# An intuitive approach to torque's dependence on radius

In summary, the torque (symbolized by T) is defined as the product of force (F) and the perpendicular distance (d) from the axis of rotation. This relationship can be understood intuitively by considering the ability of an off-center force to rotate an object and the proportionality between the ability to rotate and the point of application. There is no need for a ball and spring model or references to conservation of energy to explain this concept.

Does anyone have an intuitive explanation for why torque depends on the perpendicular distance from the axis of rotation? I understand the maths and fully accept the truth of the description, but CANNOT wrap my brain around the reason. The ball and spring model of a solid seems like a place to start? The uneven acceleration of a body's particles must play a role, since this is what defines rotational motion. References to conservation of energy seem to simply beg the question. In terms of the various linear tendencies of component particles, how does an increase in radius actually change the distribution of force on a body?

It's a definition, T = F * d

No balls or springs required, whatever they are for.

Does anyone have an intuitive explanation for why torque depends on the perpendicular distance from the axis of rotation?
I have two ideas for you:

1) It should be intuitively obvious that a force applied at the center, will not rotate the object (Which way would it rotate, if it did?). So if we accept that an off-center force does rotate the object, there must be some dependency between "the ability to rotate" and "point of application".

2) To understand why "the ability to rotate" is proportional to the lever arm, it is useful to consider a static case, where a small force counters a greater force rotationally. The proportionality can be derived from static linear forces only, without invoking conservation laws. See the PDF posted by Phillip Wood in post #10 here:

## 1. What is an intuitive approach to understanding torque's dependence on radius?

An intuitive approach to understanding torque's dependence on radius is to picture a lever. The longer the lever arm (or radius), the easier it is to lift an object with the same amount of force.

## 2. How does torque's dependence on radius affect the amount of force needed to lift an object?

The farther the object is from the pivot point (or fulcrum), the less force is needed to lift it. This is because the longer lever arm (or radius) creates a greater torque, making it easier to lift the object.

## 3. What is the relationship between torque and radius?

The relationship between torque and radius is inverse. This means that as the radius increases, the torque decreases, and vice versa.

## 4. Can you provide an example of torque's dependence on radius in everyday life?

One example of torque's dependence on radius in everyday life is using a wrench. By increasing the length of the wrench, the amount of torque (and thus force) needed to loosen a bolt decreases.

## 5. Why is understanding torque's dependence on radius important in science?

Understanding torque's dependence on radius is important in science because torque is a fundamental concept in physics and is used to explain the motion of objects. It also has important applications in engineering and mechanics.