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

Critical Questions:

  • How does something move when it’s being pushed?
  • What would it take to move the Death Star?

Newton’s First Law is all about what happens when there are no forces acting on an object. It sets everything up for us to be able to really understand everything else. But the second law is the one that gets the most press, because it tells us what forces actually do, and it even gives us a mathematical formula to let us calculate the results of various forces.

Now, I know I promised not to bring math into this blog, but this is going to be one of the exceptions, because (a) it’s arguably the most important equation in the field of physics, and (b) it’s super easy to understand.

We already know that forces move things around. As Newton’s First Law tells us, things don’t just spontaneously start moving on their own; there has to be a force that causes them to change their motion.

Picture yourself in the driver’s seat of a car. You push down on the gas pedal, and the car starts to move forwards. You push harder and pick up speed. At some point, you might ease off the gas pedal a bit and keep the car moving at a constant speed down a straight stretch of road. And when you want to slow down, you might lift your foot off the pedal for a while before using the brakes.

batmobile
Unless you own the batmobile and you NEVER BRAKE FOR ANYBODY

This is actually a complicated series of events. But Newton’s Second Law is powerful enough to explain all of it — not the inner workings of your car, necessarily, but the relationships between the amount you step on the gas and the car’s changing speeds.

In order to help sort this all out, though, I’m going to once again bring up some preconceptions you might have about the effects of forces. So leave the car behind for a moment and imagine something heavy sitting on a flat surface, and imagine yourself pushing on it with a constant amount of force for a little while. If you’ve got a heavy book or a laptop nearby, use that (if you push your laptop off your desk, don’t sue me).

What will most likely happen to the object is that it will move forwards at a constant speed. But if you’re thinking like a physicist and you didn’t skip the previous section on Newton’s First Law, you’ll know that friction has gotten in the way of our little experiment and affected its results. In other words, we still can’t say what the effect of just your force is, because in the meantime, there’s another force counteracting it.

So grab that solid object you were puhsing, strap on a Jetsons-style personal jet-pack, and head out into the profound emptiness of interstellar space. Push the object forwards a bit, then let go. We already know from our previous experiment that the object will move at a constant velocity. Next, give it another gentle push for about three seconds. The object will now be moving faster than before — it will have undergone acceleration. And the cause of that acceleration was the force you provided.

When you aren’t pushing, the object’s velocity doesn’t change, but while you are pushing, it accelerates.

And here we’ve arrived at the first idea to be found in Newton’s Second Law: a constant force does not result in a certain speed, but in an acceleration. In other words, whenever one force is being applied to an object, that object will accelerate in the direction of the force.

If you fully consider this scenario all the way through, you might imagine that at some point it will become more difficult to push the object forwards or that the object’s acceleration will start to decrease with time. There are two possible explanations for such a thought. The first is that you are still thinking of friction, even though you are in outer space. Friction does behave a bit like this, for reasons which will be discussed shortly, but once you remove friction from the equation, you find that an object’s speed can keep increasing forever even with the smallest force, provided that that force is applied for a long enough time.

The other reason you might be thinking that way is that you understand something of Einstein’s Theory of Relativity, which puts a speed limit on a moving object. But we’re still working in the area of classical mechanics, so we can leave relativity on the sidelines for now.

So we now know that if an object is pushed forwards, it will accelerate forwards. We can easily extend this rule to forces acting in other directions. Consider an object moving forwards, and push it in the other direction — it will slow down, which we already know is another kind of acceleration.

Now consider another object moving forwards, but push it to the right. It might not slow down or speed up, but it will certainly turn to the right. You’ll recall that in the previous chapter, we described this behaviour as the third type of acceleration, and now you can see another reason why: acceleration is always the result of a force, and one possible result of a force is that an object’s direction of motion changes.

Newton’s Second Law also says that if you apply a bigger force to an object, it will experience a greater acceleration — its speed will increase more quickly, or decrease more quickly, or the object will turn more sharply. Specifically, the amount of force varies directly with the amount of acceleration. For those of you who don’t remember painstakingly drawing out various curved and straight lines on graph paper in high school, this simply means that if you apply twice the force you get twice the acceleration, three times the force gives three times the acceleration, and so on.

Let’s do another experiment in space to complete the picture. Imagine two objects, one that is very big and heavy and another that is small and light. Actually, let’s take it to extremes[1]: imagine an empty cereal box on the left and that enormous spaceship from Star Wars[2] on the right. Next, push on both of them with the exact same force and watch what happens.

deathstar
What happens is that Darth Vader comes out and yells at you for pushing his Death Star.

Many people seem to find the results of this experiment difficult to imagine, and a common guess is that since this is taking place out in space, with no air resistance or other friction, it will be just as easy to push the spaceship as the cereal box. This is the same as saying that with an equal force, both will experience the same acceleration. This is incorrect, and it is another result of our lack of familiarity with low-friction scenarios.

The correct answer is that during your push, the Death Star will accelerate very slowly (almost not at all), while the cereal box’s speed will increase dramatically. In order to convince yourself that this is true, try to imagine anything you’ve seen with low friction, such as a huge, oil-covered, non-stick frying pan or very smooth ice. Picture the fattest person you know standing on an ice rink next to the smallest person you know. Both are wearing skates, but neither one knows how to skate. Which one would you rather push around for an hour or so? Unless you’re passionately in love with the fatter one and want to show them a good time, you would most likely want to push the smaller person. That’s because it will be much easier to get person number two up to a decent speed. In other words, with the same amount of force, person number two will accelerate more than person number one.[3]

Here we can make use of the term mass, which just means the amount of matter in an object.[4] The relationship between mass and acceleration is an example of an inverse relationship: the more mass you have, the less acceleration you get from an applied force. When you combine these two factors, you have the formula for Newton’s Second Law, which defines the acceleration a of an object of mass m due to a force F:

Newton's Second Law
Figure 1: Oh god, fractions

Congratulations, you just mathed! It wasn’t so bad, was it?

This is the effect of one force only: an acceleration that depends on the mass of the object being moved. In the next section, we’ll look at what happens when multiple forces affect the same object.

Big Ideas:

  • A constant force acting on an object will cause that object to constantly accelerate in the direction of the force.
  • The more mass an object has, the less it will accelerate due to an applied force.
  1. This is always a good idea when working out physics problems: imagine the most extreme version of the situation. Not extreme in the jumping-a-skateboard-over-a-helicopter sense, but more literally.
  2. Either the one that goes across the screen right at the very beginning or else the Death Star. It’s really up to you.
  3. It’s true that friction is involved here, but there is less friction than usual, and so it remains a useful experiment.
  4. For now, you can think of it as an object’s weight. These are actually two different things, but the difference isn’t really important at the moment; it’ll be explained in the chapter on gravity.

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