Physics 111: Elementary Mechanics – Lecture 12 Carsten Denker NJIT Physics Department Center for...
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Transcript of Physics 111: Elementary Mechanics – Lecture 12 Carsten Denker NJIT Physics Department Center for...
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
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