Newton’s Law of Gravitation The “4 th Law”

Quick Review NET FORCE IS THE SUM OF FORCES… IT IS NOT ACTUALLY A FORCE ON ITS OWN!

Sir Isaac Newton  Remember him?  Well… he’s very interesting.interesting  Remember him?  Well… he’s very interesting.interesting

Newton’s Universal Law of Gravitation  Every object in the universe attracts every other object with a force that is directly proportional to the masses of the bodies and inversely proportional to the square of the distance between the object.

Newton’s Universal Law of Gravitation

Example  Determine the gravitational attraction between the sun and the Earth at its mean radius of 1.5 x 10 8 m.  m Earth = 5.98 x kg  m sun = 1.99 x kg  Determine the gravitational attraction between the sun and the Earth at its mean radius of 1.5 x 10 8 m.  m Earth = 5.98 x kg  m sun = 1.99 x kg

Example  Determine the gravitational attraction between Jane (m = 100 kg) and the Earth if Jane is on the surface.  m Earth = 5.98 x kg  r Earth = 6.38 x 10 6 m  Determine the gravitational attraction between Jane (m = 100 kg) and the Earth if Jane is on the surface.  m Earth = 5.98 x kg  r Earth = 6.38 x 10 6 m

Example  If Tarzan (m = 60 kg) is 3 m from Jane (m = 100 kg), determine the gravitational force of attraction between Tarzan and Jane.

Finding ‘g’:  How is g determined?  ‘g’ is the ACCELERATION DUE TO GRAVITY.  Two equations for F g :  How is g determined?  ‘g’ is the ACCELERATION DUE TO GRAVITY.  Two equations for F g :

Example  Use Newton's law of gravitation to determine the acceleration of an 85-kg astronaut on the International Space Station (ISS) when the ISS is at a height of 350 km above Earth's surface. The radius of the Earth is 6.37 x 10 6 m. (GIVEN: M Earth = 5.98 x kg)

Example  Determine the acceleration of the Earth about the sun. (GIVEN: M sun = 1.99 x kg and Earth-sun distance = 1.50 x m)

Gravitational Motion:

Example  Determine the orbital speed of the International Space Station - orbiting at 350 km above the surface of the Earth. The radius of the Earth is 6.37 x 10 6 m. (GIVEN: M Earth = 5.98 x kg)

Example  Hercules is hoping to put a baseball in orbit by throwing it horizontally (tangent to the Earth) from the top of Mount Newton - 97 km above Earth's surface. With what speed must he throw the ball in order to put it into orbit? (GIVEN: M Earth = 5.98 x kg; R Earth = 6.37 x 10 6 m)

Kepler’s Laws Mathematics of Planetary Motion

Newton and Kepler  Newton uses gravity to explain Kepler’s mathematics.  Satellites go around in circular orbits due to gravity  Newton uses gravity to explain Kepler’s mathematics.  Satellites go around in circular orbits due to gravity

Try These:

Review of the Ellipse  An ellipse is the set of all points on a plane whose distance from two fixed points F and G add up to a constant.  F and G are focal points (foci)  An ellipse is the set of all points on a plane whose distance from two fixed points F and G add up to a constant.  F and G are focal points (foci)

Review of the Ellipse  Major Axis – longest diameter  Minor Axis – shortest diameter  Semi-Major Axis – half of the major axes (this will be important later)  Major Axis – longest diameter  Minor Axis – shortest diameter  Semi-Major Axis – half of the major axes (this will be important later)

Kepler’s Laws  3 Mathematical Laws  Took him 30 years to complete his research after Brahe dies.  Mathematical laws and described the motion.  He dies not knowing how any of it worked.  3 Mathematical Laws  Took him 30 years to complete his research after Brahe dies.  Mathematical laws and described the motion.  He dies not knowing how any of it worked.

Kepler’s First Law  The orbits of the planets are elliptical, with the sun at one focus of the ellipse.  Perihelion (Perigee) - point of closest approach distance  Aphelion (Apogee) - point of furthest approach distance  The orbits of the planets are elliptical, with the sun at one focus of the ellipse.  Perihelion (Perigee) - point of closest approach distance  Aphelion (Apogee) - point of furthest approach distance

Eccentricity  A measure of elongation of ellipse  r a = radius of aphelion  r p = radius of perihelion  if e = 0, then ellipse is a circle  if e = 1, then ellipse is a parabola  A measure of elongation of ellipse  r a = radius of aphelion  r p = radius of perihelion  if e = 0, then ellipse is a circle  if e = 1, then ellipse is a parabola

Example  The Earth has an eccentricity of 1.67%. If the aphelion distance is 1.52 x 10 8 km, what is the perihelion distance?

Example  If the aphelion distance is 5r and the perihelion distance is 3r, determine the eccentricity.