Which of the following formulas correctly shows the relation between the force acting on an object, the torque that the force produces, and the perpendicular distance of the force from the point that the object rotates around? 1) 𝑇 is equal to 𝐹 multiplied by 𝑑, 2) 𝑇 is equal to 𝑑 divided by 𝐹, 3) 𝑇 is equal to 𝐹 divided by 𝑑, 4) 𝑇 is equal to 𝐹 plus 𝑑, 5) 𝑇 is equal to 𝐹 minus 𝑑.
Okay, so in this question, we’re trying to find the correct relation between the force acting on an object, the torque that the force produces, and the perpendicular distance of the force from the point that the object rotates around. Now, this is best illustrated by considering an example.
So let’s think about something like a plank. So let’s say that this is a plank and it’s resting on a fulcrum at the middle of the plank. Now, let’s say that we exert a force on the end of the plank. We’ll call this force 𝐹. And we can say that the force is at distance 𝑑 from the fulcrum. Now, this distance is the perpendicular distance between the force and the fulcrum because as we can see the angle between the direction in which the distance is measured and the direction of the force is 90 degrees. So they’re perpendicular to each other.
Now, let’s also remember that torque is the turning or rotational force on an object. In other words, the larger the torque that we put on the object, the quicker it will turn about its fulcrum. And we need to relate this torque to the force applied 𝐹 and the perpendicular distance 𝑑.
Let’s first consider the force then. If we’re at certain distance 𝑑 away from the fulcrum and if we want to make our plank turn more quickly, then we exert a larger force. Therefore, the torque 𝑇 or turning force is directly proportional to the force 𝐹 that we apply because once again the larger the force that we apply, the larger the torque because it’s gonna turn more quickly.
Now, let’s consider the distance 𝑑. Let’s say that we can only exert a certain force 𝐹 on the plank. Well, if we’ve only got the force 𝐹 to work with, then what we can do is to increase the distance 𝑑 away from the fulcrum in order for the plank to turn more quickly about the fulcrum. This is why spanners are such a useful tool. They help us exert a larger turning force on nuts around the bolts.
In other words, let’s say that we have this nut here and we’re trying to turn it clockwise so that we can tighten it. Now, we’re trying to do this with our hands. But it’s really difficult to do. Our hands can only exert a certain force, but the turning force that they exert is not large enough. So what do we do? We use a spanner. A spanner allows us to use that same force that we could exert with our hands, but apply it at larger distance away from the nut. Because remember earlier, we were turning the nut with our hands. So we were applying the force here. And now, we’re applying that force here. So now, the perpendicular distance is increased and hence the turning force is increased.
Therefore, we can say that the torque applied is also directly proportional to the perpendicular distance. In other words, the larger the force you apply, the larger the torque and also the larger the perpendicular distance, the larger the torque. And in fact, in this case, the constant of proportionality in both cases is just one. And we find that the relationship between torque, force, and perpendicular distance is that 𝑇 is equal to 𝐹 multiplied by 𝑑.
Now, this makes sense. If you increase the force, you increase the torque. And also if you increase the perpendicular distance, again you increase the torque. And hence, our final answer is that option one gives us the correct relation between torque, force, and distance.
The torque is equal to the force exerted multiplied by the perpendicular distance of the force from the point that the object rotates around.