Newton's second law calculations | Physics | Khan Academy

Let's solve a couple of problems on Newton's Second Law.

Here's the first one: we have an elevator which is moving up, and let's say the mass of the elevator, including the passenger inside, is 1,000 kg. Now, if the force, the tension force of the cable, let's say that's about 7,800 Newtons, our goal is to figure out what the acceleration of this elevator is.

We're also given that the gravitational force acting on that elevator, including the passenger over here, is about 9,800 Newtons. So, how do we figure this out? Well, the first thing that comes to my mind is, hey, we have some forces and we have some motion variables like acceleration. What connects forces and motion variables? What connects forces and motion? Newton's Second Law.

So, the first thing I try to do before applying Newton's Second Law is I try to draw a free body diagram. So let me do that. What's the way to draw a free body diagram? Well, we try to get rid of unnecessary details. You use a box to represent your object of interest. Our object of interest is this elevator and the person over here, so that's our box. Its mass is 1,000 kg, and let's draw all the forces acting on it.

Which are the forces acting on it? Well, we have an upward force, that's tension, and we have a downward force, that's the force of gravity. Our goal is to calculate what the acceleration is, and how do we do that? Well, we use Newton's Second Law, which says the acceleration should equal the net force acting on an object divided by its mass. Of course, since we're dealing with vectors, we can put arrow marks over here. The direction of the acceleration will be the same as the direction of the net force.

Okay, now we can calculate the net force from this, and we can calculate, we know the mass. So from that, we can calculate the acceleration. So why don't you pause the video and see if you can plug in the numbers and find the acceleration yourself first?

All right, let's try. Our acceleration would be the net force. How do I figure the net force out? Well, the total force, since in the opposite direction, we subtract them. So I'll just take the bigger number, 9,800 Newtons, which is acting downwards. We need to take care of the direction. From that, I'll subtract the smaller number that is 7,800 Newtons upwards, divided by the mass which is 1,000 kg.

If we simplify, we will get 9,800 - 7,800 equals 2,000. Nice numbers, 2,000 Newtons. But what direction is it? The downward one wins, right? It's bigger. So the downward net force will be in the downward direction divided by 1,000 kg. That gives us 2, so our acceleration becomes 2 m/s² downwards.

Okay, we found our answer, but one of the best ways to gain deeper insights is to try and see if this kind of makes sense. Can you get a feeling for what's going on over here?

Okay, now the first question that could arise over here is: look, the net force is downwards because gravity is winning, right? So the total force acting on this elevator is downwards, and yet the elevator is moving up. Why is that? Well, remember, force does not dictate the direction of motion. Force tells you the direction of acceleration.

Okay, the elevator could be moving; objects can be moving whatever direction they want. When you put a force on it, it tells you what direction it should accelerate. So that's the key thing. So there's no problem that the force is acting downwards, but the elevator is moving upwards.

Secondly, what does it mean that the acceleration is downwards? See, if the net force is downwards, the acceleration has to be downward. But what does it mean? The elevator is moving up, and the acceleration is downwards. Ooh, this means, since the velocity and the acceleration are in the opposite direction, this means the elevator is slowing down.

That's what it means for velocity and acceleration to be in the opposite direction. If they're in the same direction, it means they're speeding up. So this means our elevator is going up, but it is slowing down, which probably means that, you know, it's probably about to stop. Maybe this person has reached their destination or something.

All right, on to the next problem. This time we have a sledge at rest whose mass is 70 kg. Our goal is to push it and accelerate it to 6 m/s in about 2 seconds, let's say, so that, you know, it can nicely slide down and we can enjoy the ride.

Now, there is going to be some frictional force. Even though we're on ice and everything, there will be some frictional force. Let's say the friction that this sledge will experience is about 200 Newtons. The question now is: what is the force with which we have to push on it so as to achieve all of this?

So how do we figure this out? Well, again, the first thing that comes to my mind is that, look, we're dealing with forces and we have motion. What connects forces and motion? Newton's Second Law. So the first thing I'll do is I'm going to draw a free body diagram again. I encourage you to pause the video now or anytime later on whenever you feel more comfortable, pause the video and see if you can complete it yourself.

Okay, so anyways, let's first try to draw a free body diagram. How do we do that? Well again, we'll take the object of our interest; in this case, the object of our interest is this sledge, whose mass is 70 kg, and just make it a square. Then draw all the forces acting on it. What are the forces acting on it? I know there's a frictional force acting backwards of 200 Newtons, but then I also have the applied force by this person, which I need to figure out. This is what we need to calculate—that is, the applied force.

What are the other forces? Well, I know that there's also gravity acting on it, and then there's a normal force acting on it upwards. But we know that these forces are all balanced, and so they're not going to be useful. They're not going to affect our situation over here, so I'm just not going to draw them over here. They're there, of course, but they're completely balanced; they're in the vertical, so they're not going to affect it, so we'll not draw it.

Okay, now that I have my free body diagram, let's go ahead and write down Newton's Second Law which says acceleration should always equal the net force, which let me draw using pink now, net force divided by the mass.

Okay, so what do we do next? I need to find this applied force. So this means if I can calculate the net force, then I can figure out what the applied force is, right? So, from Newton's Second Law, I just need to figure out what the net force is. For that, I ask myself: do I know m? I do; I know that m is 70 kg. Do I know the acceleration? Hmm, it's not given directly, but wait a second, I know the initial velocity, I know the final velocity, and I know that the change in velocity should happen in 2 seconds.

Oh, that means I can calculate the acceleration from this data. I can plug in, and from that, I can figure out what the total force is going to be. The net force is going to be, and from that, we can figure out what the applied force should be.

Okay, so if you haven't done this before, why don't you now pause the video and see if you can put it all together and solve the problem?

Okay, let's do this. So let me first calculate the acceleration. So our acceleration is going to be—well, how do we figure this out? This is the final velocity minus the initial velocity divided by the time taken. So that's going to be, in our case, the final velocity is 6 m/s to the right minus what's the initial velocity? Well, it's zero; it's at rest, so there is no initial velocity, divided by time is 2 seconds.

That gives us what? That gives us 6 - 0 is 6, divided by 2 is 3 m/s² to the right. So I know my acceleration has to be to the right, which is good news. It has to be to the right; we want the sled to move to the right, so that makes sense. So we're on the right track over here.

Okay, now I can plug in and figure out what the net force is going to be. So if I just simplify that, the net force is going to be, if I rearrange this equation, multiply by m on both sides, so I get net force to be mass times the acceleration.

And so now I can plug in for mass and acceleration. What do I get? Well, I get 70 kg, that's the mass, time the acceleration is 3 m/s². 70 times 3 is 210, so I get that my net force is 210 Newtons to the right. Oh, this acceleration was to the right, so this will also be to the right.

So now that I found my net force, my net force, the total force, will be to the right, and that is 210 Newtons. From that, can I calculate the applied force? Yes. So now, first of all, I know my applied force should be bigger. It has to be bigger because my total force should be towards the right, which means if I subtract the two from this, if I subtract this number, I should get this number—that's the net force, right?

So now that I know all the directions and everything, I can just subtract the numbers. So I can now say, hey, my net force, that is 210 Newtons, should equal this bigger number—the bigger force minus 200 Newtons. Now, to get the applied force, I just add 200 on both sides so that I can get rid of this, and look, the applied force becomes 210 + 200, that is 410 Newtons.

And I already know it's to the right, so I know its direction. So if I were to write down its direction, it's going to be, oops, it's okay. Okay, I mean, okay, not the most organized board over here, but let me just write it down a little bit more neatly.

So, 410 Newtons to the right. That's how much force we need to apply for all of this to happen.