Physics 111: Elementary Mechanics – Lecture 12

Carsten Denker

NJIT Physics DepartmentCenter for Solar–Terrestrial

Research

November 28, 2006 Center 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

0P

0L

November 28, 2006 Center 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.

net 0dP

Fdt

net 0dL

dt

balance of forces

balance of torques

P

November 28, 2006 Center 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.

g

gF

net

cog

cog

cog

cog

cog

cog com

1

i i gi

i i gi

g

gi

gi i gi

i i i i i

i i i

i i

x F

x F

x F

x F

x F x F

x m g x m g

x m x m

x x m xM

November 28, 2006 Center 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 m2/kg2

Superposition Principle

1 22

mmF G

r

November 28, 2006 Center 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/s2), Mt. Everest (8.8 km, 9.80 m/s2), highest manned balloon ( 36.6 km, 9.71 m/s2), Space Shuttle orbit (400 km, 8.70 m/s2), and communications satellite (35,700 km, 0.225 m/s2)

2 2 g g

Mm GMF G ma a g

r r

November 28, 2006 Center 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.

November 28, 2006 Center for Solar-Terrestrial Research

November 28, 2006 Center for Solar-Terrestrial Research

Potential and Kinetic EnergyPotential Energy

Kinetic Energy

Total Energy

GMmU

r

22

2

1

2 2

Mm v MmF G m K mv G

r r r

2

Mm Mm MmE K U G G G

r r r

November 28, 2006 Center for Solar-Terrestrial Research

Einstein and Gravitation