Part C: Concept Checks

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle before we start. Two identical balls move towards each other at the same speed. Both are clearly moving, so together they have kinetic energy. Now, what is their total momentum? Can moving bodies have zero total momentum? Hold on to your answer. Today we do quick concept checks on work, energy and collisions: one line questions, almost no calculation. Such questions do appear in JEE, and they test whether you really understand the ideas. So, let's begin.

0:36Kinetic energy and momentum: the signs

Let's warm up with the simplest check. Can kinetic energy ever be negative? And potential energy? Kinetic energy, never! Capital K equals half m v squared, with m the mass and v the speed. The mass is positive, and v squared cannot be negative. So K is greater than or equal to zero; that is the sign you see, greater than or equal. Okay? But potential energy, capital U, can be negative! We measure U from a zero that we choose ourselves, so depending on that choice, U can come out negative. So the answer: kinetic energy no, potential energy yes. Right? Now, back to our puzzle. Two equal bodies, with equal speeds, in opposite directions. What is their total momentum? Zero, exactly! Momentum, small p, is m times v, and it has a direction. The two momenta are equal and opposite, so they cancel: total p equals zero. Clear? But kinetic energy is a scalar: no direction, and never negative. So the two kinetic energies add, and total K is greater than zero; that is the greater than sign. Puzzle solved: a system can have kinetic energy with zero momentum! Got it?

1:57Does this force do work?

Now, work. For a body going round a circle, the centripetal force is perpendicular to the velocity at every instant. The upside down T sign on screen means perpendicular. So, does this force do any work? No work at all, correct! Only the part of a force along the motion does work, and here there is no such part, at any instant. So capital W, the work, is zero. With me? Next, static friction. There is no slipping, so its work is always zero, right? Think of a box resting on the floor of a truck that speeds up. Oh no, careful! Not always zero. The box does not slip, so the friction on it is static, and it is this friction that carries the box forwards. Seen from the ground, the box moves forwards, along the friction. So in the ground frame, static friction does positive work. Okay? Notice the words, ground frame. Seen from the truck, the box does not move at all. So work depends on the frame, because displacement does. Right? So in an accelerating frame, like the truck, the work energy theorem holds only if you add the work of the pseudo-force. Remember that condition. Clear?

3:14Collisions: what is conserved?

Now, collisions, and look at the figure. Two balls, masses m one and m two; the small numbers below the letters label the bodies. Top row, before: velocities u one and u two. Middle row, maximum deformation: they move together with a common velocity v. Bottom row, after: v one and v two. On the right, the momentum equation, then the restitution rule with the letter e, and the values of e for elastic, inelastic and perfectly inelastic collisions. Okay? First, an inelastic collision, with no external impulse. What is conserved? Momentum, yes! With no external impulse, momentum p is conserved, in every collision; that is the momentum equation in the figure. Got it? And total energy is always conserved. But kinetic energy is not; some of it is lost in the impact. So never use kinetic energy conservation across an inelastic impact. With me? Now, the trap. In an elastic collision, kinetic energy is conserved. So is K conserved at every instant, all through the contact? No, no, careful! Look at the middle row: maximum deformation. During the contact, the balls get squeezed, and part of the kinetic energy is stored as elastic energy of deformation. Clear? Then the balls spring back, and all of it is returned by the end. So in an elastic collision, kinetic energy is the same only at the start and at the end, not at every instant. Right? And now you know why a bouncing ball rises lower and lower. Each impact is inelastic, so at every bounce, some mechanical energy goes into heat, sound and deformation. Okay?

5:02Example 1: equal kinetic energies, which has more momentum

Example one, a quick one from the course. Two bodies have equal kinetic energies. Which has more momentum? Pause the video, and try it. Done? Okay, let's check. Same kinetic energy, so the lighter body must be moving faster. Faster means more momentum, right? Oh no, careful! Speed alone does not decide it. Momentum p is m times v, and kinetic energy can be written as p squared over two m. So p equals the square root of two m K; the tick shaped sign is the square root. Okay? Now K is the same for both bodies. So p is proportional to the square root of m; the open loop sign means proportional to. Got it? So the heavier body has more momentum. Yes, the lighter one is faster, but the heavier one wins on momentum. Perfect!

5:58Example 2: spring work from x to 2x

Now example two, a little harder. Which needs more work: stretching a spring from zero to x, or from x to two x? On the left of the figure, the top spring has its natural length, at the dashed line x equals zero. Below it, the spring is stretched by x, and pulls back with force k x, where k is the spring constant; that is why the formula below has a minus sign, F equals minus k x. Pause the video, and try it. Done? Okay, let's check. Both times, you stretch the spring by the same amount, x. So the same work, right? No, no, wait! The pull k x grows as the spring stretches. On the right is the graph of capital U, the stored energy, half k x squared. Pulling slowly from x one to x two, you do work W equals half k, times x two squared minus x one squared; the label beside the graph gives it. On the graph, this work is the rise in U, from x one to x two; the height gained, not the shaded area. The small one and two below x mark the start and the end. Okay? First stretch, zero to x; read the arrow as to. Half k, times x squared minus zero: just half k x squared. Clear? Second stretch, x to two x. Two x, squared, is four x squared. So half k, times four x squared minus x squared: that is three over two, k x squared. With me? Compare three over two with one half. From x to two x needs three times as much work! Same stretch, but the spring is already pulling harder all the way, so on the graph, the curve climbs more and more steeply. Perfect!

7:47Quick revision: KE, work by frame, what collisions conserve

So, quick revision. Kinetic energy is never negative, and a system can have kinetic energy with zero momentum. Work depends on the frame, and a force perpendicular to the velocity does no work. In an inelastic collision, momentum is conserved but kinetic energy is not; in an elastic one, kinetic energy is returned only by the end. Now go and try the practice problems on this page. Think before you calculate, and these traps will not catch you. All the best!