Newton's second law | Physics | Khan Academy

Today in the gym, when my wife was doing dumbbell curls, I started wondering. See, she's putting a force on that dumbbell upwards, right? But does that force stay constant as she moves the dumbbell up, or not? Does it change? And if it does change, how does it change? Does it increase? Does it decrease? What happens to it? Guess what, we can answer this question by the end of this video using Newton's Second Law.

So, let's start with a simpler example. We have an ice hockey ground over here, and there's a puck moving on top of it at some speed. If there are no frictional forces acting on this, if we assume that, then the forces acting on this puck would be balanced. I mean, because in the horizontal, you can see that there are no forces because we're ignoring friction, and in the vertical, the gravitational force which is pulling down on it is completely balanced by the force that the ground is pushing up on it with the normal force. They balance it out, and so since there are no unbalanced forces acting on this puck, from Newton's first law, we know that this thing will continue its state of rest or, in this particular case, the state of uniform motion. So, it'll continue to move with that same velocity.

But now comes the question: What if there was an unbalanced force acting on it? What happens because of that? Well, let's find out. For that, let's just whack it with a hockey stick now. So, if I whack it to the right, let's say in this case, I will now put an unbalanced force to the right. What will happen? Well, you can probably guess it: that puck's velocity will now be higher; it'll just get blasted off over there. So, its velocity will increase. In other words, it will accelerate. Ooh, this means when there's an unbalanced force acting on an object, in other words, if there is a nonzero net force acting on an object, which is the same thing as saying an unbalanced force. But whenever this net force acts on an object, what does it do? It accelerates our puck. The puck undergoes, or the object undergoes, an acceleration. This is the essence of Newton's second law.

Now, all we got to do is analyze the situation even more carefully and see if we can concretize this relationship. So, let's do that. The first question we could have is, yes, so a net force causes an acceleration, but how long does that acceleration last? Well, let's see. When the stick hits the puck, that's when it starts accelerating, which means as long as the stick is in contact with the puck, as long as it's in contact with it, like right now over here, during that time, there will be acceleration. But what happens once it loses contact? Well, once it loses contact, again that force goes to zero, and now going back to Newton's first law, it'll continue moving with that same increased velocity. This means the acceleration only happened during this time when the hockey stick was in contact with it. In other words, the acceleration lasts as long as the net force lasts.

Okay, next, let's think about what would happen if the net force was higher. For that, let's imagine we whacked it harder. What's going to happen now? Well, you can imagine it’ll get blasted off even faster, right? Which means it'll have a higher velocity when it loses contact. Ooh, that means there'll be bigger acceleration. So, if the net force is larger, it means you'll have a larger acceleration. If the net force is smaller, you get a smaller acceleration. In other words, we see a direct relationship between acceleration and the net force.

All right, what else can we deduce? Hey, let's think about the direction. What is the direction of the acceleration? Well, in this case, the net force is to the right, and our puck's velocity is also increasing towards the right. So, in this case, the acceleration is to the right. So, in this case, if the net force is to the right, the acceleration is to the right. What would happen if the net force was to the left? So, let's imagine we whack that puck now to the left. What would happen? Well, we can again imagine the puck would now get blasted off to the left.

But let's look at it carefully. Since the puck is already moving to the right, if we push it to the left now, we're going to slow it down. The puck will come to a stop; first it'll happen very quickly that we won't even see it, but it has to happen before going to the left. Right? Which means when you go from here to here, notice even though the puck is moving to the right, it is slowing down.

This means the acceleration is to the left. So, when the net force is to the left, we're seeing acceleration is to the left. After that, its velocity might increase to the left, which means again the acceleration is to the left. Ooh, so if the net force is to the left, the acceleration is to the left. If the net force is to the right, the acceleration is to the right. So, the acceleration will be in the same direction as that of the net force.

Okay, is there anything else that affects our acceleration? Well, let's see if we come back over here. What if you use the same bat, whacked it with the same force, but instead of a puck, let's say there was a bowling ball moving with the same velocity? What would happen now? I'm pretty sure you can feel it in your bones now; it would be much harder to stop that bowling ball and make it turn backwards, right? I mean, the same thing will happen; you will slow it down, but it'll be much, much harder. It'll take a much longer time to slow it down even though you're putting the same amount of force.

So, the net force has stayed the same, but what has happened to our acceleration? Since the velocity changed over a much longer time, the acceleration became smaller. Hey, why did the acceleration become smaller? What changed from the puck to the bowling ball? The mass changed; the mass increased. So, this means mass also affects the acceleration. But how does it affect it? Well, we saw that the mass increased right now. What did that do to the acceleration? It decreased.

And this is kind of intuitive. The bigger the mass, the harder it is to accelerate, meaning the smaller the acceleration, which means acceleration has an inverse relationship with the mass. So now, everything that we just analyzed about acceleration, its direction, its dependency on the net force, how it depends on the mass, all of it can be put down in an equation. That equation is pretty much right in front of us.

So, the acceleration will equal the net force divided by the mass. This is our Newton's Second Law. And look, the equation is saying the same thing: a direct relationship between acceleration and the net force, an inverse relationship between acceleration and the mass. And the arrowheads are saying that acceleration and the net force will always be in the same direction. Isn't it amazing that we can pack all of that information in just one beautiful equation?

And of course, you may have seen this as written as F=ma in some sources. It's the same thing. I like to write it this way because, you know, acceleration is caused by the force. So, once we decide the forces and the mass, then the acceleration gets fixed. But anyways, what will happen if the net force is zero? What if we plug in over here zero? Well, then the acceleration also goes to zero.

What does this mean? Well, this means we have all the balanced forces acting on an object. And if the acceleration is zero, it means that the velocity stays constant. In other words, this is Newton's first law, which says an object continues to stay at rest or in uniform motion; that is zero acceleration, right? When there are no unbalanced forces acting on it. So, notice Newton's first law is just a special case of Newton's Second Law, which means this equation is encompassing both the second and the first law as well.

And finally, speaking about Newton's first law, what we also noticed over here is the bigger the mass, the smaller the acceleration. In other words, if the mass is bigger, it is harder to change its velocity. It's much harder to do that, which means objects that have more mass have more inertia. That's something again we learned in Newton, right? Inertia is the property due to which objects continue to stay at rest or continue to stay in uniform motion, isn't it? It can fight acceleration.

And we can now see what inertia depends on. Inertia is the mass. More the mass of an object, more the inertia, harder it is to accelerate. Newton's Second Law could arguably be the most important equation of all of classical physics. I say classical physics because we now know that if objects are moving very close to the speed of light, then this breaks down. It doesn't work; now we'll have to resort to Einstein's theory of relativity.

On the other extreme, if we consider extremely tiny particles, like, you know, subatomic particles like electrons, protons, and neutrons, even over there, it turns out Newton's Laws don't work. So even over there, it breaks down. But as long as we don't go to such extremes, this equation will work for us.

So now, let's see if we can apply this to our original question. When she just started moving the dumbbell, the dumbbell's velocity was increasing. After that, let's say there was a small phase during which the velocity was constant. And finally, when the dumbbell is about to stop, its velocity is decreasing.

So now the question is how do we figure out what happens to the force that she's putting on the dumbbell? Well, let's apply Newton's Second Law for that. Let's first think about the acceleration. Well, over here, we're dealing with increasing velocity; therefore, the acceleration is upwards in the same direction as it's moving. Then we have a constant velocity, which means the acceleration is zero. Finally, we have a decreasing velocity since the dumbbell is still going up, decreasing velocity, which means acceleration must be down in the opposite direction.

Now, because we know the direction of the acceleration, we can figure out the direction of the net force. It has to be exactly the same: upwards here, zero here, downwards here. We're applying Newton's Second Law, the direction part over here.

Now, finally, this is the direction of the net force we want to know. We want to know what happens to the force that we are putting on the dumbbell, or she's putting on the dumbbell, actually. How do we do that? Well, let's look at all the forces acting on the dumbbell. Well, we know that there's a gravitational force acting on the dumbbell all the time. That force is a constant, and therefore, our force is in the opposite direction of the gravitational force.

Now, in this case, when she's just lifting the dumbbell, if her force, if the net force needs to be upwards, that means her force must be larger than the gravitational force. Only then her force will win out, giving a net upward force. Right? Okay, what about over here? We want the net force to be zero. Over here, how can that happen? Ah, her force has to be exactly the same as gravitational force because then, only then, the forces get balanced.

Finally, what happens over here? Well, we want the net force to be downwards, which means we want gravity to win. That means her force must be smaller than the gravitational force. Look, even though we did simplify it a little bit, I mean, I'm not really sure that her dumbbell was moving at a constant velocity, but once we simplified a little bit, we were able to analyze what happened to the force that she was putting on the dumbbell. It went on decreasing as the dumbbell moved upwards. Isn't that incredible how we use Newton's Second Law to do that? Amazing, isn't it?