Torque

@ 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??

Oh!
I forgot about answering your question. These are my views:
Look everything in Physics doesnot have proof. We accept certain Fundamental things as standard and then just verify it using other laws.
For example, in maths, you cannot define a point/line as such.
To define a point, you have to take the help of line as a standard and then define a point as the intersection of two lines.
Your question is same like asking how do you prove the equation of momentum which p = mv. Again, momentum is a defined physical quantity as the product of mass and velocity. We have to accept it as such. You can verify the equation dimensionally/ or by using other laws that p=mv.

Similar is the case of torque. We just define that quantity.

I hope its clear

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?

@swordfish

what do you mean by the word 'solely' here?

Here is another proof for
Torque=I(alpha)

We know that
F_t=ma_t where F_t is the component of force that is tangent to circular path and a_t is the tangential acceleration(for a rotating body).

Now,Torque=F_tr=ma_tr where r is the positon vector of the particle.
Using(a_t=(alpha)r),we can write

Torque=m[(alpha)r]r=(mr²)(alpha)
Torque=I(alpha) where mr² is moment of inertia denoted by I.

I completely agree with you Sir. We have to accept certain axioms without any reason.

Good.

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