halliday, resnick & walker chapter 13newt.phys.unsw.edu.au/~mgb/phys1121/lecturenotes/... ·...

Gravitation

Halliday, Resnick & Walker Chapter 13

Physics 1A – PHYS1121 Professor Michael Burton

II_A2: Planetary Orbits in the Solar System

+ Galaxy Interactions (You Tube)

21 seconds

13-1 Newton's Law of Gravitation

!  The gravitational force o  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  Holds together the Local Supercluster of galaxies

o  Attempts to slow the expansion of the Universe

o  Is responsible for black holes

!  Gravity is far-reaching and very important!


!  Gravitational attraction depends on mass of an object !  Earth has a large mass and produces a large attraction

!  The force is always attractive, never repulsive !  Bodies attract each other through gravitational

attraction !  Newton realized this attraction was responsible for

maintaining the orbits of celestial bodies !  Newton's Law of Gravitation defines the strength of this

attractive force between particles !  Between an apple & the Earth: ~0.8 N !  Between 2 people: < 1 µN


!  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)

r

m1 m2 F F


!  The shell theorem describes gravitational attraction for objects

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

!  Gravitational force forms third-law force pairs (i.e. N3L) !  e.g. Earth-apple and apple-Earth

forces are both 0.8 N

M M M


!  Earth-apple and apple-Earth forces are both ~0.8 N

!  The difference in mass causes the difference in the apple:Earth accelerations:

~10 m/s2 vs. ~ 10-25 m/s2

Consider the objects of various masses indicated below. The objects are each separated from another object by the distance indicated. In which of these situations is the gravitational force exerted on the two objects the largest?

a) #1

b) #2

c) #3

d) #2 and #3

e) #1, #2, and #3

13-2 Gravitation and the Principle of Superposition

!  The principle of superposition applies. !  i.e. Add the individual forces as vectors:

!  For a real (extended) object, this becomes an integral:

!  If the object is a uniform sphere or shell we can treat its mass as being at its centre instead

Eq. (13-5)

Eq. (13-6)

M M M

13-2 Gravitation and the Principle of Superposition

Figure 13-4

Example Summing two forces:

Consider a system of particles, each of mass m. In which one of the following configurations is the net gravitational force on Particle A the largest? The horizontal or vertical spacing between particles is the same in each case.

a) 1

b) 2

c) 4

d) 1 and 2 equally large

e) 2 and 4 are equally large

13-3 Gravitation Near the Earth's Surface

!  Combine F = GMm/r2 and F = mag:

!  Gives magnitude of gravitational acceleration at a given distance from the centre of the Earth !  Table 13-1 shows the value for ag

for various altitudes above the Earth’s surface

Eq. (13-11)

13-3 Gravitation Near the Earth's Surface

!  The calculated ag will differ slightly from the measured g at any location on the Earth’s surface

!  Three reasons. The Earth…. 1.  mass is not uniformly distributed 2.  is not a perfect sphere 3.  rotates

13-3 Gravitation Near Earth's Surface

Figure 13-6

Example Difference in gravitational force and weight due to rotation at the equator:

!

N2L : FN "mag = "mV 2

R, the centripetal acceleration.

Here V =#R with # the angular velocity, and FN = mg.Thus g = ag "#

2R.

Exercise for the student. Show this is about 0.034 m/s2. Use R=6,400km and !=2" radians/day. Question: do you weigh more or less at the equator than the Poles?

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)

!

Proof comes from integrating the force to obtain the work done.

i.e. U = "W = "! F • d! r and using F = GMm

r2.#


!  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

Figure 13-10

Eq. (13-26)


!  For a projectile to escape the gravitational pull of a body, it must come to rest only at infinity (if at all).

!  At rest at infinity: K = 0 and U = 0 (because r ! ") !  So K + U must be # 0 at surface of the body to escape:

!  This is the escape speed. The minimum value to escape. !  Rockets launch eastward to take advantage of Earth's

rotational speed, to reach vescape more easily


You move a ball of mass m away from a sphere of mass M.

1.  Does the gravitational potential energy of the system of the ball and sphere

a)  Increase, or

b)  Decrease.

2. Is the Work done by the gravitational force between the ball and the sphere

a)  Positive work, or

b)  Negative work

m M

r

In a distant solar system where several planets are orbiting a single star of mass M, a large asteroid collides with a planet of mass m orbiting the star at a distance r. As a result, the planet is ejected from its solar system. What is minimum amount of energy that the planet must receive in the collision to be removed from the solar system?

a)

b)

c)

d)

e)

m M

r

2_A3: Retrograde Motion of the Planets

13-6 Planets and Satellites: Kepler's Laws

!  The motion of planets in the solar system was a puzzle for astronomers, especially curious motions such as “retrograde motion”.

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

Figure 13-12 Figure 13-11


!  The orbit is defined by its semi-major axis a and its eccentricity e

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

Kepler2L: Kepler’s 2nd Law Equal Areas in Equal Times


!  Equivalent to Conservation of Angular Momentum !  See later in this course.


!  The law of periods can be written mathematically as:

!  Holds for elliptical orbits if we replace r with a, the semi-major axis.


A spacecraft is in low orbit of the Earth with a period of approximately 90 minutes. By which of the following methods could the spacecraft stay in the same orbit and reduce the period of the orbit?

a) Before launch, increase the mass of the spacecraft to increase the centripetal force on it.

b) Remove any unnecessary equipment, cargo, and supplies to reduce the mass and decrease its angular momentum.

c) Fire rockets to increase the tangential velocity of the ship.

d) None of the above methods will achieve the desired effect.

13-7 Satellites: Orbits and Energy

!  Relating the centripetal acceleration of a satellite to the gravitational force, we can rewrite as energies:

!  Meaning that:

!  Therefore the total mechanical energy is:

Eq. (13-38)

Eq. (13-39)

Eq. (13-40)


!  Total energy E is the negative of the kinetic energy !  For an ellipse, we substitute a for r !  Therefore the energy of an orbit depends only on its

semi-major axis, not its eccentricity !  All orbits in Figure 13-15 have the same energy

Figure 13-15

13-7 Satellites: Circular Orbit

!  Graph of variation in Energy for a circular orbit, radius r !  Note that:

!  E(r) and U(r) are negative !  E(r) = -K(r) !  E(r), U(r), K(r) all ! 0 as r!!


a)  Which orbit (1, 2 or 3?) will the shuttle take when it fires a forward-pointing thruster so as to reduce its kinetic energy?

b)  Is the orbital period T then (i) greater than, (ii) less than or (iii) the same as, that of the circular orbit?

The Law of Gravitation Gravitational Behavior of Uniform Spherical Shells !  The net force on an external object: calculate as if all the mass were concentrated at the centre of the shell

13 Summary

Eq. (13-1)

Gravitational Acceleration Superposition

Eq. (13-5) Eq. (13-11)

Eq. (13-2)

13 Summary

Gravitational Potential Energy

Eq. (13-21)

Potential Energy of a System

Eq. (13-22)

Eq. (13-19)

Free-Fall Acceleration and Weight !  Earth's mass is not uniformly distributed, the planet is not spherical, and it rotates: the calculated and measured values of acceleration differ

Gravitation within a Spherical Shell !  A uniform shell exerts no net force on a particle inside

!  Inside a solid sphere:

Escape Speed

13 Summary

Energy in Planetary Motion

Eq. (13-42)

Eq. (13-28)

Kepler's Laws !  The law of orbits: ellipses

!  The law of areas: equal areas in equal times

!  The law of periods:

Eq. (13-34)

Kepler's Laws !  Gravitation and acceleration

are equivalent

halliday, resnick & walker chapter 13newt.phys.unsw.edu.au/~mgb/phys1121/lecturenotes/... ·...

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