First of all, I should note that I'm a programmer and have only an extremely basic understanding of physics; I only know how to explain my question in layman's terms and I apologize if I'm unclear or unnecessarily verbose.
If there's a rigid square object of size 1×1 and mass 1 on a two-dimensional frictionless plane, and while stationery it is struck by a force of strength 1 at exactly halfway along one of its sides from directly perpendicular to said side, like so:
then my intuitive understanding is that it will begin traveling at a speed of 1 in the direction of the force.
Similarly, if it is struck exactly on a corner by a force of strength 1 that is exactly perpendicular to the square's diagonal (at a 45* angle to its side), like so:
then my intuitive understanding is that it will begin spinning (I don't know how fast) but remain stationery.
What I don't know is how to work out what will happen if it is struck at any other point or from any other angle. For example, if struck three-quarters of the way down a side at an angle perpendicular to the side, like so:
It seems as though it should cause it to begin moving (in some direction, at some speed) and spinning (at some speed), and the less perpendicular to the square's center of gravity the force is, the more it will spin and the less it will move. Is there a formula that, given the magnitude, angle, and impact point of the force, exactly how fast it will spin vs. how fast and in what direction it will move?
More generally, how does one work out the answer to this sort of problem when applied to objects of other shapes/sizes/masses when impacted by other forces? Does it make a difference if the object is already moving or spinning?
(Also, are any of my intuitive understandings actually entirely incorrect?)
Best Answer
When you hit anything with a force, it causes an acceleration in the direction of the force. So if you hit the square at the corner like so:
It would start moving upwards.
However, it also experiences a torque about its center of mass. This torque can be calculated as the force multiplied by the perpendicular distance between the center of mass and the line of action of the force. In this case, the distance is half of the diagonal. Remember, the line along which you measure the distance must be perpendicular to the line of action of the force, AND pass through the center of mass.
Once you have the torque $T = F \times r$, you can find the angular acceleration of the object as $T = I\alpha$ where $I$ is the moment of inertia of your object, and $\alpha$ is the angular acceleration
In the first case, i.e.
There is no rotation because the line of action of the force passes through the center of mass of the square, meaning the perpendicular distance is zero, and hence there is no torque.