2.6 Newton’s Second Law, Part 2: Multiple Forces

Critical Questions:

  • What happens when more than one forces pushes or pulls an object?
  • Why do some objects move at a constant speed even though they’re being pushed by a force?

You might have noticed that in the last section, I was careful to talk about only one force at a time. If a real physicist had been reading, steam would be coming out of her ears, because Newton’s Second Law is usually stated in terms of something called the net force. “Net” in physics means about the same thing as it does in economics: it is the total result once everything has been added and subtracted.

So what happens when there is more than one force? I like to think of net force as if two people were pulling on ropes attached to a big crate. If they pull the crate in the same direction, the crate will accelerate twice as quickly. If they pull in opposite directions with equal forces, the crate won’t move at all — these two forces cancel each other out. If one person pulls northwards and the other pulls eastwards, the crate will move to the north-east.

dog tug of war
If dogs are involved, all rules are out the window.

The net force, then, is just the sum of all of the forces going in various directions. It’s important to remember that in the formula for Newton’s Second Law, the F is not just any one force but the sum of all of the forces acting on the object. If two equal forces pull in opposite directions, the net force is zero, which means the acceleration is zero.

This leads us to an issue that many people find quite difficult, and that is the effect of a zero net force on a moving object. To start with, consider again something floating in space — say, a heavy textbook floating inside a spaceship. Now imagine that two astronauts both grab one end of the coveted book and start pulling on it with equal forces. The book, of course, doesn’t move.

Now step back and imagine that this entire scene occurred inside a spaceship that is moving at a constant speed. At first, the book was obeying the simplest form of Newton’s First Law, because it had no forces acting on it — moving at a constant speed. When the astronauts started to pull, their two forces cancelled each other out, meaning that there was still zero net force on the book. Zero net force means zero acceleration, and so the book continued to move forwards within the spaceship, at the same constant speed. In other words, having two cancelling forces and a net zero force is exactly the same as having no forces at all.

We can use logic to work backwards now and say that if something is moving at a constant velocity, the net force on it must be zero (because “constant velocity” means acceleration is zero).

To finish this off, let’s go back to where we started at the beginning of the last section, which is with a book on Earth sliding across a table. If you push it with the perfect amount of force, you’ll find that it will reach a constant speed. You now know that if your push was the only force acting on it, the book would be accelerating. I’ve already told you that friction is the other force at work. If you’re thinking like a physicist, you’ve by now figured out that the reason the book is moving at a constant speed is because your force is exactly matching the force of friction. The two forces cancel out, leaving a net force of zero, which results in zero acceleration.

You’ve finally mastered Newton’s Second Law! You can now push the gas pedal in your car while blowing down the highway and know that the reason your car is accelerating forwards is that the force provided by your engine is greater than the force of friction from the road combined with the air resistance pushing on your car, resulting in a net force that gives the car a forwards acceleration. If you notice your speedometer levelling off at 170 km/h even though you’re still pressing down on the gas, you can tell your terrified passenger that the forces of friction are now equal to the force of your engine. And if that passenger sighs a desperate sigh of relief as the speedometer starts to fall again, you can say that the engine must have overheated and cannot provide any more force, so that the net force on your car is backwards, and the car is accelerating in reverse as it slows gently to a stop.[1. Always wear your seatbelt while learning physics.]

Speeding car
If your parents drive too fast, they might turn into pigs, forcing you to work in a bath house in order to try to rescue them.

There’s one more question you might have, and it’s an excellent one, so good for you for thinking of it. The question is this: Why? That is, why does a force cause an acceleration, and why does mass resist that acceleration?

The answer to the first question is perhaps more simple. Forces are, arguably, defined by Newton’s Second Law. The only evidence that a force ever exists is that it causes something to accelerate; force is just the name we give to interactions which cause acceleration.

The second question is far more interesting. The easy answer is to say that if you are trying to push Object A, which has twice as much mass as Object B, you are essentially trying to push two Object-B’s, which should therefore require twice as much force. But the more fundamental problem remains: why does Object B resist the force in the first place? We know that things that have no mass, such as photons, constantly move around at top speed without being pushed; why doesn’t matter do that?

Here we’ve stumbled upon one of my favourite kinds of physics questions: the ones that don’t have an answer. Matter resists acceleration because… it does. (See also this great Richard Feynman video.)

In fact, it is hard (or perhaps impossible) to imagine even a hypothetical reason why matter might resist changes in movement, just as it is hard to imagine there being a reason why space changes shape around massive objects or why the universe is bound by physical laws at all. That’s not to say that answers to any of these questions might not be found someday, but for now, those mysteries will remain. One of the true joys of physics is knowing enough to understand exactly where the boundaries of our understanding may lie.

Big Ideas

  • The net force is the result of all of the forces acting on an object. If two equal forces act on the same object in opposite directions, they cancel each other out, leaving zero net force.
  • An object moving at a constant velocity has zero acceleration and therefore zero net force acting on it. This can include objects that are being pushed forwards; if such an object is still moving at a constant velocity, there must be other forces counteracting this forward push.
  • On Earth, applied forces often result in constant speeds; this occurs when friction acts to cancel out the applied force.

Next: 2.7 – Newton’s Third Law

Previous: 2.5 – Newton’s Second Law, Part 1: Effects of One Force