# Rotation of free solid object in vacuum, location of axis of rotation

1. May 6, 2012

### Holali

Hi,

I would be happy if you could solve and explain following problem:
Imagine some solid long object (wooden plank for example) being just like that in vacuum, free, without being fixed to any point of space.
Then, imagine only one forse pushing this object in a certain point of object, perpendicular to object's length. This force remains the same during time.
What I want to know is, how would such object move in space. I guess it will experience both translational and rotational motions. But where will be the axis of rotation? And how would speed of translation and rotation depend on strength of the force, on the mass of the plank and on the operation point of the force..

image for visualization
h*t*t*p://imageshack.us/photo/my-images/52/physicsimg.png/

Sorry for my bad english at some points, hope you understand the problem anyway.

2. May 6, 2012

### Staff: Mentor

The force produces both a translational acceleration of the center of mass (F = ma) and a rotational acceleration about the center of mass (Torque = Iα). The motion of the plank will be the sum of those two effects.

3. May 6, 2012

### vin300

All of the energy supplied to the plank does not translate into a single form. Some of it becomes its angular kinetic energy and the rest becomes translational. I don't know what the ratio will be.I had the exact same doubt.

4. May 6, 2012

### vin300

From the experiment I conducted myself with a pencil I concluded that the ratio of the rotational and translational energies increases as the force is applied farther from the centre of mass and is zero when it is applied at the centre of mass.

5. May 6, 2012

### Holali

Thanks for your replies, but there are still unsolved things:
How fast will the plank rotate and around what point? (are you sure it will be around centre of the mass?)
And what will be the speed of translational motion?

6. May 6, 2012

### Staff: Mentor

You'll figure out the instantaneous axis of rotation by adding the translational and rotational motion (about the center of mass).
Never said it would. But you can compute the rotation about the center of mass using Torque = Iα, then add it to the translational motion.
The acceleration of the center of mass is given by F = ma.

7. May 6, 2012

### vin300

What does this mean? The body always rotates about its centre of mass because in this state, the centrifugal forces will be balanced in the plane of rotation.The centrifugal force is proportional to the product of mass and radius, for the same angular velocity.
EDIT: The above occurs after the force has been removed. While the force is still in action, the instantaneous centre of rotation will vary from one instant to another.

This is only true if the only thing the force does is causing linear acceleration. Here the force also causes rotation.It breaks energy conservation.

Last edited: May 6, 2012
8. May 6, 2012

### Staff: Mentor

Huh? Why in the world are you introducing centrifugal forces to this discussion?
We are talking about a force acting. And even without a force, if something translates as well as rotates--as is the case here--the instantaneous axis of rotation will vary.
This is incorrect. Newton's law works just fine, as does energy conservation.

9. May 6, 2012

### cepheid

Staff Emeritus
This is false. F = ma holds true in this situation like in any other. The energy for the translational motion comes from the work done by the force $W = \int F dx$ and the energy for the rotational motion comes from the work done by the torque $W = \int \tau d\theta$. These integrals give you the expressions for translational and rotational kinetic energy respectively

10. May 6, 2012

### vin300

11. May 6, 2012

### vin300

Yes but it will always pass through the centre of mass, won't it?

12. May 6, 2012

### Staff: Mentor

No..

13. May 6, 2012

### vin300

Right.I got it.

14. May 7, 2012

### Holali

Ok, thanks for all posts, its become quite big discussion here. Can somone make some conclusion now?
The best way to understand this problem is via an example.
If the plank has the mass m=2kg and length 12m, the force pushing it is in distance d=3 meters from the centre of the mass and F=4N, and its acting for 3 seconds, whatt will be the result? I mean, what will be the angular speed and speed of translation? And what direction?
I'm sorry if I'm too annoying with so many questions. I would really like to understand this problem deeply.
Thanks anyway

15. May 7, 2012

### cepheid

Staff Emeritus
I kind of thought that your questions had been answered already. In any case, the problem you give above is not perfectly well-defined, is it? Does the force always act at a specific point on the plank that happens to be located 3 m from the centre of mass? If so, then as the plank begins to spin, the direction of the force (and hence the direction of the acceleration) changes with time. That makes it more complicated to compute the speed vs. time. If we can assume that this effect is small enough (over 3 seconds) be ignored, and we pretend that the force is constant in both magnitude and direction, then the first part of your question is easy: the acceleration will be 2 (m/s)/s and hence the velocity after 3 seconds (assuming the plank starts from rest) will be 6 m/s in the direction in which the force is being applied.

The torque around the centre of mass is easy enough to calculate: 12 Nm assuming the force is applied normal to the surface of the plank. To get the angular acceleration, we need more information: what is the shape of the plank? You have only given one dimension. If this is the only dimension that is important, and the others are negligible, then the object is more of a "thin rod" than a plank. For it to be a plank, I would think that at least one other dimension would be non-negligible i.e. you'd need a width as well as a length. In any case, we need the exact shape of the plank in order to compute its moment of inertia and hence the angular acceleration around the centre of mass, given the above torque.

16. May 10, 2012

### Holali

Well, it can be also thin rod. I originaly meant a long prism with square base. But rod is fine too. Does it matter a lot if it has square or circular base? (if the volume and mass are the same). We can use anything long with known moment of inertia.
And yes, the force is all the time acting perpendicular to the length of object, and alway 3m away from the centre of the mass. (imagine for example small rocket engine with constant force attached and fixed to it). So, the direction of the force is not constant during time, but its constantly perpendicular to the length of the object.
Hope you understand my strange description.
Is this everything you need to know to solve this problem?
I wat to know what will the force cause.

17. May 10, 2012

### A.T.

First you integrate the angular acceleration (from torque and inertial moment) around the COM twice, to get the orientation as function of time (and the final angular velocity). Then you use the orientation function and thrust force to get the linear acceleration over time, which integrated once gives you the final linear velocity of the COM.