Objectives: The student will be able to: 1.Apply the proportional relationship of the law of universal gravitation. 2.Use Newton’s second law and the law of universal gravitation to show why objects near the surface of the earth fall with the same constant acceleration. 3.Explain why a spaceship in a stable circular orbit is in free fall and why a person in that spaceship experiences weightlessness.

The Big Idea Everything pulls everything else. There is a force that pulls all objects together. It is gravity.

What Newton Knew Newton understood the concept of inertia developed earlier by Galileo. Without an outside force, moving objects continue to move at constant speed in a straight line. If an object undergoes a change in speed or direction, then a force is responsible.

Newton’s 1st Law The Law of Inertia What is it? –An object in equilibrium will remain in equilibrium unless acted on by a non zero net force. Equilibrium –Zero Net Force. –No Acceleration. Static - Object at rest. Dynamic - Object moving at a constant speed in a straight line.

The Apple and the Moon Newton saw apples falling to Earth and wondered if the moon fell towards the Earth just like the apple fell towards the Earth. Was he correct? What makes things fall towards the center of the Earth? What is different about the moon and the apple?

The Moon Falls? If something is moving and no force acts on it, how does it Keep moving? What is needed for circular motion? Newton realized that if the moon did not fall, it would move off in a straight line and leave its orbit. His idea was that the moon must be falling around the Earth.

Cannonball being shot off a Very Tall Mountain

If I could climb a mountain tall enough, could I shoot a cannonball so that it would never land back on Earth. We call this putting an object into orbit. Nev n Thought a Experiment. vton’s

From hypothesis to theory Newton thought the apple, the orbiting cannonball, and the motion of the moon were all caused by a force now called gravity. He needed to test this hypothesis. I need to test my hypothes is

The moon and the apple. Newton knew the apple fell 5m in one second. He wondered how far the moon fell in one second. The moon was 60 times away from the Earth than the apple was. The force of Gravity must dilute the farther away something is.

Using geometry, Newton calculated how far the circle of the moon’s orbit lies below the straight-line distance the moon otherwise would travel in one second. His value turned out to be about the 1.4-mm distance accepted today. But he was unsure of the exact Earth moon distance, and whether or not the correct distance to use was the distance between their centers. At this time he hadn’t proved mathematically that the gravity of the spherical Earth (and moon) is the same as if all its mass were concentrated at its center.

Because of this uncertainty, and also because of criticisms he had experienced in publishing earlier findings in optics, he placed his papers in a drawer, where they remained for nearly 20 years. During this period he laid the foundation and developed the field of geometrical optics for which he first became famous. Newton finally returned to the moon problem at the prodding of his astronomer friend Edmund Halley (of Halley’s comet fame). It wasn’t until after Newton invented a new branch of mathematics, calculus, to prove his center-of- gravity hypothesis, that he published what is one of the greatest achievements of the humankind, the law of universal gravitation.

Very fast horizontal toss t = 0s t = 1s x= 8km 5m t = 2s x=16km t = 3s x=24km 20m 45m V=8km/s

Orbital motion is free fall (stopped here) V = 4 km/sV = 6 km/sV = 8 km/sV = 10 km/s Circular Orbit!Eliptical Orbit

Centripetal acceleration a = v 2 /r for a circular orbit (v = 8km/s = 8x10 3 m/s) a = (8 x10 3 m/s) x 10 6 m = 64 x10 6 m 2 /s x 10 6 m = 10 m/s 2 Toward Earth’s center = g

Artificial satellite a v a = v 2 /r = g

Moon-earth v a=v 2 /r

Is the Moon in free-fall around the Earth? r=3.84x10 5 km v a = v 2 /r what is v? v = dist/time = 2  r 28d = 2 x 3.14 x 3.84 x 10 8 m 28d x (24h/d) x 3.6 x 10 3 s = 24 x 10 8 m 2.4 x 10 6 s = 1.0 x 10 3 m/s

Moon’s centripetal acceleration a moon = v 2 /r; v = 1.0 x10 3 m/s) a moon = (1.0 x 10 3 m/s) x 10 8 m = 1.0 x 10 6 m 2 /s x 10 8 m = 2.7 x m/s 2 Toward Earth’s center  g

Newton’s dreams Hmmmmm The Moon is in free-fall around the Earth It’s acceleration is only 1/3600 g (accel at the Earth’s surface)

Distances r=3.84x10 8 m = 60 x 6.4x10 6 m The moon is 60x further from the Earth’s center than objects on (near) the Earth’s surface = 60 x R E ()2=)2= R E = 6.4x10 6 m

Newton’s big idea The force of gravity gets weaker as distance squared The moon is 60x further from the Earth’s center than objects on (near) the Earth’s surface The strength of Earth’s gravity near the Moon is (1/60) 2 t=1/3600 times weaker

Gravity gets weaker as 1/dist 2

inverse-square law

Universal law of gravity m M r F  m F  M F  1r21r2 combine:F  mM r 2 F = G mM r 2 Proportionality constant: “Newton’s Constant”

Universal Universal: applies to all objects!!!

What is G? W= G mM E R E 2 W W= m GM E R E 2 W= m g g=g= GM E R E 2

Force of gravity between “ordinary-sized” objects 80kg 60kg 1m F = G mM r 2 F = 6.7x Nm 2 /kg 2 60 kg 80kg (1m) 2 F = 6.7 x 60 x 80 x N Boy’s weight = mg = 80kg x 10m/s 2 = 800 N  30x10 9 times bigger! F = x N = 3.2x10 -7 N

Measuring G G was first measured 150 years after Newton’s discovery of universal gravitation by an English physicist, Henry Cavendish.

Henry Cavendish’s experiment determined the proportionality constant G in Detailed clip on experiment

Newton’s Law of Universal Gravitation The gravitational force on you is one-half of a Third Law pair: the Earth exerts a downward force on you, and you exert an upward force on the Earth. When there is such a disparity in masses, the reaction force is undetectable, but for bodies more equal in mass it can be significant.

Measuring Weight mg N

Weightlessness N=mg N>mgN<mg N=0

Weightlessness means =0 N compensating upward

Weightlessness in action

“Floating” is space is really free-falling in space

What is g on the moon? W= G mM M R M 2 W= m GM M R M 2 W= m g M gM=gM= GM M R M 2 W R M =1.7 x 10 6 m m M

g M on the Moon g = GM M R M 2 6.7x N m 2 /kg 2 x 7.4x10 22 Kg (1.7 x 10 6 m) 2 = g M = 1.7 m/s 2 1/6 x g Earth

© 2004 Pearson Education Inc., publishing as Addison-Wesley How does the acceleration of gravity depend on the mass of a falling object? It does not. All falling objects fall with the same acceleration (on a particular planet). Now see why… F = ma and on Earth acceleration due to gravity denoted “ g ” so F=mg or g=F/m If mass of earth is M 1 then F g =GM 2 /d 2

Thus, the weight of an object of mass m at the surface of the Earth is obtained by multiplying the mass m by the acceleration due to gravity, g, at the surface of the Earth. The acceleration due to gravity is approximately the product of the universal gravitational constant G and the mass of the Earth M, divided by the radius of the Earth, r, squared.

Gravity Near the Earth’s Surface; Geophysical Applications The acceleration due to gravity varies over the Earth’s surface due to altitude, local geology, and the shape of the Earth, which is not quite spherical.

Satellites and “Weightlessness” Satellites are routinely put into orbit around the Earth. The tangential speed must be high enough so that the satellite does not return to Earth, but not so high that it escapes Earth’s gravity altogether.

Satellites and “Weightlessness” The satellite is kept in orbit by its speed – it is continually falling, but the Earth curves from underneath it.

Satellites and “Weightlessness” Objects in orbit are said to experience weightlessness. They do have a gravitational force acting on them, though! The satellite and all its contents are in free fall, so there is no normal force. This is what leads to the experience of weightlessness.

Satellites and “Weightlessness” More properly, this effect is called apparent weightlessness, because the gravitational force still exists. It can be experienced on Earth as well, but only briefly:

The velocity of a satellite keeps it in orbit Even when moving, the satellite is actually accelerating toward the Earth (this is what keeps it in its circular path) Its acceleration results in a curved path which is the same as the curve of the Earth Gravity is providing the centripetal force

Perception of Weightlessness There is still gravity acting in a satellite (about 8.9 m/s 2 ), so why do we feel weightless? In an free falling elevator, if the F A is equal to the F G, there is no F N No force is felt feel weightless – called apparent weightlessness

Weightlessness that you feel in a satellite is like the weightlessness in an elevator The satellite and everything on it are all accelerating toward the earth at the same rate

Video Period 3 The Apple and the Moon

Problem 1 Two spheres of mass 35kg are 30m apart. A)What force does one exert on the other? B)If the mass of one is tripled and the radius is quadrupled how does the force change?

Problem 2 Two spheres of equal mass have a force of gravity of 7x10 -9 N exerted on each other. If the distance between them is 7m, find the mass.

Problem 3 Find the value of the gravitational acceleration g. The mass of the Earth is 6.0 x kg. The radius of the Earth is 6.38 x 10 6 m.

Elaboration Gravitation worksheet Law of Universal Gravitation.

Homework – Chapter 7 Kahoot