# Can there be force without motion?

I am confused. Can you have a force or tension without motion?

Take for instance two robots with jet packs connected via a cord, each is flying in opposite directions.

The tension of the cord is measured through a sensor of some kind. At some point, the net forces of the robots becomes zero and they no longer are moving, yet the sensor of the cord is still reading a force.

So what is causing this force if there is no relative motion?

• Yes! There is indeed no motion when you apply force on the wall. – user36790 Oct 30 '14 at 6:04
• i don't follow...what wall? – fernando pessoa Oct 30 '14 at 6:11
• Even though the robots aren't flying any further apart, they are trying to (with the same amount of force but held back), this is creating the tension in the rope and there for the force picked up via the sensor. – Harry David Oct 30 '14 at 6:28
• As I type this there is a force of about 650N being applied to my buttocks by the chair I'm sitting on, but I'm not moving. – John Rennie Oct 30 '14 at 6:48
• Wall means walls of your school building . You apply force to them. Will they move? Definitely not! Static friction opposes it. Thus there is no motion albeit you are applying force. – user36790 Oct 30 '14 at 6:52

In a system, the total sum of forces when added together equals mass times acceleration: $$\sum F = \frac{\mathrm{d}p}{\mathrm{d}t} = \frac{\mathrm{d}mv}{\mathrm{d}t} = m\frac{\mathrm{d}v}{\mathrm{d}t} = ma$$

Since the sum of the forces on the robots is zero, there is no acceleration. However, the tension of the string is not contingent on the movement. I will assume that there is an arbitrary force of 500 newtons being pulled by each robot. It is simple to calculate this. Since force is a vector quantity (because acceleration is a vector quantity), the direction is important. $$Robot \space 1:\\ \sum F = ma\\ 500 - T = ma$$ where $T$ is Tension. Since acceleration is $0$, we know that $ma$ must be $0$. Therefore, $T = 500$. The same can be done for the other robot, which is pulling the string the opposite way: $$Robot \space 2:\\ \sum F = ma\\ T - 500 = ma$$ Again, if they are not moving, acceleration is zero and $ma$ is $0$. Therefore $T = -500$ (relative to Robot 1). The tension in the string would be $500 \mathrm{N}$.

There is acceleration if, and only if, the sum of the forces is not zero. In this case it is. The sensors are not measuring force, per se, but rather tension. They are still measuring force, but this force is offset by the other robot, so you must think of the system as a whole.

In your example there is movement - though it may be so small that it appears to be zero.

The cord would stretch. Even a steel bar would stretch by a tiny amount. Obviously if we pull hard enough then the material would break/rip/snap. Up until that point, the material is still being stretched. And it will keep stretching until the molecular forces keeping it together match the robots pulling it apart. Try this thought experiment with a rubber band instead of a cord and you can see what is happening. You can imagine other materials as being less stretchy than a rubber band, but the same thing is happening.

Regarding other examples, pushing a wall etc, there is also movement. even it is infinitesimal.

A net force acting on a body should do some work. For the simplest conservative case, the applied work will appear as the change in potential energy of the system. The object is not moving. But the potential energy creates a tension on the system. An example is a spring with its one end fixed. You apply a force that causes the spring to extend. The applied work appears as the increase in potential energy of the spring system which causes a tension on the spring.

Why tension is there even there is no net force?

The applied force causes the potential energy of the robots-cord system to increase. This increase in potential energy has to be accounted to some kind of force; otherwise, it will violate the energy conservation theorem. The net force is zero. However, each robot is accelerating in order to maintain that state.

... if there is no relative motion?

Obviously, there are no absolute motions. We can only speak about relative motions and relative measurements. There is no universal fixed reference for any motion or measurement.