# A Question About the Physical Explanation Behind Torque

• I
• mvhpets
mvhpets
TL;DR Summary
I'm looking for an explanation for why increasing the distance between a force and the center of rotation makes the force better at causing rotation (torque); Basically, I'm trying to explain the 'Torque = force * radius' formula.
Hello!
I was wondering if anyone knew a good explanation behind the physical reason for torque. As in why a force applied from a greater distance to the center of rotation is better at turning an object than a force applied closer to the center. The question seems obvious, but all I've been able to find so far is the formula 'torque = Fr ' behind it, without a good physical explanation.

Currently, the best I could come with up in my head is that the closer a force is to the "outside edge" of an object the better it is at making the outer part of an object move relative to the stationary center, hence rotatory motion. However, this explanation does not seem to be concrete/ well put together.

Suspend a meter stick with a string attached at the 50 cm mark. It will be balanced.
Put a 10 g weight at the 20 cm mark and another 10 g weight at the 80 cm mark. It will be balanced. Note that there are equal forces applied at equal distances (30 cm) from the 50 cm mark. Also note that the product of force time distance from the 50 cm mark is (in SI units, with g = 10 m/s2)

P = 0.3 (m)×0.01 (kg)×10m/s2 = 0.03 N.

This is the same for the point to the left of the suspension at 20 cm and to the right of the suspension at 80 cm.
Now leave the 10 g weight at the 20 cm mark to the left alone. Add another 20 g on top of the 10 g at the 80 cm mark to the right for a total of 30 g. There is no balance. By trial and error, you find that there is balance when you move the 30 g weight at the 60 cm mark which is 10 cm from the 50 cm mark.

The product of force times distance from the suspension point at the 20 cm mark is the same as before
Pleft = 0.3 (m)×0.01 (kg)×10m/s2 = 0.03 N.
The product of force times distance at the 60 cm mark is
Pright = 0.1 (m)×0.03 (kg)×10m/s2 = 0.03 N.

Aha!

You note that the product of force times distance (left-to-right) from the point of suspension, which also acts as the the axis of rotation when equilibrium is lost, is the same when there is equilibrium. You conclude that this product is responsible for the angular acceleration when it's not balanced the same way the linear acceleration is caused by a force imbalance.

This product cannot be called "force" because the name is already taken. It cannot be called "Fred" or "Laura"either, so it ends up being called "torque."

Note that door designers cleverly place the doorknob, push-plate or handle as far away from the hinge as possible without being esthetically offensive. That's because most people naively think that to open a door they need to apply a force. Nothing could be farther from the truth. To open a door, you need to apply a torque because you want the door to acquire angular acceleration about the hinge. If you don't believe me, apply a force on the door very near the hinge and see how easily the door opens.

Last edited:
berkeman
mvhpets said:
TL;DR Summary: I'm looking for an explanation for why increasing the distance between a force and the center of rotation makes the force better at causing rotation (torque); Basically, I'm trying to explain the 'Torque = force * radius' formula.

I was wondering if anyone knew a good explanation behind the physical reason for torque.
"..., a lever long enough, and a place to stand...." ---Archimedes (allegedly)

berkeman
mvhpets said:
I was wondering if anyone knew a good explanation behind the physical reason for torque. As in why a force applied from a greater distance to the center of rotation is better at turning an object than a force applied closer to the center.
It's not based on reasoning, it's based on the observations you noted.

Welcome, @mvhpets !

For the same length of arc along which a tangential force is applied (same input work), a longer lever produces less angle of rotation; therefore, more torque (same output work).

Last edited:
russ_watters and Mister T
mvhpets said:
TL;DR Summary: I'm looking for an explanation for why increasing the distance between a force and the center of rotation makes the force better at causing rotation (torque); Basically, I'm trying to explain the 'Torque = force * radius' formula.
The physical explanation can be found in Isaac Newton's second law of motion.

https://en.wikipedia.org/wiki/Newton's_laws_of_motion

The small difference, and therefore the subtlety that makes it difficult to understand, is that the mechanical couple does not involve a moving mass.

The moment of inertia therefore does not appear clearly because, unlike Newton's law, there is no longer any question of mass.
It is therefore easy to get fooled by the concept of force if it has not been understood.
A force is not a physical object, but the consequence of a physical phenomenon.
So here, with the inertial change (of a supposedly massless bar... because the torque theoretically works with massless equipment), we can assume that we can take into account the change in orientation of a force instead of the change in trajectory of a mass.
Since BEHIND the force, there is necessarily a mass (or even an ‘energy’, but let's keep it simple).
If there is a change in trajectory then there is a force and if there is a force then there is a change in trajectory, it's the same thing.

Standard said:
The physical explanation can be found in Isaac Newton's second law of motion.

https://en.wikipedia.org/wiki/Newton's_laws_of_motion
If one contemplates only contact forces then I agree that Newton's laws of motion entail conservation of angular momentum. Every third law contact force pair has zero net torque.

However, if one contemplates instantaneous action at a distance (e.g. electrostatic repulsion or gravitational attraction) then one needs an additional caveat on the third law: "the third law force pair between interacting point-like bodies acts in a direction parallel to the separation between the bodies". Interaction forces at a distance would have a non-zero net torque if this condition were not upheld.

One could justify such a caveat on the grounds that the laws of physics should be predictive and isotropic (the same in every direction). If the force pair between two objects does not align with their separation, in what direction does it point? If the laws of physics are predictive and isotropic, the only permitted direction is parallel to the separation.

[In the modern view, one eliminates the idea of instantaneous action at a distance in favor of approaches such as electromagnetic fields or curved space time where interactions are local and disturbances propagate at finite speeds]

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mvhpets said:
I'm looking for an explanation for why increasing the distance between a force and the center of rotation makes the force better at causing rotation (torque)
There is a static way to understand torque based on linear forces only, by considering simplest truss structures (beams connected with torque-free joints), and solving the equilibrium equations for them.

Also note that torque is not necessarily about rotation. You can accelerate two masses in opposite directions on parallel straight paths. Nothing is rotating here, but the masses gain angular momentum, and the rate of that angular momentum change is torque.

When you use a wrench to tighten a bolt the longer the wrench the easier it is for you to tighten it since F the force you apply < f the force tightening the bolt.

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