Momentum and the Second Law

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Picture a block, released from rest, half a metre from a point, and pulled back towards it by a force. But the force keeps changing: the closer the block comes, the weaker the pull. How fast is the block moving when it reaches the point? Your favourite equation, v squared equals u squared plus two a s, needs a constant acceleration. Here, the acceleration changes every moment. Today's tool handles this: Newton's second law, written with momentum. It comes up often in JEE physics. So, let's begin.

0:41Momentum: mass times velocity

So, first a question. Picture a loaded truck and a bicycle, moving at the same speed. Which one is harder to stop? The truck, of course! To stop it in the same time, you need a much bigger force. So mass and velocity both matter. Together, they give linear momentum. p vector equals m times v vector: mass times velocity, okay? The small arrow on top of p and v says, this is a vector. Mass is positive, so momentum points the same way as the velocity. Right? Now, the unit: kilogram metre per second. The small minus one up there means, per second. And the same unit can be written as newton second, the capital N with an s. Clear?

1:27Newton's second law

Now, the main law. Newton's second law: F net vector equals d p over d t. The small word net, written below the F, means the total of all the forces on the body, okay? The upright d means a tiny change, so d p over d t is the rate of change of momentum: how fast the momentum changes with time. So force is about how fast momentum changes, not about momentum itself. Got it? Now, a question for you. Momentum is m times v vector. If the mass stays the same, what does d p over d t become? Yes, exactly! The constant mass comes out, and we get m times d v over d t. And d v over d t is the acceleration vector. So F net equals m a. But careful! This form needs a constant mass. If the mass itself changes, go back to d p over d t. With me? Next, a useful fact. The law holds for each component separately: F x equals m a x, and F y equals m a y. The small x and y written below are subscripts: the part along x, and the part along y, okay? Topper's trick: resolve every force, and solve each direction on its own. Now, the newton itself. One newton gives one kilogram an acceleration of one metre per second squared; the minus two up there means, per second squared. And the weight of a body is W equals m g: mass times g, the acceleration due to gravity.

3:05When the force keeps changing

Clear? Alright, now the most useful part. In many problems, the force keeps changing. Then, can we still use v equals u plus a t? No, no! That equation needs a constant acceleration. If the force depends on time t, then d v equals F over m, times d t. Velocity on one side, time on the other, okay? Now, if the force depends on position x. Here is a neat trick: the acceleration can also be written as v times d v over d x. So, v d v equals F over m, times d x. Now every v is on one side, and every x on the other. Right? And if the force depends on velocity, like air drag. F, then v in brackets, means the force written in terms of v, okay? Then m d v over F of v equals d t, for the time. Or m v d v over F of v equals d x, for the distance. So the rule is really simple. See what the force depends on, pick the matching form, separate the variables, and integrate both sides. Got it?

4:14Example 1: average force from a change in velocity

Now, Example one. The velocity of a five kilogram body changes from three i hat plus four j hat, to six i hat plus eight j hat, metres per second, in three seconds. i hat and j hat, with little caps on top, are unit vectors along x and y, okay? Find the average force on it. Pause the video, and try it first. Done? Okay, let's check. Force is the rate of change of momentum. So, which do we need: the final velocity, or the change in velocity? The change, correct! Delta v vector, with the little triangle, means final minus initial. Six minus three is three, and eight minus four is four. So delta v is three i hat plus four j hat. Now its size. The two straight bars mean the magnitude. Root of, three squared plus four squared, is root twenty five: five metres per second. This change takes three seconds. So the average acceleration, a with the small av for average, is five over three metres per second squared. Right? Now, the second law. F average equals m times a average: five times five over three, that is twenty five over three newtons. The wavy equals sign means approximately: about eight point three newtons. Oh no, careful! Students often stop at five over three. That is only the acceleration. Multiply by the mass! And the direction. The force acts along the change in velocity, delta v, that is along three i hat plus four j hat. Perfect!

6:04Example 2: speed under the force F = -8x

Now, our puzzle from the start, with numbers. The body has mass two kilograms, and moves along x under the force F equals minus eight x, in S I units. The minus sign means the force always pulls it back towards x equals zero, okay? It is released from rest at x equals zero point five metres. Find its speed as it passes x equals zero. Pause the video, and try it. Done? Okay, let's check. At the start, the acceleration is minus eight times zero point five, over two: minus two metres per second squared. So, with v squared equals u squared plus two a s, over half a metre, v comes out as root two, right? No, no, wait! That is the trap. As the body moves in, the force gets weaker, so the acceleration is not constant. The force depends on x, so we use the position form. v d v equals F over m, times d x. F over m is minus eight x over two, so we get minus four x d x. Now integrate both sides. The tall S shape is the integral sign, okay? On the left, v goes from zero, because it starts from rest, up to v. On the right, x goes from zero point five, where it starts, to x. With me? The left side gives v squared over two. The right side gives minus two times, x squared minus zero point two five. The double arrow means, so. Multiply by two: v squared equals four times, zero point two five minus x squared. Clear? Now put x equals zero. v squared equals four times zero point two five, that is one. So the speed, the size of v, is one metre per second. Its direction is towards minus x, but the question asks only for the speed. Puzzle solved: a changing force, and still just a few lines. Very good!

8:09Quick revision: p = mv, F = dp/dt, changing forces

So, quick revision. Momentum is m v. Force is the rate of change of momentum, which is m a when the mass is constant. The law works for each component separately. And when the force keeps changing, match it to time, position or velocity, separate the variables, then integrate. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!