Question Video: Studying the Equilibrium of a Rod under the Action of Four Parallel Forces | Nagwa Question Video: Studying the Equilibrium of a Rod under the Action of Four Parallel Forces | Nagwa

Question Video: Studying the Equilibrium of a Rod under the Action of Four Parallel Forces Mathematics • Third Year of Secondary School

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In the figure, forces having magnitudes of 61, 43, 100, and 𝐹 newtons are acting on the light rod, and the rod is in equilibrium horizontally. Determine the length of line segment 𝐷𝐴 and the magnitude of 𝐹.

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Video Transcript

In the figure, forces having magnitudes of 61, 43, 100, and 𝐹 newtons are acting on the light rod, and the rod is in equilibrium horizontally. Determine the length of the line segment 𝐷𝐴 and the magnitude of 𝐹.

The key to answering this question is to spot that the rod is in equilibrium. So what does that mean? Well, it means two things. Firstly, the sum of all the forces acting on the rod is equal to zero. In this case, we’ll consider the sum of the forces acting in a vertical direction. Secondly, it also means that the sum of the moments of our forces is also equal to zero, where the moment is calculated by multiplying the force acting at a point by the perpendicular distance from the line of action of this force to the point about which the object will turn.

So let’s begin by considering the sum of the forces acting on our diagram. Let’s define the positive direction to be upwards so that 𝐹 is acting in the positive direction, whilst 61, 43, and 100 are acting in the negative direction. We can therefore say that the sum of our forces must be 𝐹 minus 61 minus 43 minus 100. And of course we know that this sum is equal to zero.

Now, in fact, the expression on the left-hand side simplifies to 𝐹 minus 204. So 𝐹 minus 204 is equal to zero. We’ll solve for 𝐹 by adding 204 to both sides. So 𝐹 is equal to 204 or 204 newtons. So we’ve now calculated the value of 𝐹 and we’ve done everything we can with the sum of our forces. So we move on to the second criteria. The sum of the moments of our forces is equal to zero.

Now, we’ve been defined a positive direction here. That’s the counterclockwise direction. And we know that a moment is calculated by multiplying the force by the perpendicular distance of this force from the point about which the object is trying to rotate. We have a few distances on here, but there is one distance that we’re missing.

Let’s define the distance between 𝐴 and 𝐷, so the length of the line segment 𝐷𝐴, which coincidently we’re trying to find, to be 𝑥 centimeters. And then once we have that information, we pick a point about which to calculate our moments. Now, we can pick any point on the rod itself. We’re going to pick 𝐷 here. Now, the reason we’re going to pick 𝐷 is because the force 𝐹 is acting at this point. And had we not calculated 𝐹 yet, we still would’ve been able to calculate the moments about this point since the moment of 𝐹 would’ve been zero.

It’s also worth noting that we generally choose to work in newton meters when calculating moments. But actually, our dimensions are in centimeters. So we’re going to be working in newton centimeters throughout. And that’s absolutely fine as long as we’re consistent. So let’s find the moment of our 61-newton force. This force is trying to turn the object in a counterclockwise direction. So its moment will be positive. It’s 100 centimeters away from 𝐷, so the moment is 61 times 100.

Moving from left to right, we’re now going to deal with the force at 𝐶. Once again, this is trying to move the object in a counterclockwise direction. So its moment is positive. But now it’s 43 times 50. As we said, the force 𝐹 is zero centimeters away from 𝐷. So we don’t need to worry about that moment. And instead we move on to the force at 𝐴.

Now, this force is trying to move the object in a clockwise direction. So its moment is actually going to be positive. And it’s 100 times the distance from 𝐷, which is 𝑥. We know of course that the sum of these moments is zero. So we can form an equation in 𝑥. This equation simplifies to 8250 minus 100𝑥 equals zero. Then we add 100𝑥 to both sides and finally divide through by 100. So 𝑥 is 82.5. And we see that the length of line segment 𝐷𝐴 is 82.5 centimeters.

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