Energy of Satellites

Transcript

0:00Introduction

Hello everyone! Okay, a puzzle to start. Picture a satellite going round the Earth in a low orbit. Up there, there is still a little air, so air drag slowly takes energy away from it. So what happens to its speed? It slows down, obviously, right? Hmm, hold on to that answer. The truth surprises almost everyone. Today's topic, the energy of satellites, settles it, and it comes up often in JEE gravitation. So, let's begin.

0:31Kinetic and potential energy in an orbit

So, take a satellite of mass small m, in a circular orbit of radius small r, measured from the centre of the Earth, okay? On the graph, the horizontal axis is small r, and the vertical axis is energy. Three curves; we will build each one. With me? From the last concept, the orbital speed v o, with a small o written below, equals the square root of capital G M over r. Capital G is the gravitational constant, and capital M is the Earth's mass. Right? Now, the kinetic energy, capital K, is one half m v squared. Can you write it using capital G, capital M, small m and small r? Very good! Since v o squared is just G M over r, K equals G M m over two r. Positive, of course: the green curve, above the axis. Clear? And the potential energy, capital U, from chapter three: minus G M m over r, taking zero at infinity. Negative, so it is the dark blue curve, the lowest one. Got it?

1:38Total energy: always negative

Now, the total energy, capital E, is simply K plus U: G M m over two r, minus G M m over r, okay? One positive half, and one negative whole. So tell me, is the total energy positive, or negative? Negative, exactly! E equals minus G M m over two r. Look carefully: that is just minus K, and also U over two, half the potential energy. On the graph, the orange curve, halfway between the axis and the blue curve. Clear? So the ratio K to U to E is one, to minus two, to minus one; the colons mean ratio. Know any one of these energies, and you get the other two at once. With me? And because E is negative, the satellite is bound to the Earth. The energy needed to set it completely free is the binding energy: G M m over two r, simply minus E, right?

2:34Moving to a different orbit

Now, to move a satellite into a different circular orbit, the energy needed is E final minus E initial, with final and initial written small below, okay? Look at the orange curve on the graph. If the satellite moves out to a bigger orbit, does its total energy go up, or down? Up, yes! The orange curve rises towards zero as r grows, so we must supply energy. From r one to r two, with one and two written small below, and the arrow meaning to: minus G M m over two r two, plus G M m over two r one. Got it? Take out G M m over two: G M m over two, times the bracket, one over r one, minus one over r two. When r two is bigger than r one, this is positive: energy goes in. Clear?

3:23The trap: air drag speeds a satellite up

Now, back to our puzzle. Air drag takes energy away, so the total energy E decreases. It becomes more negative, okay? So what happens to the orbit? Does small r grow, or shrink? It shrinks! E equals minus G M m over two r. To make it more negative, small r must get smaller; the double arrow means so. On the graph, the satellite slides to the left, down along the orange curve. With me? And now the beautiful part. K equals minus E, so when E goes down, K goes up! See, at smaller r, the green curve is higher. The satellite speeds up. Puzzle solved! So if you said it slows down, that was the trap. Right? One more trap from your course. Astronauts float in orbit, and many students say gravity must be zero up there. No, no, careful! Gravity is very much there. They feel weightless because they and their craft are in free fall together. Okay?

4:23Example 1: total energy of a 1000 kg satellite at height R

Example one, from your course. Find the total energy of a one thousand kilogram satellite, at a height capital R above the surface. Small g is nine point eight, and capital R, the Earth's radius, is six point four times ten to the power six metres. Pause the video, and try it first. Done? Okay, let's check. The height is capital R, so we put small r equal to capital R in the formula, right? No, no, careful! Classic trap. Small r is measured from the Earth's centre, but height from the surface. So small r is capital R plus the height capital R, which makes two R, okay? So E, minus G M m over two r, becomes minus G M m over four R, clear? But we are given small g, not capital G and capital M, right? So, a topper's trick from chapter two: at the surface, small g equals G M over R squared, so G M equals g R squared. Then E becomes minus m g R over four. Much easier! Now the numbers. One thousand, times nine point eight, times six point four times ten to the power six, all over four. About minus one point five seven times ten to the power ten joules; capital J is joules, and the wavy equals sign means approximately. Negative, as in any circular orbit. Got it?

5:54Example 2: energy to orbit vs energy to escape

Now a harder one, example two. Compare the energy needed to put a satellite into a very low circular orbit, with the energy needed to make it escape from the Earth. Pause and try. Hint: final energy minus initial energy. Done? Okay, let's check. At rest on the surface, there is no kinetic energy, only potential. So E surface, with surface written small below, is minus G M m over capital R, okay? Now, a very low orbit is just above the ground; the satellite hardly rises. So it needs almost no energy, right? No, no, careful! It must also move at orbital speed. Very low means small r is approximately capital R, so E orbit, with orbit written small, is minus G M m over two R, clear? Energy needed for the orbit: E orbit minus E surface. Minus G M m over two R, plus G M m over R, gives G M m over two R, right? Now, escape. To escape, the satellite must reach infinity, where U is zero, and at the very least it arrives there with zero speed. So its total energy must reach zero. Energy needed: zero minus E surface, which is G M m over capital R, okay? That is exactly the energy behind the escape speed of chapter three. With me? So, orbit to escape is one to two. Look, a low orbit costs only half the energy of escaping the Earth! Easy to remember, right?

7:31Quick revision: K, U, E ratio, change of orbit, air drag

So, quick revision. In a circular orbit, K is G M m over two r, U is minus G M m over r, and E is minus G M m over two r: one, minus two, minus one. To change orbit: G M m over two, times one over r one, minus one over r two. And air drag: E falls, the orbit shrinks, and the satellite speeds up. Now go and try the practice problems on this page. All the best!