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Gravitation

Chapter 13

13-1 Newton's Law of Gravitation

The gravitational forceo Holds us to the Earth

o Holds Earth in orbit around the Sun

o Holds the Sun together with the stars in our galaxy

o Holds together the Local Group of galaxies

o Is responsible for black holes Gravitation is the tendency for bodies to attract each

other The force is always attractive, never repulsive Newton's law of gravitation defines the strength of

this attractive force between particles

13-1 Newton's Law of Gravitation

The magnitude of the force is given by:

Where G is the gravitational constant:

The force always points from one particle to the other, so this equation can be written in vector form:

Eq. (13-1)

Eq. (13-2)

Eq. (13-3)

13-1 Newton's Law of Gravitation

Earth is a nesting of shells, so we feel Earth's mass as if it were all located at its center

E.g., Earth-apple and apple-Earth forces are both 0.8 N

We can apply Newton’s law of gravitation to real objects as long as the size of the objects are small relative to the distance between them.

13-2 Gravitation and the Principle of Superposition

Find the net gravitational force by the principle of superposition: the net is the sum of individual forces

For n interacting particles, gravitational force on particle 1:

Gravitational force on particle 1 from a real object:

Eq. (13-5)

Eq. (13-6)

13-2 Gravitation and the Principle of Superposition

Sample Problem 13.01:

The figure below shows an arrangement of three particles, particle 1 of mass m1 = 6.0 kg and particles 2 and 3 of mass m2 = m3 = 4.0 kg, and distance a = 2.0 cm. What is the net gravitational force on particle 1 due to the other particles?

13-2 Gravitation and the Principle of Superposition

The figure shows three arrangements of three particles of equal masses. Rankthe arrangements according to the magnitude of the net gravitational force on theparticle labeled m, greatest first.

13-3 Gravitation Near Earth's Surface

Combine F = GMm/r2 and F = ma

g:

This gives the magnitude of the gravitational acceleration at a given distance from the center of the Earth

Table 13-1 shows the value for a

g for various altitudes

above the Earth’s surface

Eq. (13-11)

Table 13-1

13-3 Gravitation Near Earth's Surface

The calculated ag will differ slightly from the measured

g at any location Therefore the calculated gravitational force on an

object will not match its weight for the same 3 reasons:

1. Earth's mass is not uniformly distributed

2. Earth is not a sphere

3. Earth rotates

13-3 Gravitation Near Earth's Surface

Sample Problem 13.02:

An astronaut whose height h is 1.70 m floats “feet down” in an orbiting space shuttle at distance r = 6.77×106 m away from the center of Earth. What is the difference between the gravitational acceleration at her feet and at her head?

13-4 Gravitation Near Earth's Surface

The shell theorem also means that:

Forces between elements do not disappear, but their vector sum is 0

Let's find the gravitational force inside a uniform-density Earth:

Eq. (13-17)

13-4 Gravitation Near Earth's Surface

The constant density is:

Substitute in to Eq. 13-17:

Eq. (13-19)

13-5 Gravitational Potential Energy

Note that gravitational potential energy is a property of a pair of particles

We often speak as of the “gravitational potential energy of an baseball” in the ball-Earth system

13-5 Gravitational Potential Energy

Gravitational potential energy for a two-particle system is written:

Note this value is negative and approaches 0 for r → ∞

The gravitational potential energy of a system is the sum of potential energies for all pairs of particles

Eq. (13-21)

Eq. (13-22)

Figure 13-8

13-5 Gravitational Potential Energy

The gravitational force is conservative

The work done by this force does not depend on the path followed by the particles, only the difference in the initial and final positions of the particles

Since the work done is independent of path, so is the gravitational potential energy change

Eq. (13-26)

Figure 13-10

13-5 Gravitational Potential Energy

Newton's law of gravitation can be derived from the potential energy formula by taking the derivative

Escape speed: minimum initial speed that will cause a projectile move upward forever (stops only at infinity) At rest at infinity: K = 0 and U = 0 (because r → ∞)

Eq. (13-28)

13-5 Gravitational Potential Energy

Sample Problem 13.03:

An asteroid, headed directly toward Earth, has a speed of 12 km/s relative to the planet when the asteroid is 10 Earth radii from Earth’s center. Neglecting the effects of Earth’s atmosphere on the asteroid, find the asteroid’s speed vf when it reaches Earth’s surface.

13-6 Planets and Satellites: Kepler's Laws

Johannes Kepler (1571-1630) derived laws of motion using Tycho Brahe's (1546-1601) measurements

Figure 13-12

13-6 Planets and Satellites: Kepler's Laws

The orbit is defined by its semimajor axis a and its eccentricity e

An eccentricity of zero corresponds to a circle Eccentricity of Earth's orbit is 0.0167

Equivalent to the law of conservation of angular momentum

13-6 Planets and Satellites: Kepler's Laws

The law of periods can be written mathematically for circular orbits:

Holds for elliptical orbits if we replace r with a

Figure 13-13

Eq. (13-34)

13-6 Planets and Satellites: Kepler's Laws

Sample Problem 13.04:Comet Halley orbits the Sun with a period of 76 years and, in 1986, had a distance of closest approach to the Sun, its perihelion distance Rp, of 8.9 x m. This is between the orbits of Mercury and Venus.(a) What is the comet’s farthest distance from the Sun, which is called its aphelion distance Ra?(b) What is the eccentricity e of the orbit of comet Halley?

13-7 Satellites: Orbits and Energy

If a satellite orbits Earth in circular orbit (m―mass of satellite, M―mass of Earth):

Meaning that:

Therefore the total mechanical energy is:

Eq. (13-38)

Eq. (13-39)

Eq. (13-40)

13-7 Satellites: Orbits and Energy

For an ellipse, we substitute a for r Therefore the energy of an orbit depends only on its

semimajor axis, not its eccentricity

Eq. (13-42)

Sample Problem 13.05:A playful astronaut releases a bowling ball, of mass m = 7.20 kg, into circular orbit about Earth at an altitude h of 350 km.(a) What is the mechanical energy E of the ball in its orbit?(b) What is the mechanical energy E0 of the ball on the Launchpad at Cape Canaveral (before it, the astronaut, and the spacecraft are launched)? From there to the orbit, what is the change ∆E in the ball’s mechanical energy?

13-8 Einstein and Gravitation

The general theory of relativity describes gravitation Its fundamental postulate is the principle of

equivalence Gravitation and acceleration are equivalent

Figure 13-18

The experimenter inside this box is unable to tell whether he is on Earth experiencing g = 9.8 m/s2, or in free space accelerating at 9.8 m/s2.

13-8 Einstein and Gravitation Gravitation is due to a curvature of space that is

caused by the masses. Analogies: In (a) and (b), paths that appear to be

parallel, along the surface of the Earth or falling toward the Earth's center, actually converge

(c) Far from Earth, space is flat and parallel paths remains parallel. Close to Earth, the parallel paths begin to converge.

Figure 13-19

13-8 Einstein and Gravitation

We can observe the curvature of space Light bends as it passes by massive objects: an effect

called gravitational lensing In extreme cases we observe the light coming from

multiple places, or bent to form an Einstein ring

Figure 13-20