Force and acceleration

- Welcome back. So now we know if a net force is acting on a particle then it will accelerate in that direction. By how much will it accelerate? To answer the question of how force and acceleration are related, Newton observed that if you increase the net force by, say, a factor of two, then the acceleration increases by that same factor. This means that force and acceleration are proportional to one another. But that's not all that matters. Next, let's consider the mass of our particle. Imagine we have two particles floating in space, which are the same size but have different masses, like if one is a ping pong ball and the other is made of lead. If we applied an equal force, like wind, to both particles, what would happen? Both particles would experience the same net force in the direction of the wind, but they wouldn't accelerate at the same rate. The less massive particle, the ping pong ball, would accelerate faster than the one made of lead. So less mass results in more acceleration and more mass results in less acceleration, meaning that mass and acceleration are inversely proportional to one another. And we already know that acceleration is proportional to force. Putting these together we see that acceleration depends on the magnitude of net force, which is proportional to acceleration, and the mass of the object, which is inversely proportional to acceleration. This gives us a is proportional to f divided by m. Multiplying both sides by m gives m times a is proportional to f. And if we flip this, we get f is proportional to m times a. Newton found that f isn't just proportional to ma, it's in fact equal to ma. This is Newton's second law, f equals ma. To recap, f is the net force acting on the particle, m is the mass of the particle, and a is the acceleration of the particle. Now let's consider the force of gravity. You made have heard of the famous story about Galileo's experiment in 1589, where he dropped two balls off the Leaning Tower of Pisa. One was made of a light material, the other a heavy material. You might be surprised to know that he observed that the two balls accelerated at exactly the same rate. That blew everyone away. At the time, everybody, starting with the ancient Greeks, just assumed that heavier objects fell faster than lighter objects. So unlike wind, the force of gravity seems to be independent of mass. The interesting question is why. Newton gave us the answer. His first law of gravity said that more massive objects experience greater gravitational force and his second law says that mass is a resistance to acceleration. These two competing trends, one encouraging acceleration and one resisting it, cancel each other out. To see why this happens mathematically, Newton theorized that force due to gravity, call it big F, is proportional to the mass of the particle. Big F is proportional to ma. Think of gravity as an acceleration vector, call it g, such that big F is equal to mg. So we have two equations. Newton's second law, little f is equal to ma where little f is the net force and Newton's law of gravity where big F is equal to mg. For a particle being acted on by only gravity, the net force little f is big F. Little f is equal to mg is equal to big F is equal to ma. Or more simply, mg is equal to ma. Notice the m cancels, leaving just g is equal to a. That is, the acceleration of a particle, when acted on only by gravity, is independent of the mass of the particle. This is why objects of different mass fall at the same rate. An equation like this one, that allows us to compute the acceleration of particles, is called an equation of motion. We've covered a bunch of new and important concepts in this video. So let's stop here for some practice, using the next exercise.