Question Video: Finding the Magnitude of the Moment of a Horizontal Force Acting at the Endpoint of a U-Shaped Rod Mathematics

In the figure, determine the moment about point 𝑜, given that the force 11 is measured in newtons.


Video Transcript

In the given figure, determine the moment about point 𝑜, given that the force 11 is measured in newtons.

Remember, a moment is the turning effect of a force. The magnitude of a moment can be calculated using the equation moment of the force is equal to force times distance. But we must remember that the distance is the perpendicular distance from the pivot to the line of action of the force. And so, we’re going to need to work out the distance from our pivot, zero, to the line of action of the force and the magnitude of the force that acts perpendicular to this line we’ve drawn.

To calculate the value of 𝑑, let’s draw out the right-angled triangle. We can see that this horizontal measure in this triangle must be four centimeters. And then, since this side is eight centimeters and this side is three centimeters, the left-hand side in our triangle must be eight minus three, which is five centimeters. So, let’s begin by using the Pythagorean theorem to find the value of 𝑑.

It says that the sum of the squares of the two shorter sides in our right-angled triangle must be equal to the square of the hypotenuse. The hypotenuse is the longest side. It’s the side opposite the right angle. So, here is the side we’ve labeled 𝑑 centimeters. And so, we see that 𝑑 squared must be equal to five squared plus four squared or 𝑑 squared must be equal to 41. By finding the square root of both sides and remembering that we’re only taking the positive square root of 41 since it’s a length, we see 𝑑 is equal to the square of 41.

And so, we found the distance. But what about the force? Well, we’re going to make another right-angled triangle. This time, though, we add an included angle 𝜃. That angle 𝜃 also appears in our earlier right-angled triangle. And that’s really useful because we know that the hypotenuse in this triangle is 11 newtons. We want to work out the component of the force that’s acting perpendicular to the distance we calculated. In our triangle, let’s label that 𝐹 or 𝐹 newtons. Since the hypotenuse of our triangle is 11 newtons and we’re trying to find the adjacent, we’ll use the cosine ratio to link these. That is cos 𝜃 is adjacent over hypotenuse, or cos 𝜃 is 𝐹 over 11.

Now, if we go back to the earlier triangle, we see that the hypotenuse is root 41. And we calculated the length of the adjacent to be equal to five centimeters. In this case, cos 𝜃 is five over root 41. Remember, cosine is a ratio, so we know that this ratio holds for our other right-angled triangle, meaning five over root 41 must be equal to 𝐹 over 11. And then, we’ll solve this equation for 𝐹 by multiplying both sides by 11. Of course, that’s the same as multiplying by 11 over one. So, we find that 𝐹 is equal to 55 over the square root of 41.

Going back to our earlier equation for the moment of a force, we know we now need to multiply our value of 𝐹 by our value of 𝑑. That’s 55 over root 41 times root 41, meaning our root 41s cancel. This leaves us with 55. Since our force is in newtons and our distance is in centimeters, we see that the moment about point 𝑜 is 55 newton centimeters. Note that since the direction given was counterclockwise and that’s the direction we’ve worked in, our value is positive. It’s 55 newtons centimeters.

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