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Lecture 4: Gravity and Motion
Describing Motion
Speed (miles/hr; km/s) Velocity (speed and direction) Acceleration (change in velocity)
Units: m/s2
Acceleration of gravity: 9.8 m/s2
All objects feel the same acceleration due to gravity, regardless of their mass
Momentum and Force
Momentum is mass times velocity Force causes a change in momentum
(usually a change in velocity) A net force causes acceleration
Mass and Weight
Mass refers to the amount of matter in an object (universal)
Weight is the force that acts on a body depends on strength of gravity, or other
forces present
Orbits and Escape Velocity
Units of Force, Mass and Weight
Mass: grams (g) or kilograms (kg) units of force are kg m/s2
1 kg m/s2 = 1 Newton Weight is the force exerted on an object
by gravity so weight also has units of kg m/s2
Newton’s Laws of Motion
First Law: in the absence of a net force, an object moves with constant velocity
Second Law: Force = mass times acceleration
Third Law: For any force, there is an equal and opposite reaction force
centripetal force
Conservation of Momentum
The total amount of momentum in the Universe does not change
Momentum can only be transferred, not destroyed
Torque and Angular Momentum
A torque is a twisting force Torque = force x
length of lever arm Angular momentum is
torque times velocity For circular motion,
L = m x v x r
Laws for Rotational Motion
Analogs of all of Newton’s Laws exist for rotational motion
For example, in the absence of a net torque, the total angular momentum of a system remains constant
There is also a Law of Conservation of Angular Momentum
Conservation of Angular Momentum during star formation
Newton’s Universal Law of Gravitation
Every mass attracts every other mass through a force called gravity
The force is proportional to the product of the two objects’ masses
The force is inversely proportional to the square of the distance between the objects’ centers
Universal Law of Gravitation
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
Remember Kepler’s Laws?
Orbits of planets are ellipses, with the Sun at one focus
Planets sweep out equal areas in equal amounts of time
Period-distance relation:
(orbital period)2 = (average distance)3
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 Motion
Ellipses are one possible solution, but there are others (parabolas and hyperbolas)
Conic Sections
Bound and Unbound Orbits
Unbound (comet)
Unbound (galaxy-galaxy)
Bound (planets, binary stars)
Understanding Kepler’s Laws:conservation of angular momentum
L = mv x r = constant
r
smaller distance smaller r bigger vplanet moves faster
larger distance
smaller v
planet moves slower
Understanding Kepler’s Third Law
42 a3 p2 =
G(M1 + M2)
Newton’s generalization of Kepler’s Third Law is given by:
42 a3 p2 =
GMsun
Mplanet << Msun, so
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
An Astronaut and the Space Shuttle have the same orbit!
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!
The End