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In the following diagram:

enter image description here

Point(c) is a going into the page and attached to the disc, Point(c) applies a torque($\tau$) to the disc, and it starts to rotate due to that torque.

And if point(c) was move from the center to the far left like so: enter image description here

The disc will still rotate, around point(c). What if we added an equal point with the same torque in the opposite position(far right), what is the result? enter image description here

I think the disc now will be at rest, unable to move. However, I could not explain why...

Pupil
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3 Answers3

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If you attached two flywheels through a motor to the disk at the positions you show, and the motors start spinning in the same direction, then conservation of angular momentum tells us that as the flywheels spin clockwise, the disk must (and will) rotate counterclockwise.

However - if you attach the motors to an external structure, you are preventing rotation. I think that's what you inadvertently did in your original experiment and why you were getting confused (per your comment under ja72's answer).

UPDATE

The laws of motion that apply in this instance can be summarized as follows:

  • the center of mass will move as though the vector sum of all forces acts there
  • the angular momentum of the object will change with the sum of all torques multiplied by their duration: $$\Delta L = \sum \Gamma \Delta t$$ regardless of the axis about which the torque acts.

This second point is the one you appear to be struggling with. But you can think of it as follows: if you have a floating disk, and you have people standing on that disk with everything stationary, there is no net angular momentum. Now regardless of what the people do, the SYSTEM (people plus disk) must still have net zero angular momentum. So if two people start to spin clockwise about their own axis, the disk must start to spin counterclockwise. Let's see how that works in practice - put them with their feet apart on the disk:

enter image description here

Both are trying to rotate, and so they push forward with their right foot and back with their left. But if you look closely you see that there are now two sets of torque: the left foot of the right hand person and the right foot of the left hand person make a pair that is centered about the center of the disk, and the other feet make another pair in the opposite direction. Now the force is the same but the distance is different, so there will be a net torque on the disk equal to the sum of the two torques applied by the two persons.

Take a bit of time and look at this picture; I think it should clear up your confusion.

Floris
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  • Not sure what is meant by: "if you attached two flywheels through a motor to the disk ..." ? – Pupil Apr 01 '15 at 04:10
  • Attach two motors to the disk. Attach one flywheel to each motor. Allow the whole thing to rotate about the axle of the disk (the motors are not attached to anything except their own flywheel and the disk). – Floris Apr 01 '15 at 10:22
  • The experiment that I will conduct is diagramed here :http://i.imgur.com/poqHlT7.jpg the motors are placed on position's m1/m2 and their gears/flywheels facing out of the page, assuming no load(initially) if the motor's are powered the disc does not rotate due to the reaction force = 0, now attach a load to gears/flywheels of the motors the reaction torque(as diagramed red) would not rotate the disc, because its not acting at the rotation axis, am I right here? – Pupil Apr 02 '15 at 01:00
  • No I don't think that's right. It is perfectly OK to apply a torque off axis and generate a rotation. – Floris Apr 02 '15 at 02:47
  • I want to study this, could you guide me to the subjects I need to study and understand to grasp my system and it's outcome with my planned configuration? What Subjects should I study? – Pupil Apr 02 '15 at 14:08
  • This is Newtonian Mechanics. See if my update helps you. – Floris Apr 02 '15 at 14:27
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On a free floating body, if a pure torque is applied (with net zero force) then the body is going to rotate about it's center of mass (see https://physics.stackexchange.com/a/81078/392). This is regardless of where the torque is applied, or how many torques are applied. If the net force is zero then the center of mass will not move.

Now if C or A or B are joints, besides torques they also supply reaction (support) forces in order to maintain the kinematic relationships. So a joint not at the center of mass will need an in-plane force to make the center of mass rotate about the joint.

In the last case, there is no motion which can make the disk rotate about A and B at the same time. To impose that the linear velocity is zero on A and on B at the same time means the body will not move. The body is overconstrained.

For planar bodies their only possible kinematics are:

  1. No motion at all (fixed)
  2. All points move with the same non-zero velocity (translation)
  3. All points rotate about a single point (rotation about instant center)

The instant center may lie on the body or outside the body. Its location can change with time, but at any instant of time there is only one point a body can rotate about.

See http://en.wikipedia.org/wiki/Instant_centre_of_rotation for more details.

Community
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John Alexiou
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  • So if I build an experimental model of this using a large disc and two motors that are connected to the disc(mechanically in a efficient way) at points a&b as diagrammed in #3 the disc and the motors should not move at all correct(Assuming no friction)? This question came about from me placing a pencil on a center of a paper and rotated the pencil and the paper rotated as well, then placed to pencils as diagram#3 and rotated them the paper does not rotate and it feel intuitive not that it didn't. – Pupil Mar 30 '15 at 14:40
  • The only way to get any motion would be to put the two motors on the same axis. It is like you are taking two gears on two shafts and gluing them together. Nothing will move. – John Alexiou Mar 30 '15 at 18:08
  • Yes, that makes sense. And that was my initial assumption that the disc will not rotate in my final diagram, because the two motors are not attached on the same axis but they are separated on points a&b. – Pupil Mar 30 '15 at 20:58
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By saying point(c) (or any of the other points) applies a torque on the disc, it sounds like point(c) is a small physical body (If point(c) exerts a torque on the disc, then disc must exert a torque on point(c)) I'm going to assume that the mechanism by which point(c) causes a torque on the disc is a motor connecting point(c) and the disc.

So, if points (a) and (b) are not connected to anything else apart from the disc via motors, then each point will exert a torque on the disc, and the disc will exert equal and opposite torques on the points. Both torques applied to the disc are in the same direction, and so will add together. As there are no other forces or moments being applied anywhere, this means the disc will rotate one way, and the points will rotate the other way (assuming both points apply torques of the same direction on the disc). This seems to make sense if you consider the conservation of angular momentum (not moment of momentum).

If, however, the two points are fixed in space (as if those points are both attached to a rigid wall), then those points will have to be in equilibrium. This means there will be reaction forces to keep the points fixed, and if the points are held fixed, then the points will not only apply torques but also forces on the disc, such that the disc is also held in equilibrium. Net force and torque of that system = 0.

Here is a diagram showing the forces and moments applied to the disc for both cases:

enter image description here

Involute
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  • Im thinking about to motors that are connected to the disc via point a&b, and the two motors are activated how can the disc rotate?(Assuming the motors are placed on the ground). It more about the rotational axis. Because if you look at the first diagram where point(c) is the center and a motor is applying a torque to it it will rotate or even diagram 2, however if there are two motors on point a&b applying the same torque in the same direction it's confusing seeing the disc rotate. – Pupil Mar 30 '15 at 14:36
  • Well, you need a motor to be attached to two different bodies to operate. I assume that when you mean the motors are "placed on the ground", you mean they are fixed to the ground. In which case, you will get the second scenario I provided. In which case, nothing can rotate, unless you have the motors apply such a high torque that one of the motors break from either the rotor of ground, and that motor no longer functions, giving you the one-motor problem again. – Involute Mar 30 '15 at 14:43
  • I want to go back to your first case, assume that we connected motors to points a&b. Then activated the motors(they are not fixed to anything free to move just to understand the theory), the disc cannot move because the two motors are connected to two separate rotational axis? – Pupil Mar 30 '15 at 21:10
  • This is the interesting thing though: if the motors aren't fixed to anything, then the disc is no longer constrained and has all six degrees of freedom (3 translational, 3 rotational). Because a net torque is applied to the disc (and no net force), the disc will start to spin and speed up spinning about an axis through its centre of mass. If you want to know why, you should look into Newton's 2nd Law extended to rotational dynamics (torque = polar moment of inertia * angular acceleration, for a lamina). – Involute Mar 30 '15 at 21:28