Conservation of Angular Momemtum

[shockwave121]

Homework Statement

First off, Thanks for taking the time to explain to me why how I'm looking at this scenario is incorrect! I already asked my professor about this, and while I certainly don't think he was incorrect in saying that the conservation of momentum laws prevent what I'm asking about, he didn't provide a detailed explanation of why.

Any guidance is greatly appreciated!

Scenario:
^ (Z- axis)
M (motor)
| (rod)
O-------|--------0 --> (X- axis)
m1 x1 | x2 m2
|
M (base)

A rod, attached to a device (such as an electric motor, although not important), is providing an increasing angular acceleration, resulting in an increasing angular velocity. Attached to that rotating rod is a mass (m1) normal to the rod, at some distance from the axis of rotation (x1). Also attached to the same rod is another mass (m2) normal to the axis of rotation, but opposite of m1, at some distance (x2) such that (m1)(x1) = (m2)(x2).

Another device (again, details of this device, as I understand, are not important) moves m2 from distance x2 to the axis of rotation, such that (m1)(x1) >> (m2)(x2) taking all other masses and components necessary to make the situation feasible into account.

This second device, which is built-into / attached to the apparatus moves m2 into the axis of rotation during half the rotational period and moves the mass m2 back out to x2 during the other half of the rotation. (as a function of the angular velocity).

If I'm doing my equations properly, it would seem that the apparatus described above would be exerting an un-cancelled (in the Newton’s third law sense) force on the rotating rod during the unbalanced portion of the rotation, when (m1)(x1) >> (m2)(x2). But during the other period, when the mass m2 is extended back out to its x2 position, there is no net tangential force on the rod.

I’m sure you see the question already, if we attached this apparatus to some object, and let it run for some time, could this apparatus exert a net force on the attached object (in the plane of rotation)? The answer is obviously no, since this would violate conservation of momentum, but I’m having a really hard time understanding why. If there is no force to cancel the generated tangential force during the unbalanced period, wouldn’t this apparatus generate a net linear force (again, obviously, not all in one direction, but the net linear force would be in one direction)?

I’m looking for something more than “that’s impossible because it violates global conservation of momentum”. Feel free to use whatever terms or examples or references that you need to explain, if I don’t know it or don’t understand it, I’ll look it up.

Thanks!

Homework Equations

L = r x p

p = (m)(v)

F = (m)(a)

I used other equations here, but I'm not really sure what else to include!

The Attempt at a Solution

Again, when I worked out the torque equations for the above apparatus, I'm getting a force exerted on the rotating rod during the unbalanced period of rotation.

 

[jbsteele]

The net torque is not zero, which should result in a force (F=ma) being exerted on the rotating rod.

 

[thomate1]

This goes against the idea of conservation of angular momentum, which states that the total angular momentum of a system remains constant unless acted upon by an external torque.

In this scenario, the external torque would be the force exerted by the second device on the mass m2, which would then be transferred to the rotating rod. However, during the balanced period of rotation, the force is no longer present, and thus the angular momentum of the system would remain constant.

To put it simply, the force exerted during the unbalanced period is cancelled out by the force exerted during the balanced period, resulting in no net change in angular momentum. This is the fundamental principle of conservation of angular momentum.

Additionally, the force exerted during the unbalanced period is not a violation of Newton's third law, as it is still being exerted by the second device on the mass m2, which is then transferred to the rotating rod. The force may appear to be uncancelled, but it is still being balanced by an equal and opposite force.

In conclusion, the scenario described in the question is not possible due to the fundamental principle of conservation of angular momentum. The forces and torques involved in the system are all balanced and cancel each other out, resulting in no net change in angular momentum.