Torque

whatever i can gather is that when i start proving the moment of inertia i come across r cross f which has been designated as torque .so may be it has been defined as the rule of the game.

for ex you cannot prove F=ma
(by the way how many of you doubt this formula. i seriously do because i think
F=maⁿ where n varies from one part of universe to another.although what i am speaking may be useless stuff but i think that is why Newtonian mechanics fails in front of relativistic mechanics.the difference comes because of the fourth dimension time.)

for ex you cannot prove F=ma
Very wrong.
This is Newton's second law and we have a very good proof of it.
F=maⁿ
Can you prove this?

@ jeetopper

I completely agree with swordfish that F=ma corresponds to newton's second law.However,there is no law for which we obtain a equation for torque.So,i want to know how the equation for torque is derived.

ANYONE??

Actually, this question has two parts in it:

(a) What is the motivation for bringing in a product that involves r and F to define a quantity called torque, and

(b) Why is that written in the precise fashion r X F

This post will deal with the first part

Think about opening a door. Your experience tells you that you need to expend less effort by applying force as far away from the hinge as possible. How does this happen? Why does the point of application of force make a difference?

Notice that for the same angular displacement, a point further away from the hinge moves through a greater displacement than a point closer to the hinge. That means the same force does greater work when it is applied further away from the hinge. This provides us with a starting point.

Let us say it is applied at a distance R from the hinge and at the time interval dt it moves through an angle dθ. Let the force be applied at an angle φ

The work done is F \sin \phi \times R d \theta

Now, this example has been chosen so so that the effect of the force on the body is pure rotation. The work done causes only a change in KE. Hence, now an element of mass dm at a distance r undergoes a change in kinetic energy of

\frac{1}{2} dm \times r^2 (\omega + d \omega)^2 - \frac{1}{2} dm \times r^2 \omega ^2 = dm \times r^2 \omega d \omega

Now if you take this for the entire body, you will get

d \omega \times \int r^2 dm.

This quantity \int r^2 dm is denoted by I.

So the change in KE is just I d \omega

So now we have F \sin \phi \times R d \theta = I \omega d \omega

Then, F \sin \phi \times R \frac{d \theta}{dt} = I \omega \frac{d \omega}{dt}

and hence F \sin \phi \times R \omega = I \omega \alpha \Rightarrow F \sin \phi \times R = I \alpha

Denoting F \sin \phi \times R by \tau we have

\tau = I \alpha.

for ex you cannot prove F=ma
Very wrong.

I again don't agree with you.
This is based on observation and experiments. These are Fundamental Laws.

And if insisted to prove this you would unknowingly take the help of observations/experiments based on the laws itself. There should be other laws more fundamental than these, I order to be proved, and I think they aren't there right now.

P.S. *No* scientific theory can be proven true. Newton's laws are no exception. Compare this to mathematical theorems, which can be proven true. In fact, mathematical conjectures are not called theorems until they have been proven to be true.

Eg. ' a '
You cannot prove that no. of letters in above word is 1 ! It is defined and fundamental.

@the Prophet sir,

Thanks for letting us know the derivation. And for the second part of your question viz., Why is that written in the precise fashion r X F, You have given enough hints in your first answer already.

@Prophet Sir,
You are infact using the conservation of angular momentum to derive torque = I(alpha). How do you derive it solely?

You can get really fundamental and say that forces arise as a result of breaking of symmetries and stuff, but you need something more workable than that. We can take the path of Galileo and Newton and work with what we can see - velocity and acceleration. Notice that acceleration happens under certain special conditions, call that as force. By observation and definition F = ma.

All the rest is then derived from this framework, including conservation of momentum, mechanical energy etc. There is no reason otherwise to arbitrarily designate mv as momentum 1/2 mv² as kinetic energy, work as ∫F.ds etc. These quantities are so defined because there is a mathematical motivation to it. They like to examine what is invariant under certain conditions and things like that. There is no Theory of Everything yet that tells us what mass, acceleration are related this way, so you have to accept that kind of axiomatic definition and then work your way up.

I was aiming to provide in the light of what is known up to that stage, the motivation for looking at rXF in the first place.

Good.

Thank you all;especially you,Prophet (SIR) for your detailed explaination.