[MUSIC] Newton's constant, big G, is the universal constant of gravitation. It's constant universally. Little g, the acceleration due to gravity, is not so constant. And nor is the force that we measure with these gadgets. Objects at the surface of the Earth are not exactly in mechanical equilibrium. Their centripetal acceleration, while it's only a few centimeters per second per second, but it's not zero, except at the poles. Here there's no centripetal acceleration so the normal force measured by the scales equals the gravitational force. At other latitudes the normal and gravitational forces add up to give the object's mass times centripetal acceleration. Well, that raises some interesting questions which we'll ask you. So, to recap, little g is smallest at the equator. And the vertical and horizontal directions are indeed at right angles. But because of the centripetal acceleration the Earth is not exactly spherical. Earth's diameters at zero latitude are slightly larger than the diameter from pole to pole. Because the centripetal acceleration is very much smaller than g, the effect is very small. If the acceleration were larger than g, things would indeed be awkward. So little g varies with latitude. It also varies altitude. As we get further from the Earth, the gravitational force gets smaller as one on r squared. And so does the gravitational acceleration, little g. Now that raises a problem. Previously, we wrote potential energy, Ugrav equals mgh. Assuming that little g was constant. Well, g is almost constant if we stay near the Earth's surface. But at large altitude the attraction gets smaller so the potential energy rises less quickly with altitude. Because this course doesn't use calculus, we shant do the derivation. But here is the expression for the gravitational potential energy of two masses, big M and little m at separation r. And here's what it looks like using the Earth as an example, frame one. Qualitatively make sense. Close to the earth the attraction is strong so we have to do lots of work to rise into a given distance. So the slope of potential energy versus altitude is large. Further away weaker attraction, smaller slope. Remember that we have to define the zero of potential energy. Previously, we used floor level or ground level or whatever was convenient. For the astronomical version, we see that the potential energy goes to zero at very large separation. This makes sense. Astronomers want one reference that works for every planet, star, galaxy. Note that for astronomers, the potential energy is always negative. It takes work to pull planets and stars apart. Actually, going back to the beginning, astronomical bodies lost lots of potential energy when atoms came together to form stars, galaxies, and planets. Well, I hope you're thinking, how can we have two very different expressions for potential energy? The exact one is U equals minus G M one M two over r. And let's call the approximate one U dashed equals MGH. Remember that they have different references. The astronomers put U equals zero a long, long way away, while we could put U dashed equals zero at the Earth's surface. So both axes are different on their graph. Here is the linear approximation to the full expression. And these two are consistent, provided we stay very close to the Earth's surface. What goes up, must come down. [MUSIC] Probably the first person to know that this isn't always true was Isaac Newton. If we give a projectile enough mechanical energy, it can go very far and still have some kinetic energy left over. Let's do the calculation. Initial separation is one Earth radius. Final separation is very large. In space there's no air resistance or friction. So let's neglect those losses and mechanical energy is conserved. Suppose we give our spacecraft with mass m, just enough kinetic energy, so that it has zero mechanical energy overall. Initially, the large positive kinetic energy exactly cancels the negative potential energy. Finally, it will have tiny kinetic energy a long, long way away. Call this initial speed the escape speed, and use the mass of the Earth and the Earth's radius in the initial potential energy. So, how fast is escape speed? Good question for you. Just to recap, cancelling the m, and rearranging, we have escape speed equals square root, 2Gm over r. Substitution gives, 11 kilometers per second which is 33 times faster than the speed of sound, or 40 thousand kilometers per hour. Looking at this equation we can see that the escape speed is large if the mass is large. And also if the radius of the planet or star is small. So an obvious question. How massive and small does a star have to be if the escape speed equals the speed of light? Well, that question was obvious to John Mitchell who did the analysis in 1783. Now, the speed of light is large, and big G is very small. So the required ratio is huge. But what if a star collapsed to a small enough size? Mitchell called this case a dark star. And we now call it a black hole. An object so dense, that even light can't escape its gravity. But the numbers are astonishing. Do a calculation to get a feeling for this. So rearranging Mitchell's expression, our sun would have to collapse to a diameter of six kilometers in order to become a black hole. And it's density would have to be 20 million tons per cubic millimeter. Well, it turns out that our sun is not big enough to collapse into a black hole, though that does happen to many larger stars, once they've used up their nuclear fuel. Our local star is not even big enough to form a supernova. [LAUGH] Whew. Our sun is just the right size to live for several billion years, become a red giant, expand to engulf what is left of the earth. Well, back to cheerier topics. Another obvious questions is, if no light gets out of a black hole, how do we know it's there. Here are two ways. Suppose we look at a distant galaxy or quasar and there just happens to be a black hole in our line of sight. The gravity of the black hole bends the light from the more distant galaxy, and so we see images on either side of the black hole. Another way, we know that there is a gigantic black hole at the center of our galaxy. Observations of the motions of the stars near the center of the galaxy show that these stars are orbiting a mass equal to about 3.6 million times that of our sun. A huge black hole. All of which gives us interesting material for our quiz. [MUSIC]
