# Why does torque exist?

## Main Question or Discussion Point

More specifically, why does the torque go up proportional to the distance from the fulcrum. I know that you can balance a 5 pound weight with a 1 pound weight if the 1 pound weight is 5 times closer to the fulcrum, but why? I can't think of a good reason.

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the reason torque "exists" is because if you apply a force to an object that is not through the objects center of mass it will try to rotate. The further from the center the mass is it the force could be weaker but provide the same amount of torque

the reason torque "exists" is because if you apply a force to an object that is not through the objects center of mass it will try to rotate. The further from the center the mass is it the force could be weaker but provide the same amount of torque
Right- but why? Can you derive it from F = ma or something?

I do not know if it can be derived from F=ma though, as a and f are always in the same direction... And if you apply a force on an object not through its center off mass then the object will rotate, not in the direction of the force. I believe F=ma assumes the force is through the center of mass and hence torque cannot be derived from it. I stand to be corrected though

russ_watters
Mentor
It is probably easier to derive using a dynamic case and conservation of energy. As a lever pivots, one end moves further than the other, so in order for conservation of energy to be satisfied, the force has to be lower on that side.

It is probably easier to derive using a dynamic case and conservation of energy. As a lever pivots, one end moves further than the other, so in order for conservation of energy to be satisfied, the force has to be lower on that side.
You know what? That makes so much sense. Thank you.

m1gh1 + m2gh2 = m1gh1' + m2gh2'

"Can you derive it from F = ma or something?"
Yes, f=ma on lots of rigidly bound point particles leads to rotational acceleration which needs describing in an angular equivalent: torque = inertia tensor * angular acceleration. It is just the same equation but with rotation instead of translation.

... why does the torque go up proportional to the distance from the fulcrum.... I can't think of a good reason.
... conservation of energy. As a lever pivots, one end moves further than the other, so in order for conservation of energy to be satisfied, the force has to be lower on that side.
If you want to understand the lever [torque], think of it as an invention, a tool that concentrates mechanical energy. Just as a burning mirror, a magnifying lens concentrates thermal energy [the sun's rays]. If you apply a force [weight] of 5 N on one arm at 1 m [r1] from the fulcrum, and rotate the lever 1 r[adian], energy [mechanical work] is
J = [N * m] = F1 * r1 = 5 x 1: E1 = 5 J
energy is concentrated on the other arm [r2], because F1 * r1 = F2 * r2
F2 = E1 / r2 , ($\frac{5}{r2}$)
if you put a weight [50 N] at 10 cm from the fulcrum the lever is balanced because energy is the same E1 = E2 = 5 J, but force is 10 times greater at r2:
F2 = [$\frac{5J}{0.1m}$] = 50 N

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Torque is defined by cross product of radial distance and force r×F, since F exists torque exists. It could be defined by angular momentum. Since angular momentum is r×p=r×vm. Take time dirivative of both sides, dp/dt=F, v=ωr, dω/dt=α the angular acceleration. For point mass, moment of inertia I=mr2. Therefore the equation transform into r×F=Iα, or torque τ=Iα.

This equation means more radius would lead to more angular momentum change and thus more effectively change the angular acceleration α. They follow the definition.