The Apple & the Moon Isaac Newton realized that the motion of a falling apple and the motion of the Moon were both actually the same motion, caused by the same force - the gravitational force. Isaac Newton realized that the motion of a falling apple and the motion of the Moon were both actually the same motion, caused by the same force - the gravitational force.

Universal Gravitation Newton’s idea was that gravity was a universal force acting between any two objects. Newton’s idea was that gravity was a universal force acting between any two objects.

At the Earth’s Surface Newton knew that the gravitational force on the apple equals the apple’s weight, mg, where g = 9.8 m/s 2. Newton knew that the gravitational force on the apple equals the apple’s weight, mg, where g = 9.8 m/s 2. W = mg

Weight of the Moon Newton reasoned that the centripetal force on the moon was also supplied by the Earth’s gravitational force. Newton reasoned that the centripetal force on the moon was also supplied by the Earth’s gravitational force. F c = mg ?

Weight of the Moon Newton’s calculations showed that the centripetal force needed for the Moon’s motion was about 1/3600 th of mg, where m is the mass of the Moon. Newton’s calculations showed that the centripetal force needed for the Moon’s motion was about 1/3600 th of mg, where m is the mass of the Moon.

Weight of the Moon Newton also knew that the Moon was about 60 times farther from the center of the Earth than the apple. Newton also knew that the Moon was about 60 times farther from the center of the Earth than the apple. And 60 2 = 3600 And 60 2 = 3600

Universal Gravitation Newton concluded that the gravitational force is: Newton concluded that the gravitational force is: Directly proportional to the masses of both objects. Directly proportional to the masses of both objects. Inversely proportional to the distance between the objects. Inversely proportional to the distance between the objects.

Law of Universal Gravitation In symbols, Newton’s Law of Universal Gravitation is: In symbols, Newton’s Law of Universal Gravitation is: F grav = G F grav = G m is the mass of each object m is the mass of each object r is the distance between the two masses r is the distance between the two masses G = 6.67 x N m 2 /kg 2 G = 6.67 x N m 2 /kg 2 m1m2m1m2 r 2

Law of Universal Gravitation In symbols, Newton’s Law of Universal Gravitation is: In symbols, Newton’s Law of Universal Gravitation is: F grav = G F grav = G m is the mass of each object m is the mass of each object r is the distance between the two masses r is the distance between the two masses Where G is a constant of proportionality. Where G is a constant of proportionality. G = 6.67 x N m 2 /kg 2 G = 6.67 x N m 2 /kg 2 m1m2m1m2 r 2

scale 150 lbs scale 25.6 lbs scale 406 lbs g = 9.81 m/s 2 g = 1.67 m/s 2 g = 26.6 m/s 2

An Inverse- Square Force Since gravitational force is inversely proportional to the square of the distance between the two objects, more separation distance will result in weaker gravitational forces. Distance = 1, force =1 Distance = 2, force = ¼ Distance = 3, force = 1/9

Experimental Evidence The Law of Universal Gravitation allowed extremely accurate predictions of planetary orbits. The Law of Universal Gravitation allowed extremely accurate predictions of planetary orbits. Cavendish measured gravitational forces between human-scale objects before His experiments were later simplified and improved by von Jolly. Cavendish measured gravitational forces between human-scale objects before His experiments were later simplified and improved by von Jolly.

Inertial vs. Gravitational Mass

Black Holes When a very massive star gets old and runs out of fusionable material, gravitational forces may cause it to collapse to a mathematical point - a singularity. All normal matter is crushed out of existence. This is a black hole. When a very massive star gets old and runs out of fusionable material, gravitational forces may cause it to collapse to a mathematical point - a singularity. All normal matter is crushed out of existence. This is a black hole.

Black Hole Gravitational Force

The black hole’s gravity is the same as the original star’s at distances greater than the star’s original radius. The black hole’s gravity is the same as the original star’s at distances greater than the star’s original radius. Black hole’s don’t magically “suck things in.” Black hole’s don’t magically “suck things in.” The black hole’s gravity is intense because you can get really, really close to it! The black hole’s gravity is intense because you can get really, really close to it!

Black Hole Gravitational Force The black hole’s gravity is the same as the original star’s at distances greater than the star’s original radius. The black hole’s gravity is the same as the original star’s at distances greater than the star’s original radius. Black hole’s don’t magically “suck things in.” Black hole’s don’t magically “suck things in.” The black hole’s gravity is intense because you can get really, really close to it! The black hole’s gravity is intense because you can get really, really close to it!

Earth A satellite is a projectile shot from a very high elevation and is in free fall about the Earth.

Velocity Vectors Acceleration Vectors Force Vectors

Centripetal Acceleration and Force The time it takes to make one revolution is the period (T), and the displacement is 2  r The velocity is v = d = 2  r t T t T acceleration a c = v 2 /r = 4  2 r\T 2 Force F c = ma c

Example A 13 g rubber stopper is attached to a 0.93 m string. The stopper is swung in a horizontal circle, making one revolution in 1.18 s. Find the centripetal force exerted by the string on the stopper. A 13 g rubber stopper is attached to a 0.93 m string. The stopper is swung in a horizontal circle, making one revolution in 1.18 s. Find the centripetal force exerted by the string on the stopper. a c = v 2 = 4  2 r= 4  2 (0.93m )= 26 m/s 2 a c = v 2 = 4  2 r= 4  2 (0.93m )= 26 m/s 2 r T 2 (1.18s) 2 r T 2 (1.18s) 2 F c = ma= (.013 g) 26 m/s 2 F c = ma= (.013 g) 26 m/s 2 = N Fc

Planet Force of gravity F g Centripetal force F c Orbital path Fg Fg = FcFc Fg Fg = G m1 m1 m2m2 r2r2 Fc Fc = m v2v2 r G m1 m1 m2 m2 = m1 m1 v2v2 r 2 r Canceling m1 m1 & r on both sides V2 V2 = G m2m2 r

Gravitation & Satellite Problems (a)What is the velocity of a satellite 5000 km above the Earth? (b)What is it period of rotation ? 5000 km ReRe r = r e + h m e = 6 x kg R e = 6.4 x 10 6 m (a) V 2 = G m 2, v = G m e 1/2 r r V = (6.67 x )(6 x ) 1/2 (6.4 x x 10 6 ) V = 5900 m/s (b) V = 2π r / T, T = 2π r/ v T = 2π (11.4 x 10 6 ) / 5900 = 1.2 x 10 4 sec = 3.4 hr. () ()

Why Two Tides? Tides are caused by the stretching of a planet. Tides are caused by the stretching of a planet. Stretching is caused by a difference in forces on the two sides of an object. Stretching is caused by a difference in forces on the two sides of an object. Since gravitational force depends on distance, there is more gravitational force on the side of Earth closest to the Moon and less gravitational force on the side of Earth farther from the Moon. Since gravitational force depends on distance, there is more gravitational force on the side of Earth closest to the Moon and less gravitational force on the side of Earth farther from the Moon.

Why Two Tides? Remember that Remember that

Why the Moon? The Sun’s gravitational pull on Earth is much larger than the Moon’s gravitational pull on Earth. So why do the tides follow the Moon and not the Sun? The Sun’s gravitational pull on Earth is much larger than the Moon’s gravitational pull on Earth. So why do the tides follow the Moon and not the Sun?

Why the Moon? Since the Sun is much farther from Earth than the Moon, the difference in distance across Earth is much less significant for the Sun than the Moon, therefore the difference in gravitational force on the two sides of Earth is less for the Sun than for the Moon (even though the Sun’s force on Earth is more). Since the Sun is much farther from Earth than the Moon, the difference in distance across Earth is much less significant for the Sun than the Moon, therefore the difference in gravitational force on the two sides of Earth is less for the Sun than for the Moon (even though the Sun’s force on Earth is more).

Why the Moon? The Sun does have a small effect on Earth’s tides, but the major effect is due to the Moon. The Sun does have a small effect on Earth’s tides, but the major effect is due to the Moon.