Lecture 4: Gravity and Motion

Describing Motion Speed (miles/hr; km/s) Velocity (speed and direction) Acceleration (change in velocity) Units: m/s 2 Acceleration of gravity: 9.8 m/s 2 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/s 2 1 kg m/s 2 = 1 Newton Weight is the force exerted on an object by gravity so weight also has units of kg m/s 2

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 m 3 /(kg s 2 ) 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 v  planet moves faster larger distance  smaller v  planet moves slower

Understanding Kepler’s Third Law 4  2 a 3 p 2 = G(M 1 + M 2 ) Newton’s generalization of Kepler’s Third Law is given by: 4  2 a 3 p 2 = GM sun M planet << M sun, 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 M 1 << M 2

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