# How can acceleration $a$ be zero when applied force $F$ is non-zero?

An object is at rest and stays at rest, but an attached rope pulls with increasing force. What is the acceleration?

The correct solution is that acceleration is constant and 0. Obviously $a= dv/dt =0$, since the object is at rest.

But if force is increasing, since $F=ma$, isn't acceleration increasing as well? If the two are directly related, why does acceleration not increase along with increasing force?

-
You have to consider the SUM of the forces, not just the force from the rope. – ja72 May 27 '13 at 19:32
"An object is at rest and stays at rest, but an attached rope pulls with increasing force." There must be some other force acting on the object that cancels out the force from the rope. Newton's second law refers to the total force acting on one object. In all of this, I don't see how it's relevant whether $F$ and $a$ are increasing. The issue is whether they're nonzero. – Ben Crowell May 27 '13 at 19:32
This is impossible unless there is another opposing force – juan May 27 '13 at 19:40

If you know the motion is fixed $a=0$ there there must be a reaction force enfoncing this constraint. The reaction force would be equal in magnitude by opposite in direction as the applied force $F$.

If you don't know for sure the motion is fixed, but the body is free to move then $a=\frac{F}{m}$.

Which one is it?

-
The body cannot move because of friction – Bogus Roads May 27 '13 at 19:43

The force in F = ma is net force. When you push an object and it stays at rest, it is because there is an opposite drag force (in this case friction) that is pushing against it. So the net force on the object is 0, and a = 0. The force you are thinking about is the applied for you put on with the rope, but it is not the total force. For an object at rest with a force acting on it: Fnet = Fapplied + Ffriction a = 0 Fnet = ma = 0 Ffriction = - Fapplied

So the force of friction is equal and opposite to the force, and is increasing as well as the force applied.

-

Why acceleration does not increase? Because the force does not. You have to include all forces that influence the body. Simply put, if you are acting with a force and the object is still at rest, there must be some other force that cancels the first one. This extra force you forgot about is friction. That is exactly the same but opposite as the force from the rope so that the resulting force is zero and the object stays at rest.

-

Friction is the kind of situation you must take into account. Apparently, you have not yet overcome static friction, which prevents an object from moving. ma is the sum of the forces, and if the sum of the forces is zero, the acceleration is zero.When both the applied force and friction force are increasing, you are approaching the threshold of static friction. Once you have overcome that threshold, the object will start moving either at constant speed or accelerating, as you would normally expect.

-

## protected by Qmechanic♦May 27 '13 at 21:06

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).