Video: Analyzing the Forces on an Object in Rotational Equilibrium

An object is subject to a 72 N⋅m torque about a point when a 4.5 N force is applied to a second point along the object’s length. How far apart are the two points?

04:53

Video Transcript

An object is subject to a 72-newton-meter torque about a point when a 4.5-newton force is applied to a second point along the object’s length. How far apart are the two points?

Okay, so in this question, we’re told that we’ve got an object which is subject to a 72-newton-meter torque, and the object is subject to that torque about a point when a 4.5- newton force is applied to a second point along the object’s length. What we’ve been asked to do is to find out how far apart are the two points.

Okay, so let’s start by drawing a diagram of our object. Now we don’t know what our object is, so it could be any old thing. It could be this big blob, or it could be any other blob or object. Feel free to be creative with your blobs. The important thing is not the shape or size of our blob. The important thing is the next bit.

We need to label on the object a point. Let’s say this is the point. Again, we don’t know where on the object the point is, so we can just label it anywhere. But this is the point about which the object experiences a torque, and it experiences this torque when a force is applied to a second point along the object’s length. So let’s say that the second point is here.

Now this second point is where the force is acting. So we need to draw an arrow representing the 4.5-newton force on the second point. However, what direction is the force acting in? That’s an important question to answer. Well to do this, we need to record the definition of torque.

So we recall that torque is defined as the force applied to an object multiplied by the perpendicular distance between the fulcrum and the point at which the force acts. Now in this particular question, we’ve been told that there’s a point about which the object experiences a torque. That’s this point here. And so, that point has to be where the fulcrum of the object is.

This could be because it’s literally balancing on a fulcrum or maybe it’s nailed to the wall at this point. Again, we don’t know what the object is. The point is thought that is rotating about the orange point, and it’s doing this when a force is applied at this pink point here. Now a torque is defined as the force applied to the object multiplied by the distance between these two points when that distance is perpendicular to the direction of the force.

Therefore, the distance between these two points is this, and this distance has to be perpendicular to the direction of the force. In other words, on our diagram, the force can only be acting in one of two directions. Either the force is acting this way or the force is acting this way, because only then is the force applied perpendicular to or at right angles to the distance between the two points.

Now we can choose what direction we want, so let’s go for the downward pointing arrow, at which point we have a force acting on the object, which happens to be 4.5 newtons. And now this force is perpendicular to the distance between the pink point and the orange point. Now we’ve been told in the question what the torque on the object actually is. We’ve been told that the torque is 72 newton-meters.

But remember, the torque is equal to the force applied to the object multiplied by the perpendicular distance between the fulcrum and the point at which the force is acting. So this 72-newton-meter torque is equal to the force applied, which is 4.5 newtons, multiplied by the distance between the orange and the pink points, which we’ll call 𝑥 meters.

So we multiply the 4.5 newtons by 𝑥 meters. Now at this point, we’re trying to find out the value of 𝑥. So all we need to do is to rearrange the equation. We do this by dividing both sides of the equation by 4.5 newtons. Doing this results in the cancellation of 4.5 newtons on the right-hand side of the equation. And hence, we’re just left with 𝑥 meters on the right hand side and 72 newton-meters divided by 4.5 newtons on the left.

Now interestingly, we’ve left the units in our calculation here. This is not something that’s necessary, but it does help us to see that on the left-hand side of the equation, we’ve got newton-meters in the numerator and just newtons in the denominator. So we can cancel out the newtons unit, and we’re just left with meters on the left-hand side and meters on the right-hand side.

Therefore, the units on both sides of the equations match up and our calculation is more likely to be correct this way. But if we simplify things a little bit and leave out the units completely, all we’ve got is 72 divided by 4.5 on the left-hand side. And on the right-hand side, we’ve got 𝑥, at which point all we have to do is to work out the fraction on the left-hand side.

And it turns out that 72 divided by 4.5 is the same as 16. And remember, 16 represents 𝑥, where 𝑥 is the distance in meters between the orange point and the pink point. So when the question asks us how far apart are the two points, we can pretty confidently say that the two points are 16 meters apart and that is our final answer.

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