2 Describing Motion and Forces speed, velocity and accelerationmomentum and forcemass and weightNewton’s Laws of Motionconservation of momentumanalogs for rotational motion
3 Torque and Angular Momentum A torque is a twisting forceTorque = force x length of lever armAngular momentum is torque times velocityFor circular motion, L = m x v x r
4 Laws for Rotational Motion Analogs of all of Newton’s Laws exist for rotational motionFor example, in the absence of a net torque, the total angular momentum of a system remains constantThere is also a Law of Conservation of Angular Momentum
6 Conservation of Angular Momentum during star formation
7 Newton’s Universal Law of Gravitation Every mass attracts every other mass through a force called gravityThe force is proportional to the product of the two objects’ massesThe force is inversely proportional to the square of the distance between the objects’ centers
9 The Gravitational Constant G The value of the constant G in Newton’s formula has been measured to be G = 6.67 x 10 –11 m3/(kg s2)This constant is believed to have the same value everywhere in the Universe
10 Remember Kepler’s Laws? Orbits of planets are ellipses, with the Sun at one focusPlanets sweep out equal areas in equal amounts of timePeriod-distance relation:(orbital period)2 = (average distance)3
11 Kepler’s Laws are just a special case of Newton’s Laws! Newton explained Kepler’s Laws by solving the law of Universal Gravitation and the law of MotionEllipses are one possible solution, but there are others (parabolas and hyperbolas)
13 Bound and Unbound Orbits Unbound (comet)Unbound (galaxy-galaxy)Bound(planets,binary stars)
14 Understanding Kepler’s Laws: conservation of angular momentum L = mv x r = constantsmaller distancesmaller rbigger vplanet moves fasterrlarger distancesmaller vplanet moves slower
15 Understanding Kepler’s Third Law Newton’s generalization of Kepler’s Third Law is given by:4p2 a3p2 =G(M1 + M2)4p2 a3p2 =GMsunMplanet << Msun, so
16 This has two amazing implications: The orbital period of a planet depends only on its distance from the sun, and this is true whenever M1 << M2
17 An Astronaut and the Space Shuttle have the same orbit!
18 Second Amazing Implication: If we know the period p and the average distance of the orbit a, we can calculate the mass of the sun!
19 Example: How can we use this information to find the mass of the Sun? Io is one of the large Galilean moons orbiting Jupiter.It orbits at a distance of 421,600 km from the center ofJupiter and has an orbital period of 1.77 days.How can we use this informationto find the mass of the Sun?