1. Physics 111: Elementary Mechanics – Lecture 12 Carsten Denker NJIT Physics Department Center for Solar–Terrestrial Research

2. November 28, 2006Center for Solar-Terrestrial Research Equilibrium  The linear momentum P of the center of mass is constant.  The angular momentum L about the center of mass, or about any other point, is constant.  Static equilibrium: and  Stable and unstable static equilibrium

3. November 28, 2006Center for Solar-Terrestrial Research Requirement of Equilibrium  The vector sum of all external forces that act on a body must be zero.  The vector sum of all external torques that act on a body, measured about any possible point, must also be zero.  The linear momentum of the body must be zero. balance of forces balance of torques

4. November 28, 2006Center for Solar-Terrestrial Research Center of Gravity  The gravitational force on a body effectively acts on a single point, called the center of gravity.  If is the same for all elements of a body, then the body’s center of gravity is coincident with the body’s center of mass.

5. November 28, 2006Center for Solar-Terrestrial Research Newton’s Law of Gravitation A uniform spherical shell shell of matter attracts a particles that is outside the shell as if all the shell’s mass is concentrated at its center! G = 6.67  10 –11 N m 2 /kg 2 Superposition Principle

6. November 28, 2006Center for Solar-Terrestrial Research Gravitation Near Earth’s Surface Gravitational acceleration 1.Earth is not uniform 2.Earth is not a sphere 3.Earth is rotating Mean Earth surface (0 km, 9.83 m/s 2 ), Mt. Everest (8.8 km, 9.80 m/s 2 ), highest manned balloon ( 36.6 km, 9.71 m/s 2 ), Space Shuttle orbit (400 km, 8.70 m/s 2 ), and communications satellite (35,700 km, 0.225 m/s 2 )

7. November 28, 2006Center for Solar-Terrestrial Research Kepler’s Laws 1.The Law of Orbits: All planets move in elliptical orbits, with the Sun at one focus. 2.The Law of Areas: A line that connects the planet to the Sun sweeps out equal areas in the plane of the planet’s orbit in equal times, i.e., the rate dA/dt at which it sweeps out area A is constant. 3.The Law of Periods: The square of the period is proportional to the cube of the semimajor axis of its orbit.

8. November 28, 2006Center for Solar-Terrestrial Research

9. November 28, 2006Center for Solar-Terrestrial Research Potential and Kinetic Energy Potential Energy Kinetic Energy Total Energy

10. November 28, 2006Center for Solar-Terrestrial Research Einstein and Gravitation