1. CAPSTONE Lecture 2 Gravity and its uses 07.06.2010 07/06/2010CASTONE 2010. Lecture 21

2. 07/06/2010CASTONE 2010. Lecture 22 Lecture 2: Gravity and its uses I.The laws of gravity II. Galileo, g III.Newton, G IV. Orbital velocity

3. 07/06/2010CASTONE 2010. Lecture 23 From studies of the motion of bodies, Newton concluded that three laws governed motion. a) Every action on a body has an equal and opposite reaction. b) Bodies at rest stay at rest until acted on by a force. Bodies in uniform motion maintain that motion until acted on by a force. c) The force is the mass of a body times its acceleration (uniform change in velocity per unit time. Newton’s Laws of Gravity

4. 07/06/2010CASTONE 2010. Lecture 24 --For gravity as a force, Newton concluded that F=- GMm/r 2, where G is a constant and M and m are the inertial masses of two bodies separated by a distance, r. --Newton convinced himself that the ability of a body to attract another body was directly proportional to the intertial mass of the same body and that the proportionality constant was G. --He also showed that for an object far from the attracting body, the latter could be treated as a point source of gravity and that neither the shape, density or type of material affected this law.

5. 07/06/2010CASTONE 2010. Lecture 25 Circular velocity for the gravitational force Huygens showed (mid 17 th century) that for objects in circular orbits, no matter the nature of the force constraining one body to orbit another, F=-mv 2 /r, where v is the velocity of the orbiting body and the other terms are as defined above. The minus sign is a convention that means an attractive force (which will act to reduce the distance between the two bodies.)

6. 07/06/2010CASTONE 2010. Lecture 26 Equating Huygen’s expression with Newton’s expression for the specific force of gravity: -GMm/r 2 =-mv 2 /r, where M and m are the two masses involved, G is the universal gravitational constant, r is the separation of the two bodies. So, the orbital velocity for perfect circular motion is v=(GM/r) 1/2, assuming m<<M.

7. 07/06/2010CASTONE 2010. Lecture 27 Newton’s force law, F=-GmM/r 2, reduces to F=- mg, where g=GM/r 2, M is the mass of the Earth and r is the radius of the Earth (onto which a body is falling). Galileo (~1570 – 1642) derived the equation F=-mg for bodies falling on the Earth, and found g=980 cm/sec 2. Newton’s law is a generalization of Galileo’s result, for bodies of any M and r. G and M always appear together. To measure M, one must know G. BIG G and little g

8. 07/06/2010CASTONE 2010. Lecture 28 Newton had to show, mathematically, that for perfect, uniform sphere, one can treat the entire mass of the Earth as if it is a point source at the center of the Earth. (For objects like the Earth, this rule is a very good approximation, even though the Earth is not perfectly homogeneous.) He had to assume that the nature of the material of Earth did not affect the value of G or the (1/r 2 ) form of the force law.(No one has ever been able to show this assumption to be wrong.