Angular Vs Linear Inertia, (Basic) Theoretical mechanics question

Hi guys!

Imagine a rigid rod/bar with no forces acting on it. Then you apply a force. Depending on where that force is applied, the rod will start to move in the direction of the force and will also start to spin.

If the force is applied in the dead centre, intuition tells me it won't spin at all all; the entire force will be used to accelerate the rod linearly. My guess is that if the force was applied (perpendicular) to the very tip of the rod, all the force would go into spinning to rod without moving it (ie, the centre would remain in one spot).

What I want to know is, given a location that a perpendicular force is applied to a rod, how can we tell how much it will accelerate angularly vs how much it will accelerate linearly? What's the (name of the) theory behind this?

I'm guessing it's got something to do with inertia and moment of inertia...

Ballistics.

You sure? Ballistics seems to be quite niche discipline of trajectories rather than (what I assume is) a fairly general kinematics problem?

The force applied accelerates the object linearly, regardless of where it's applied.

It also causes a moment, corresponding to the distance between the center of mass and the point of force. If the force is pointing right at the COM, the moment is zero. If it's applied perpendicularly at the end, the moment is F * L/2, for a force F and end-to-end length L (assuming COM is in the middle).

For a force applied through a fixed distance, the object will be moving faster (linearly) if the force is applied to the center, because all the energy delivered (E = F*d for a distance d) goes into linear motion. The maximum amount that can go into angular rotation depends on the distribution of mass; a ring, with most of the mass at the periphery, will rotate slower and move faster than one with all the mass in the center (think potato with a skewer sticking out of it).

Exact values can be calculated easily. A ring of radius R has inertia M*R^2 and mass M. A force F applied tangentially will accelerate the mass with acceleration F/M = a. The applied moment is F*R, so the angular acceleration is alpha = F*R / (M*R^2) = F / (M*R). The tangential acceleration is R*alpha = F / M, so the edge will rotate at the same speed the ring moves forward. That means one edge will move at twice the forward rate, while the other edge has zero velocity (once in motion, you could put this hoop against a wall, and it would simply roll along against it with no change in rotation).

A different distribution, like a rod, with inertia 1/3 * M*R^2, will rotate much faster, because more mass is in the center. In this case, the pushed edge will rotate at 3 times the forward velocity, for a total of 4 times, while the opposite edge has a total velocity of 2 times forward, in the opposite direction! (In this case, if you bring a wall near it, it will dig into it and kick off, because it's rotating faster than it's moving.)

Tim

Ballistics deals mostly with free flight & the effects of gravity. I suppose a more accurate term for the theory you are looking for would be "classical mechanics".

Hi Sch3m, thanks for the description! Much appreciate the time...