Calculating Angular Accerlation for 3D Rotation

[davikrehalt]

I am having some problems understanding rotational motion in three dimensions. If a torque is applied to a rigid body that is already under rotation but τ-torque's direction is different from ω-angular velocity, how do I calculate the angular accerlation of the body. In particular, I am unable to decide which axis I should calculate the moment of inertia with respect to, any help is appreciated

 

[Simon Bridge]

That's a very nasty problem - yes.
It helps if you know how the torque is being applied.
Basically you have to use conservation of angular momentum and $$\vec{\tau} = \frac{d\vec{L}}{dt}$$.

... the resulting motion can get quite complicated.

Start by looking at what happens if you apply the torque about an axis 90deg to the established rotation.

 

[D H]

I am having some problems understanding rotational motion in three dimensions. If a torque is applied to a rigid body that is already under rotation but τ-torque's direction is different from ω-angular velocity, how do I calculate the angular accerlation of the body. In particular, I am unable to decide which axis I should calculate the moment of inertia with respect to, any help is appreciated

You don't have to worry if all three eigenvalues of the inertia tensor are the same. You can treat the inertia tensor as if it were a scalar in this case. If not, you're asking about a regime where you can no longer treat moment of inertia as a scalar.

What level physics background do you have?

 

[davikrehalt]

You don't have to worry if all three eigenvalues of the inertia tensor are the same. You can treat the inertia tensor as if it were a scalar in this case. If not, you're asking about a regime where you can no longer treat moment of inertia as a scalar.

I don't think that's the case for which I am dealing with. I am considering a system of connected particles each experiencing different forces which sum to zero. The angular momentum and torque are quite easy to calculate but the problem is on angular acceleration and individual acceleration. They might be already spinning in an arbitrary axis.

It helps if you know how the torque is being applied.

From my problem, the torque is just the sum of torques from the individual particles.

What level physics background do you have?

My physics background in mechanics is just a freshmen physics course, but I am open to learn how to deal with this with tensors.

Side note: I just realized I spelled acceleration wrong before, oops

 

[D H]

I am considering a system of connected particles each experiencing different forces which sum to zero.

That doesn't make sense. Those particles will either remain at rest or move at a constant velocity. Newton's first law. They certainly won't be rotating, as that requires a non-zero net force.

 

[Redbelly98]

That doesn't make sense. Those particles will either remain at rest or move at a constant velocity. Newton's first law. They certainly won't be rotating, as that requires a non-zero net force.

I think he means the net force on the system of particles is zero, but the individual force on a given particle in the system is not necessarily zero.

Just for example, this happens if you could grab a bicycle wheel with both hands on opposite sides of the wheel and give it a spin, and the forces exerted by your two hands are equal in magnitude and opposite in direction.

 

[AlephZero]

I think the OP means a system modeled as point masses connected by rigid massless rods.

If your mechanics course covered bending of beams, you know about the idea of "principal axes" for a beam cross section. Any arbitrary shape has two principal axes at right angles such that the cross-product of area is zero, i.e. $\int xy\,dA = 0$. If you apply a force along one of the principal axes, the beam bends in the same direction as the force. If you apply a force in a different direction, in general the beam does NOT bend in the same direction. You can deal with that by resolving the force into components along the principal axes and summing the displacements from each component.

The same idea applies to your system of particles. For any system, there will be a set of 3 orthogonal coordinate directions through the center of mass, such that all the cross-products of inertia are zero. If you resolve your applied moment into those three components, you can find the anguilar acceleration in each component direction and sum them.

But there is another complication that didn't arise in the beam problem: if your system will rotate through arbitrary large angles (i.e. it's not just vibrating with small amplitude about some fixed position) the directions of the principal axes will also change with time... and that's a good place to stop describing the situation in words, and say "read a dynamics textbook".

Of course if the moment starts off aligned with just one of the principal axes, the system rotates about that axis and that axis does not change direction, so things are much simpler - and that's probably the only case you will meet in a first dynamics course (Redbelly's bicyvle wheel, for example)

A "not quite so simple" example is a gyroscope, where the angular acceleration depends not only on the applied moment, but also on the motion of the system (i.e. on the speed of rotation of the gyro).