; those are forces of 11 newtons. But we don’t yet know 𝑑.
To start solving for it, let’s consider that if we started at this vertical dashed line and we moved all the way around the center point, we would have gone through an angular displacement of 360 degrees, knowing that it’s helpful because it means if we drew dashed lines from our center point to all the vertices of our pentagon, then the angular extent of any one of those five sections would be 360 degrees divided by five. And then going further, if we consider just one-half of this angle, that is, the angle that’s also part of this right triangle in orange, then that would be 360 over 10. In other words, this angle right in here is 360 over 10 or 36 degrees. So what we have then is a right triangle where one of the other interior angles is 36 degrees and the length of the side adjacent to it is the one we want to solve for.
We know something else, though, about this triangle. We can also solve for the length of this opposite side here. Let’s recall we’re told that each of the side lengths of this pentagon is 16 centimeters. So that means, in our original diagram, that this length here is 16 centimeters, which means that half of that is eight. And therefore, since the tangent of this angle is equal to the ratio of this side length to this side length, we can write that the tangent of 36 degrees is equal to eight over 𝑑 or 𝑑 equals eight over the tangent of 36 degrees.
Now that we’ve solved for 𝑑 which we’ll use for all of the moments created by all five of the forces acting on our pentagon, we can move right ahead with calculating the overall moment. To start doing that, let’s clear some space and compute the overall moment, we’ll call it 𝑀, around the center of our pentagon. The simplifying thing about this calculation is that the contribution to that overall moment from all five of our forces is identical. That’s because all these forces have the same magnitude, act in the same clockwise direction as we’ve called it, and are also the same perpendicular distance 𝑑 away from the center.
In calculating 𝑀, we can take 11, multiply it by 𝑑, and then multiply that by five. Entering this expression on our calculator, when we round the result to two decimal places, we get 605.61. This result is positive, according to our sign convention. And the units involved are newtons centimeters. To this level of precision then, the magnitude of the movement acting on our pentagon is 605.61 newton centimeters.
