# Force, torque and velocity

I know that when I apply a Force on the center of gravity of a body, I generate a linear motion and no rotation around any axe. (supposing here there is no drag)

Now when I apply the same force off COG, I get a torque T = F.d making the body rotate
AND a linear velocity, which is strictly the same that I get when applying the force on the COG.

So in the second case, it seems that the system has MORE energy (translational + rotational) than in the first case (translational) for the same amount of applied force.
Can someone explain me this apparent violation of energy conservation ?

## Answers and Replies

berkeman
Mentor
The violation comes when you say "AND a linear velocity, which is strictly the same that I get when applying the force on the COG." The linear velocity will be less.

Take the example of a stick laying left-right on a frictionless surface in front of you. Push in the center, and F=ma and the stick accelerates away from you with no rotation. But push 1/2 way between the center and the right end, and you get less than the original a of the COM, and you impart a CCW rotation. The energy that is invested in the rotation takes away from the overall COM acceleration.

Doc Al
Mentor
I know that when I apply a Force on the center of gravity of a body, I generate a linear motion and no rotation around any axe. (supposing here there is no drag)
The force produces a linear acceleration of the center of mass.

Now when I apply the same force off COG, I get a torque T = F.d making the body rotate
AND a linear velocity, which is strictly the same that I get when applying the force on the COG.
Right, now you have both a linear acceleration of the center of mass and an angular acceleration about the center of mass. And, yes, the linear acceleration of the center of mass is the same, since the force is the same.

So in the second case, it seems that the system has MORE energy (translational + rotational) than in the first case (translational) for the same amount of applied force.
Can someone explain me this apparent violation of energy conservation ?
It requires more work to maintain that off-center force, since the point of application moves more. It's work--force through a distance--not just force, that determines the energy required.

Hi Davidoux!
Your analysis if the problem is the good one.
When you apply the same force off COG, the object does acquires the SAME linear speed plus rotational speed. The object acquires more energy.
But, no panic, the supplementary energy comes from supplementary work that your force has done. The reason is that the point of application of the force has displaced through a grater distance than in the case of the force applied on the COG. See why?

Hi Doc Al, lpfr,

thanks for your explanations, now I fully understand And yes lpfr, I see why the distance is greater, it is because it moves along a circular path !

i was just reading this thread and I have a similar problem. Would the center of mass accelerate at the same magnitude AND direction with a centered force as well as an off centered force?

Yes. You can "feel" it when you unreel a spool laying flat on a table.

Doc Al
Mentor
i was just reading this thread and I have a similar problem. Would the center of mass accelerate at the same magnitude AND direction with a centered force as well as an off centered force?
Yes. Per Newton's 2nd law, the same net force acting on the same mass produces the same acceleration of the center of mass (both magnitude and direction), regardless of where it acts on the object.