# Describing forces mathematically

Let's say you wanted to write a differential equation for the forces acting on an orange sitting on the table (or any other object) using Newton's second law. Somehow you would have to add up all the forces acting on the orange. The orange isn't accelerating anywhere, so the net force acting on it must be $0$. How would you describe all of the forces acting on it (including their directions)? For example, lets say you have a force (of gravity) of $0.2N$, and a normal force of $0.2N$. You can't just add up these forces and say they equal a total of $0.4N$, because they are in opposite directions! Somehow, I would think, you would have to describe, matematically, the directions of these forces. How do you do this or is it even necessary at all?

• A force is a vector, so it always has magnitude and direction. Did they teach you how to work with vectors, yet? To go from vectors to differential equations we usually chose a coordinate system with three linearly independent axes (and it makes life easier if they are orthogonal). Now we can represent any vector with three numbers, which describe its components parallel to the coordinate system axes. In this case you would have a z-coordinate axis that usually points upwards. The z component of the gravity is then negative, that of the force holding up the orange is positive. – CuriousOne Apr 28 '15 at 23:49
• @CuriousOne That's a correct and adequate answer for the question, I think. – Asher Apr 28 '15 at 23:50
• @Asher. Thanks. If nobody wants to write a better one I might make it an answer. I am sure there is a much better one in a previous article, already, anyway. – CuriousOne Apr 28 '15 at 23:52

It involves vector math. To add vectors, we must take into account their direction. We can measure this line, but there is another way. Taking the line R, we can make it the hypotense of a right triangle.

Using the Pythagorean theorem , we can find the value of R.

The orange isn't accelerating anywhere ...

It most certainly is accelerating! The orange is fixed with respect to the Earth. The Earth rotates about its axis (so the orange is accelerating). The Earth also orbits about the Sun (so the orange once again is accelerating), and the Sun (along with the Earth) orbits the galaxy (so the orange yet again is accelerating).

I'll put all that aside for a moment and ignore the fact that the orange is accelerating. Consider an orange sitting at rest on a table on a non-rotating rogue planet that is far from any massive object. It is still accelerating, but by such a tiny amount that one can safely ignore that tiny acceleration toward remote masses.

I'll assume a typical orange with a mass of 200 grams. I'll also assume gravitational acceleration on this rogue planet is 10 m/s2 (about 2% stronger than Earth standard gravity). Suppose we stick a needle that acts as a harpoon into the orange, and tie a string to the expose end of the needle. Nothing much happens if you pull up on that string with a force of one newton. Increase the force to 1.9999 newtons, and still nothing much happens.

Increase the force by a tiny, tiny bit to 2.0001 newtons and the orange lifts off the table.

What's happening is that the normal force, the force that keeps the orange from sinking into the table, is a constraint force. It does what it needs to do to keep the orange from sinking into the table. The normal force can only prevent the orange from sinking into the table; it cannot stop the orange from lifting off the table. The normal force ceases operating at the moment the upward force exerted on orange by the string exceeds the orange's weight.

You can see this in action when a large spacecraft takes off. There's an obvious delay (one second or more, depending on the rocket) between ignition and liftoff. It takes a little while for the engines to build up thrust from nothing to equal to the rocket's weight.

Finally, I'll get back to the orange on a rotating and orbiting planet. From a kinematics perspective, you know the orange is sitting still on the table. You see it doing so. Also from a kinematics perspective, you know that the orange is accelerating from the perspective of an inertial frame.

Kinematics tells you what the net force has to be. Dynamics tells you what the individual forces must be. The connection between the two perspectives is that forces are vectors; they add vectorially. It's the constraint forces such as the normal force and static friction that make this connection happen.