# Why Does Torque Exist & Increase with Distance?

• nhmllr
In summary: The longer the lever arm, the more torque is generated. In summary, torque is the result of applying a force to an object at a distance from its center of mass, causing it to rotate. This can be derived from the equation torque = inertia tensor * angular acceleration, or by considering conservation of energy in a lever system. The longer the lever arm, the more torque is generated.

#### nhmllr

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.

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

Trevormbarker said:
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

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.

russ_watters said:
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.

nhmllr said:
... why does the torque go up proportional to the distance from the fulcrum... I can't think of a good reason.
russ_watters said:
... 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.

## 1. What is torque?

Torque is a measure of the force that causes an object to rotate around an axis. It is commonly represented by the symbol "τ" and is measured in units of newton-meters (Nm).

## 2. Why does torque exist?

Torque exists because of the fundamental laws of physics, specifically Newton's first and second laws of motion. These laws state that an object will remain at rest or in motion at a constant velocity unless acted upon by an external force, and that the acceleration of an object is directly proportional to the force applied to it.

## 3. How is torque calculated?

Torque is calculated by multiplying the force applied to an object by the distance from the axis of rotation to the point where the force is applied. This can be represented by the equation τ = r x F, where τ is torque, r is the distance, and F is the force.

## 4. Why does torque increase with distance?

Torque increases with distance because the longer the distance from the axis of rotation to the point where the force is applied, the greater the lever arm and the more torque that can be generated. This is why using a longer wrench or lever can make it easier to loosen a tight bolt.

## 5. What are some real-world applications of torque?

Torque has many applications in everyday life, such as in vehicles where it is used to turn the wheels and in tools like wrenches and screwdrivers. It is also important in sports like baseball and tennis, where it is used to generate power and spin. Additionally, torque is crucial in machines and engines, where it is used to create rotational motion and perform work.