Part C: Concept Checks

Transcript

0:00Introduction

Hello everyone! Okay, a quick question before we start. You have seen astronauts floating inside a spacecraft, with their pens and water drops drifting around them. Why do they float? Most people answer at once: there is no gravity up there. Is that really true? Hold that thought; we settle it in Example one. This is Part C of the unit problem bank: five concept checks. Your course says these ideas are often tested in statement form, true or false. No long calculation, but one wrong idea is enough to lose the marks. So, let's begin.

0:37Check 1: does orbital speed depend on mass?

So, check one. Read it with me: true or false, the orbital speed of a satellite depends on its mass. Heavier satellite, different speed? What do you say? Let's check with the formula. For a circular orbit of radius small r, measured from the planet's centre, the orbital speed v o, with a small o written below, is the square root of capital G M over r. The tick-shaped sign is the square root, and capital G is the gravitational constant. Okay? Now look closely. Capital M is the planet's mass. And the satellite's own mass, small m? It is not there at all! The speed depends only on the planet's mass and the radius. So the statement is false. Right? Now, one more thing from the same formulas. The period, capital T, has no small m either. But the energies, and the angular momentum, capital L, do carry small m. Statement questions often mix up these two lists, so keep them apart.

1:40Check 2: a geostationary satellite

Clear? Now, check two, about a very special satellite. Your course says a geostationary satellite has a period, capital T, of twenty four hours; the small h is for hours. It moves from west to east, and stays above one point on the Earth. With me? So, here is the check. Can a geostationary satellite stay above New Delhi? Twenty four hours, same as the Earth, same direction. Why not, right? No, no, careful! The answer is no. The orbit of a geostationary satellite must lie in the equatorial plane, the flat plane through the equator. Got it? Why? In any other plane, the orbit is tilted to the equator, so the satellite would appear to move north and south, never fixed in the sky. And New Delhi is well north of the equator, so no geostationary satellite can stay above it. Okay?

2:34Check 3: at the centre of the Earth

Alright, check three: the centre of the Earth. True or false: small g is zero at the centre, but the gravitational potential, capital V, is not. The equals sign with a slash through it means not equal to. So think: if there is no field at all, can there still be a potential? Let's check. Treat the Earth as a uniform solid sphere, as the course does. Inside, small g equals capital G, capital M, small r, over capital R cubed. Capital M is the Earth's mass, capital R is its radius, and small r is the distance from the centre. At the centre, r is zero, so small g is zero. That part is correct. Right? Now the potential inside: minus G M over two R cubed, times the bracket, three R squared minus r squared. Look at the graph. The solid curve is our solid sphere, and it keeps going down as we move in, all the way to the centre. With me? Put r equal to zero. V at the centre is minus three G M over two R, one and a half times the surface value. And, in the bracket on the right, G M equals g R squared, so this is minus three halves g R, not zero at all! Clear? So the statement is true. Yes, exactly! Zero field only means the potential curve is flat there; see how it levels off at the centre. It does not mean the potential is zero. Same with the dashed line, a thin shell: inside, the field is zero, but the potential is constant, not zero. Got it?

4:11Example 1: are astronauts weightless because g is zero?

Now, remember our floating astronauts? Here they are, example one. True or false: astronauts in an orbiting spacecraft are weightless because gravity is zero there. Pause the video, and decide. Done? Okay, let's check. Is gravity zero up there? No! Your course says it is only slightly less than at the surface. In fact, gravity is what holds the craft in its orbit. Without it, the craft would fly off in a straight line, right? So gravity is very much there. Then why do the astronauts float? Because the astronauts and the craft are in free fall together. Both fall with the same acceleration, all the time. Okay? Remember the lift from laws of motion? In a freely falling lift, a weighing machine reads the normal force, capital N, and that reading is zero. Nothing pushes up on you, so you feel weightless. The spacecraft is just like that lift. With me? So the statement is false: weightless because of free fall, not because gravity is zero. Puzzle solved! And look out for the word because in a statement; very often, the trap hides right there. Perfect!

5:27Example 2: does air drag slow a satellite down?

Now a harder one, example two. True or false: a satellite losing energy to air drag slows down. Sounds obvious, doesn't it? Pause the video, and decide. Done? Most students think: drag slows down a car, drag slows down a ball, so of course it slows down a satellite. True, right? No, no, wait! Use the energies. In a circular orbit of radius r, the total energy, capital E, is minus G M m over two r, where small m is the satellite's mass. The kinetic energy, capital K, is G M m over two r, which is exactly minus E: the same size, with the opposite sign. On the graph, K is the top curve, E is the middle one, and the potential energy, capital U, is the lowest. Okay? Now, drag takes energy away, so E falls; it becomes more negative. Look at the E curve: it is more negative only at a smaller r. The double arrow on screen is read as: so. So the satellite drops to a lower orbit. Right? And since K equals minus E, when E becomes more negative, K becomes bigger! More kinetic energy means more speed. The satellite speeds up! Surprising, isn't it? So the statement is false: it moves to a lower orbit, and speeds up. This is a classic trap. Drag does take energy away, but the satellite falls lower, where it moves faster. And the reverse is also true: supplying energy moves a satellite outward, and slows it. Very good!

7:07Quick revision: five gravitation concept checks

So, quick revision. Orbital speed and period do not depend on the satellite's mass; the energies and angular momentum do. Geostationary means twenty four hours, west to east, and the equatorial plane only. At the centre of a uniform Earth, small g is zero, but the potential is minus three halves g R, not zero. Astronauts float because of free fall, and air drag sends a satellite to a lower orbit, where it moves faster. Now try the practice problems on this page, and read every because very carefully. All the best!