# How do objects move in a linear path if you can never apply a force to the center of mass of it?

When you apply a force to an object, you are applying it to somewhere other than the center of mass since you cannot perfectly push it to the point. Centimeters, millimeters, micrometers---whatever it may be, it won't be the exact CoM (or whatever axis it may be). And when you apply a force to somewhere that is not the axis, you are applying a torque, which means it causes angular acceleration not linear. How is this possible? How are we able to move it linearly then? Do torques also cause linear acceleration? If so, is there some sort of mathematical formula?

(I feel like I missed some information on angular motion and stuff, I do not know what to google about this and all of them are not relevant)

Edit: I am asking this to simulate forces and torques in my computer. Please consider that I am ignoring the friction, air drag, how humans probably don't need to consider that, etc. etc.

Usually this is because you are not pushing at a single point, or because there are other forces that contribute to the total torque.

You don't give any examples, but usually I don't push things with a force concentrated at a single point. I'll push a box with a splayed hand (or two hands). If one side were to move further, I would push less hard on it for it to stay in balance. I wouldn't try to push a large box with a stick because keeping it centered would be too difficult.

Also when I push a box on the floor, I don't have to worry too much about the vertical center. If I apply a small torque forward on the box, the weight of the box will cause the normal forces from the floor to shift so that the floor provides a counter-torque.

Do torques also cause linear acceleration?

A pure torque would not, but it's very difficult to find a pure torque on an object. An external force generally can apply a linear force and an angular torque simultaneously. Whether it causes a linear acceleration or an angular acceleration depends on what other forces are present.

• But wouldn't that "external force" also not be on the CoM therefore applying pure torque and not a force towards CoM? Commented May 1 at 20:55
• A "pure torque" would apply torque but no linear force. For linear forces in $F_{net} = ma_{net}$, we don't care if the line of force goes through the center of mass. Toward the center, over at the edge, makes no difference to F=ma. Commented May 1 at 21:04
• So a pure torque will only apply angular acceleration since it has a balance of force but it has torque? And if we apply two forces at the same distance from the axis in parallel, it will accelerate linearly but not angularly since the forces aren't balanced but the torque is? Commented May 1 at 22:20

When you apply a force of 1N to the center of mass of a 1 kg object, the CoM accelerates 1 m/s^2.

When you apply a force of 1N to somewhere other then the center of mass of a 1 kg object, the CoM accelerates 1 m/s^2.

The acceleration of center of mass doesn't depend on where you are applying the force, because force is a flow of momentum, and the momentum has to be stored in the object, and stored momentum is calculated by multiplying mass with velocity of center of mass.

• So the acceleration of CoM doesn't matter on where you are applying the force? Commented Apr 29 at 23:10
• Yes, the acceleration of CoM doesn't depend on where you are applying the force, because force is a flow of momentum, and the momentum has to be stored in the object, and stored momentum is calculated by multiplying mass with velocity of center of mass. Commented Apr 30 at 4:59
• Didn't you forget about angular momentum? For example, in a chase, the police often try to ram the fugitive's car off-centre in order to make it spin. Commented Apr 30 at 11:14
• @JinsuJang provided you ensure a constant force vector, it does indeed not matter where you apply the force for acc-of-CM. However, in reality it is seldom the case that you have full control over the force vector: when hitting an object far off-center, you'll typically slip off somewhat, resulting in a force in an unintended direction, or the object starts spinning too easily and you can't follow, or some combination of effects. However, as long as you hit it nearly central, it's usually ok. E.g., look at billiards players imposing spin on the white ball but still potting the object ball. Commented Apr 30 at 13:15
• @stuffu Am I reading this right? You say that a force tangent to the surface of a sphere will have the same effect as a force in-line with the CoM ? You're saying that won't impart any spin??? Commented Apr 30 at 16:08

What I was really asking was if the distance from the axis mattered to linear acceleration as it is to angular acceleration when applying force. It does not. F=ma only cares about the magnitude of force and its direction, not its distance from the axis.