Gravity

Newton’s Law of Gravitation

Kepler’s Laws of Planetary Motion

Gravitational Fields

Newton’s Law of Gravitation

m1

m2r

There is a force of gravity between any pair of objects anywhere. The force is proportional to each mass and inversely proportional to the square of the distance between the two objects. Its equation is:

FG = G m1 m2

r 2

The constant of proportionality is G, the universal gravitation constant. G = 6.67 · 10-11 N·m2 / kg2. Note how the units of G all cancel out except for the Newtons, which is the unit needed on the left side of the equation.

Gravity Example

FG = G m1 m2

r 2

How hard do two planets pull on each other if their masses are 1.23 1026 kg and 5.21 1022 kg and they 230 million kilometers apart?

This is the force each planet exerts on the other. Note the denominator is has a factor of 103 to convert to meters and a factor of 106 to account for the million. It doesn’t matter which way or how fast the planets are moving.

(6.67 · 10-11 N·m2 / kg2) (1.23 · 1026 kg) (5.21 · 1022 kg) =(230 · 103 · 106 m)

2

= 8.08 · 1015 N

3rd Law: Action-ReactionIn the last example the force on each planet is the same. This is due to to Newton’s third law of motion: the force on Planet 1 due to Planet 2 is just as strong but in the opposite direction as the force on Planet 2 due to Planet 1. The effects of these forces are not the same, however, since the planets have different masses.

For the big planet: a = (8.08 · 1015 N) / (1.23 · 1026 kg) = 6.57 · 10-11 m/s2.

For the little planet: a = (8.08 · 1015 N) / (5.21 · 1022 kg) = 1.55 · 10-7 m/s2.

5.21 · 1022 kg

1.23 · 1026 kg8.08 · 1015 N 8.08 · 1015 N

Inverse Square Law

FG = G m1 m2

r 2

The law of gravitation is called an inverse square law because the magnitude of the force is inversely proportional to the square of the separation. If the masses are moved twice as far apart, the force of gravity between is cut by a factor of four. Triple the separation and the force is nine times weaker.

What if each mass and the separation were all quadrupled?

answer: no change in the force

Calculating the Gravitational ConstantIn 1798 Sir Henry Cavendish suspended a rod with two small masses (red) from a thin wire. Two larger mass (green) attract the small masses and cause the wire to twist slightly, since each force of attraction produces a torque in the same direction. By varying the masses and measuring the separations and the amount of twist, Cavendish was the first to calculate G.

Since G is only 6.67 · 10-11 N·m2 / kg2, the measurements had to be very precise.

Calculating the mass of the EarthKnowing G, we can now actually calculate the mass of the Earth. All we do is write the weight of any object in two different ways and equate them. Its weight is the force of gravity between it and the Earth, which is FG in the equation below. ME is the mass of the Earth, RE is the radius of the Earth, and m is the mass of the object. The object’s weight can also be written as mg.

=FG = G m1 m2

r 2

G ME m RE

2 = m g

The m’s cancel in the last equation. g can be measured experimentally; Cavendish determined G’s value; and RE can be calculated at 6.37 · 106 m (see next slide). ME is the only unknown. Solving for

ME we have: g RE

2

GME = = 5.98 · 1024 kg

Calculating the radius of the Earth

RE

This is similar to the way the Greeks approximated Earth’s radius over 2000 years ago:

Red pole blocks incoming solar rays and makes a shadow. is measured.

The green pole is on the same meridian as the red one, but it casts no shadow at this latitude. s is the measured distance between the cities in which the poles stand.

s

is also the central angle of the arc. s = RE

RE = s / 6.37 · 106 m

Earth

Net Force Gravity Problem3 106 kg

2.7 106 kg 2.5 106 kg

40 m

60 m

3 asteroids are positioned as shown, forming a right triangle. Find the net force on the 2.5 million kg asteroid.

Steps: 1. Find each force of gravity on it and draw in the vectors.2. Find the angle at the lower right.3. One force vector is to the left; break the other one down into components.4. Find the resultant vector: magnitude via Pythagorean theorem; direction via inverse tangent.

answer: 0.212 N at 14.6 above horizontal (N of W)

Falling Around the Earth v

Newton imagined a cannon ball fired horizontally from a mountain top at a speed

v. In a time t it falls a distance y = 0.5 g t 2 while

moving horizontally a distance x = v t. If fired fast

enough (about 8 km/s), the Earth would curve downward the same amount the cannon ball falls downward. Thus, the projectile would never hit the ground, and it would be in orbit.

The moon “falls” around Earth in the exact same way but at a much greater altitude. .

x = v t

y = 0.5 g t 2 {

continued on next slide

Necessary Launch Speed for OrbitR = Earth’s radiust = small amount of time after launchx = horiz. distance traveled in time ty = vertical distance fallen in time t

RR

y = g t 2 / 2

x = v t

(If t is very small, the red segment is nearly vertical.)

x 2 + R

2 = (R + y) 2

= R

2 + 2 R y + y 2

Since y << R,

x 2 + R

2 R

2 + 2 R y

x 2 2 R y

v 2 t

2 2 R (g t 2

/ 2)

v 2 R g. So,

v (6.37 · 106 m · 9.8 m/s2)

½

v 7900 m/s

Early AstronomersIn the 2nd century AD the Alexandrian astronomer Ptolemy put forth a theory that Earth is stationary and at the center of the universe and that the sun, moon, and planets revolve around it. Though incorrect, it was accepted for centuries.

In the early 1500’s the Polish astronomer Nicolaus Copernicus boldly rejected Ptolemy’s geocentric model for a heliocentric one. His theory put the sun stated that the planets revolve around the sun in circular orbits and that Earth rotates daily on its axis.

In the late 1500’s the Danish astronomer Tycho Brahe made better measurements of the planets and stars than anyone before him. The telescope had yet to be invented. He believed in a Ptolemaic-Coperican hybrid model in which the planets revolve around the sun, which in turn revolves around the Earth.

Early Astronomers

In the late 1500’s and early 1600’s the Italian scientist Galileo was one of the very few people to advocate the Copernican view, for which the Church eventually had him placed under house arrest. After hearing about the invention of a spyglass in Holland, Galileo made a telescope and discovered four moons of Jupiter, craters on the moon, and the phases of Venus.

The German astronomer Johannes Kepler was a contemporary of Galileo and an assistant to Tycho Brahe. Like Galileo, Kepler believed in the heliocentric system of Copernicus, but using Brahe’s planetary data he deduced that the planets move in ellipses rather than circles. This is the first of three planetary laws that Kepler formulated based on Brahe’s data.

Both Galileo and Kepler contributed greatly to work of the English scientist Sir Isaac Newton a generation later.

Kepler’s Laws of Planetary Motion

1. Planets move around the sun in elliptical paths with the sun at one focus of the ellipse.

2. While orbiting, a planet sweep out equal areas in equal times.

3. The square of a planet’s period (revolution time) is proportional to the cube of its mean distance from the sun: T

2 R 3

Here is a summary of Kepler’s 3 Laws:

These laws apply to any satellite orbiting a much larger body.

Kepler’s First LawPlanets move around the sun in elliptical paths with the

sun at one focus of the ellipse.

An ellipse has two foci, F1 and F2. For any point P on the ellipse, F1 P + F2 P is a constant. The orbits of the planets are nearly circular (F1 and F2 are close together), but not perfect circles. A circle is a an ellipse with both foci at the same point--the center. Comets have very eccentric (highly elliptical) orbits.

F1 F2

Sun

PlanetP

Kepler’s Second Law

Sun

While orbiting, a planet sweep out equal areas in equal times.

C

D

A

B

The blue shaded sector has the same area as the red shaded sector. Thus, a planet moves from C to D in the same amount of time as it moves from A to B. This means a planet must move faster when it’s closer to the sun. For planets this affect is small, but for comets it’s quite noticeable, since a comet’s orbit is has much greater eccentricity.

(proven in advanced physics)

Kepler’s Third LawThe square of a planet’s period is proportional to the

cube of its mean distance from the sun: T 2 R

3

Assuming that a planet’s orbit is circular (which is not exactly correct but is a good approximation in most cases), then the mean distance from the sun is a constant--the radius. F is the force of gravity on the planet. F is also the centripetal force. If the orbit is circular, the planet’s speed is constant, and v = 2 R / T. Therefore,

Sun

PlanetF

RM

m

G M mR

2

m v 2

R =

G M R

2

m [2 R / T] 2

R =

Cancel m’s and simplify:

4 2 R

T 2

=

Rearrange:G M

T 2

4 2

= R3

Since G, M, and are constants, T 2 R

3.

Third Law Analysis

G M T

2 4 2

= R 3We just derived

• It also shows that the orbital period depends on the mass of the central body (which for a planet is its star) but not on the mass of the orbiting body. In other words, if Mars had a companion planet the same distance from the sun, it would have the same period as Mars, regardless of its size.

• This shows that the farther away a planet is from its star, the longer it takes to complete an orbit. Likewise, an artificial satellite circling Earth from a great distance has a greater period than a satellite orbiting closer. There are two reasons for this: 1. The farther away the satellite is, the farther it must travel to complete an orbit; 2. The farther out its orbit is, the slower it moves, as shown:

G M mR

2

m v 2

R = v =

G M

R

Third Law ExampleOne astronomical unit (AU) is the distance between Earth and the sun (about 93 million miles). Jupiter is 5.2 AU from the sun. How long is a Jovian year?

answer: Kepler’s 3rd Law says T 2 R

3, so T 2 = k R

3, where k is the constant of proportionality. Thus, for Earth and Jupiter we have:

TE 2 = k RE

3 and TJ 2 = k RJ

3

k’s value matters not; since both planets are orbiting the same central body (the sun), k is the same in both equations. TE = 1 year, and RJ / RE = 5.2, so dividing equations:

TJ 2

TE 2

RJ 3

RE 3= TJ

2 = (5.2) 3 TJ = 11.9 years

continued on next slide

1 year 365 days

Third Law Example (cont.)

What is Jupiter’s orbital speed?

answer: Since it’s orbital is approximately circular, and it’s speed is approximately constant:

2 (5.2) (93 · 106 miles)v = dt

= 11.9 years

Jupiter is 5.2 AU from the sun (5.2 times farther than Earth is).

· ·1 day

24 hours

29,000 mph. Jupiter’s period from last slide

This means Jupiter is cruising through the solar system at about 13,000 m/s ! Even at this great speed, though, Jupiter is so far away that when we observe it from Earth, we don’t notice it’s motion.

Planets closer to the sun orbit even faster. Mercury, the closest planet, is traveling at about 48,000 m/s !

Third Law Practice Problem

Venus is about 0.723 AU from the sun, Mars 1.524 AU. Venus takes 224.7 days to circle

the sun. Figure out how long a Martian year is.

answer: 686 days

Uniform Gravitational FieldsWe live in what is essentially a uniform gravitational field. This means that the force of gravity near the surface of the Earth is pretty much constant in magnitude and direction. The green lines are gravitational field lines. They show the direction of the gravitational force on any object in the region (straight down). In a uniform field, the lines are parallel and evenly spaced. Near Earth’s surface the magnitude of the gravitational field is 9.8 N / kg. That is, every kilogram of mass an object has experiences 9.8 N of force. Since a Newton is a kilogram ·

meter per second squared, 1 N / kg = 1 m/s2. So, the gravitational field strength is just the acceleration due to gravity, g.

Earth’s surface

continued on next slide

Uniform Gravitational Fields (cont.)

Earth’s surface

10 kg

98 N

A 10 kg mass is near the surface of the Earth. Since the field strength is 9.8 N / kg, each of the ten kilograms feels a

9.8 N force, for a total of 98 N. So, we can calculate the force of gravity by multiply mass and field strength. This is the same as calculating its weight (W = mg).

Nonuniform Gravitational FieldsNear Earth’s surface the gravitational field is approximately uniform. Far from the surface it looks more like a sea urchin.

Earth

The field lines

• are radial, rather than parallel, and point toward center of Earth.

• get farther apart farther from

the surface, meaning the field is weaker there.

• get closer together closer to the surface, meaning the field is stronger there.