# Why do things rotate? I know what torques are .the question is deeper than that

1. Jul 15, 2010

### metalrose

Why do things rotate? I know what torques are.....the question is deeper than that...

This is not any homework or coursework problem. This question has been troubling me since it first popped into my head.

Suppose there are two point masses both of mass m, connected to each other by a massless rigid rod. Suppose we have the rod in a horizontal position in gravity free sapce. Now a force , F, is applied to the right mass(could be left mass too) in the forward direction. Now according to the newton's laws, the entire system should have a linear acceleration of F/2m. So, both the balls move ahead with a=F/2m, to say the least. So far so good.

I read in some book a long time back that the concepts of torque and angular momentum are not independent laws of nature, and that they have been derived from the more fundamental newton's laws of motion.

That means, that the rotational motion is simply a manifestation of the newton's laws.

If it is really so, then in the above example of two masses, why does the system of these two masses also rotate along with moving forward with a linear acceleration ,a=F/2m ?

The two masses, apart from having a forward component of acceleration, a=F/2m, which was derived from the newton's laws, also have another component of acceleration, namely, the angular acceleration.

Where did this new component of acceleration, i.e. the angular acceleration come from?

Please explain how the newton's laws gave rise to this angular acceleration component?

P.S. Please don't use the concepts of torques and angular momentum. I want an answer purely in terms of Newton's laws. If the torques are really not independent laws of nature, and if rotation is really a manifestation of the newton's laws, there should be an answer to the above in terms of newton's laws.

2. Jul 15, 2010

### Gear.0

Re: Why do things rotate? I know what torques are.....the question is deeper than tha

But there is no rotational motion in the system you described...
At least if I understood you correctly, you have two masses on a massless rod like this:
O--------O Then you apply a force on the right side in the (forward?) direction like so:
O--------O <---Force
The force could be going the other way, but basically this is like pushing a pencil across a desk longways. If you want rotational motion you would need to apply a force at a non-parallel direction: like
O--------O
^
|
Force

Do you like my pictures? :D

So if you applied force in a non-parallel direction like that, then according to Newton's laws you have one side that wants to remain at rest because of "inertia" (Newton's first law), and one side that begins to move because of the "force" (Newton's second law) being applied to it.
Then once the object is already in motion (using my last picture) you have the left mass pulling on the right mass (via the rod) so the right mass will begin to move inward toward the left (a little bit, also some of the "pull" would be in the upwards direction), but because of the "reaction force" (Newton's third law) the right mass must also be applying a somewhat inward force on the left mass and it moves to the right.

Repeat over and over, and the two masses appear to move around each other.
Hope that is the answer you were looking for.

3. Jul 15, 2010

### mg0stisha

Re: Why do things rotate? I know what torques are.....the question is deeper than tha

Either way you look at it you're applying a torque to your system if you're giving supplying a force perpendicular to the system...

4. Jul 15, 2010

### Gear.0

Re: Why do things rotate? I know what torques are.....the question is deeper than tha

An easier way is to just think of the moon around the Earth. Let's say that the ONLY force acting on the moon is the Earth, and let's pretend that the Earth has an infinite mass so that the Earth won't move because of the moon's gravity.

Now the Earth want's to pull the moon downward. and in fact, the moon would fall right into the Earth, the ONLY thing causing it from not falling in is that it already has a velocity. So if you just imagine the path that the moon would travel if there were no gravity, it moves out horizontally into space, but just imagine that it moves a little bit along that line. Now we know that the Earth also moves it down towards it.

So you have this balance between inertia, the moon wanting to move outward on a tangential direction, and the force of gravity pulling it inward which changes the velocity direction.

5. Jul 15, 2010

### Jack21222

Re: Why do things rotate? I know what torques are...this is deeper than that...

I think the answer lies in inertia. The other point mass and the rod connecting them don't experience a force in the direction of you pushing on it, at least not directly. They want to stay still while the mass that you pushed on moves.

6. Jul 15, 2010

Re: Why do things rotate? I know what torques are.....the question is deeper than tha

I don't think you have studied enough Lagrange mechanics, so there are one or two things that I won't explain any deeper, but this is how it works:

You can define a rotation of a single particle by the angle it turns around the origin of the coordinate system. So a particle in straight flight will rotate around the origin unless it is coming straight at it. The fact that the angular momentum of a straight flying particle doesn't change, you can check yourself with high school math.

If you have two point masses attached to each other by an infinitely thin stick. If you push both of them at the same time in the same direction, then we don't have a problem they just fly straight ahead together.

Now we just push one particle. The particle wants to move but the second particle is attached to it, through the stick. The stick can only produce forces in its direction and it will produce as much as it takes to keep the particles at the same distance.

If you push the particle in an arbitrary direction, you can decompose the force into two parts: one along the stick's direction and one perpendicular to it. The one along the sticks direction gets transmitted to the second particle and will lead to an acceleration of both particles or better of their center of mass, making the whole thing move in one direction. The perpendicular force will give the particle some speed.

Now the interesting part, where we show that the force along the stick will lead to a rotation: First we get rid of the average motion, so we transform into an inertial system where the center of the stick is not moving (we are moving as fast as the rotator). In this system the masses move in opposite directions perpendicular to the stick. We now have the two masses moving with the same speed up and down, and we would normally be forced to solve the differential equation under the constraint that the distance of the masses stays the same. The solution would be a circular rotation.

But if we don't want to do the math, we can use some infinitesimal arguments, and split the different things that are happening into imaginary, infinitely small steps:

Step 1:
Both masses move along their way up and down a tiny bit.

Step 2:
The stick stretches, and turns a bit.

Step 3:
It is not amused that the particles are coming apart and pulls them closer until the distance is the same again.

Step 4:
But now the particles are in a slightly rotated position. But what is their new speed?

Step 5:
The stick ensures that the acceleration is always perpendicular to the direction of flight. But this means that there is no energy transfer into the particle. So the kinetic energy and thus the speed must be the same. It just has a new direction.

7. Jul 15, 2010

### PhanthomJay

Re: Why do things rotate? I know what torques are.....the question is deeper than tha

Newton's 2nd Law of Rotation is derived from his 2nd Law of Motion. If the point masses, m, are separated by a distance of 2r, the force, F, applied to one mass, may be represented (from rigid body dynamics) as a force, F, and a couple (torque), T = F(r), applied at the center of mass of the system. It's translational acceleration is F/2m, as you noted. Regarding the Torque, since from Kinematics, angular acceleration , α, = a/r, or a = α(r), and since F = T/r, then using Newton 2:
F = ma
substituting,
T/r = mα(r)
or
T= mr^2(α)
Where mr^2 is the particle mass Moment of Inertia, I, thus,
T= I(α), which is Newton 2 for rotation.