GRAVITY LAB. This procedure takes you through the law of gravitation to Black Holes in a series of problems you are supposed to solve. Enjoy!

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Cecil Dickerson
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1 GRAVITY LAB This procedure takes you through the law of gravitation to Black Holes in a series of problems you are supposed to solve. Enjoy! 1. Gravitational law of attraction: The gravitational force F g between two spherically symmetric objects of mass m 1 and m 2 is proportional to the product of their masses, and falls off inversely as the square of the distance r between their centers of masses: F = G Nm 1 m 2 r 2 where G N is the Gravitational (or Newton s) Constant. (a) Find the physical dimensions of G N (in terms of [M], [L] & [T]). (b) What is the value for G N? Include proper SI units. (c) Two objects are moved twice their distance apart. The gravitational force between them diminishes by a factor of. They are moved back to a third of their original distance apart. Now the gravitational force between them increases by a factor of.

2 2. The gravitational force between a body of mass m 1 and the Earth, m 2 = M E, is called the weight of the body m 1. But the weight of the body on Earth s surface is also given by F = m 1 g E, where the value with units of g E =. By equating the two expressions for force, find g E in terms of G N, M E, R E. (Thus, knowing the size of the Earth and the acceleration due to gravity, it is possible to weigh the Earth i.e., find its mass. This is how we know the Earth s mass!) As a balloon ascends into the atmosphere, it surface, the smaller the value of. weight, because the higher we go from Earth s 3. The average density ρ E of the Earth is defined to be its mass M E divided by its total volume. Find an expression relating g E to average density and radius of the Earth.

3 Given that g E = 9.8 m/s 2, R E = m, and using the value of G N from Q.1, part(b), find the average density of the Earth. A planet half the radius of the Earth, R P = 1 2 R E, is determined to have (using a pendulum) a g P = 4g E. How does its average density compare to that of Earth s?

4 4. The gravitational force of the Earth causes a Gravitational potential energy on an object of mass m a distance r from the center, where r R E, given by U g (r) = G NM E m. r Derive the (approximate) expression for potential energy at a height h above Earth s surface by evaluating the difference between the gravitational PE at the two places; that is, show that when h << R E, then U g (R E + h) U g (R E ) = G NM E m R E + h ( G NM E m R E ) mg E h. Hint: you would want to employ the Binomial Theorem Approximation, (1 + x) n 1 + nx, provided that x << 1. When you use U g = mg E h in physics problems, you are assuming that height h is compared with R E, and that for this height, g E is.

5 5. The total energy of a body is the sum of its gravitational potential energy and its kinetic energy. Write an expression for the total energy of a satellite orbiting the Earth in a circular orbit at a distance R from Earth s center, where R R E, assuming its orbit speed to be v. Find the satellite s orbit speed v as follows: equate the centripetal force needed for going in a circular orbit to the gravitational force between the satellite and the Earth.

6 Suppose the orbit is completed in time T ; find an expression relating T and R (Kepler s third law of motion!). Geostationary Orbits: When the period of orbit of a satellite coincides with the earth s rotational period, then it is said to be in an orbit that s stationary with respect to earth. (Such a satellite will remain in a fixed location in the sky.) Find the relationship between R and R E for this to happen. Evaluate the height above the earth s surface in kilometers.

7 Ground-hugging Orbits: Find the period of a satellite which barely clears the ground as it orbits the Earth. Determine the speed of such a satellite, and give the answer in both m/s and mph. Substitute back the expression for satellite speed v into the expression for its total energy. You should find that the total energy is always negative, a universal property of gravitating systems. The orbit is called a bound state orbit. The satellite will be trapped in this orbit unless enough extra kinetic energy is given to it to escape.

8 6. Escape velocity. If enough KE is given to an object on Earth s surface to equal its gravitational PE with respect to the Earth, then the body would escape into space without ever falling back. The total energy of the body is zero, and at infinity, its speed will eventually be zero as well. Find escape speed ve e in terms of G N, M E and R E. Find an expression for the velocity of escape, v E e in terms of g E at the surface, & R E. Does the escape velocity depend on the mass of the ejected object? Circle: Y / N Knowing g E & R E compute v E e the escape speed for Earth, in both km/s and miles per second. How long at this speed will an object take to go across continental USA?

9 The Moon is such that R M = 0.27 R E & g M = 1 6 g E. Find the ratio of the escape velocity from the Moon, v M e to that from the Earth v e E. Knowing v E e from the previous question, find v M e in miles/sec.

10 7. An object that is such that even light v e = c = m/s cannot escape from its grasp is called a Black Hole, abbreviated henceforth BH. Take a beam of light of mass m and equate its escape KE to its gravitational PE on the BH surface. This gives a relationship between R BH & M BH that involves in it constants c & G. The spherical surface of radius R BH is called the event horizon of the BH. The more massive a black hole, the is its size. Find R BH for (i) the Earth (ii) the Sun, (iii) the entire Universe. In each case, look up the data for mass M BH of the object as it exists now. Note that the BH radius of the Universe is also the radius of the observable Universe! We live in a big black hole.

11 8. Find a relation between density of BH material and the mass of a black hole. The more massive a black hole, the is its BH material. Evaluate solar BH density. Use M = kg, and R BH as determined in the previous page.

12 For fun, let us find the volume of a rectangular lake with the same mass of water as the mass of one teaspoonful (= 5 cm 3 ) of solar black hole material. If the lake is one mile by one mile square, find its depth. Water density is ρ W = 1.0 gm/cm 3. Show that the surface (i.e., event horizon radius) value of the acceleration due to gravity of a BH is given by g BH = c2 2R BH. Evaluate the radius of a BH for which this is equal to g E = 9.8 m/s 2. Find also its mass in terms of M. The existence of such supermassive black holes proves that statements such as as you enter a black hole, you will be torn apart are not always true. That only applies if the black holes are small. The End. Submit at the rate of 10 extra credit points per page!

Chapter 13: universal gravitation

Chapter 13: universal gravitation Newton s Law of Gravitation Weight Gravitational Potential Energy The Motion of Satellites Kepler s Laws and the Motion of Planets Spherical Mass Distributions Apparent