Chapter 13 - Gravity

Newton’s Law ofGravitation

Gravitational PotentialEnergy

Kepler’s Laws

“The moon is essentially gray,no color. It looks like plasterof Paris, like dirty beach sandwith lots of footprints in it.”

-James A. Lovell(from the Apollo 13 mission)

David J. StarlingPenn State Hazleton

PHYS 211

Kepler’s Laws

Newton’s Law of Gravitation

The gravitational force is a mutual force between

two separated objects (distance r) of masses m1

and m2 given by

F = Gm1m2

G = 6.67× 10−11 m3/kg-s2.

Kepler’s Laws

The gravitational force is a mutual force between

two separated objects (distance r) of masses m1

and m2 given by

F = Gm1m2

G = 6.67× 10−11 m3/kg-s2.

Kepler’s Laws

From Newton’s third law, we know that this force

must have an equal but opposite pair.

Kepler’s Laws

Shell Theorem: a uniform sphere of matter

attracts a particle that is outside as if all the

sphere’s mass were concentrated at its center.

Uniform spherical objects just become points.

Kepler’s Laws

Shell Theorem: a uniform sphere of matter

attracts a particle that is outside as if all the

sphere’s mass were concentrated at its center.

Uniform spherical objects just become points.

Kepler’s Laws

Principle of Superposition: If N objects interact

with particle 1 gravitationally, the total force is

just the vector sum.

~F1,net = ~F12 + ~F13 + · · ·+ ~F1N

~F1,net =

N∑i=2

Kepler’s Laws

We can apply Newton’s Law of Gravitation to an

object (m) near the surface of the Earth (M):

~F = GMmr2 = mag

ag =GMr2

Kepler’s Laws

The force due to gravity on the surface of the

earth is not consistently 9.83 m/s2.

I Earth is not a perfect sphere;

I the mass within the Earth is not uniformly distributed;

I Earth rotates.

Kepler’s Laws

The force due to gravity on the surface of the

earth is not consistently 9.83 m/s2.

I Earth is not a perfect sphere;

I the mass within the Earth is not uniformly distributed;

I Earth rotates.

Kepler’s Laws

Lecture Question 13.1If an object at the surface of the Earth has a weight W, whatwould be the weight of the object if it was transported to thesurface of a planet that is one-sixth the mass of Earth andhas a radius one third that of Earth?

(a) 3W

(b) 4W/3

(d) 3W/2

(e) W/3

Kepler’s Laws

Gravitational Potential Energy

Gravity is a conservative force, so lets find its

potential energy using ∆U = −W.

∆U = −W

U(∞)− U(R) = −∫ ∞

~F(r) · d~r

= −GMm∫ ∞

1r2 dr

= −[

− U(R) = −0 +GMm

U(R) = −GMmR

Kepler’s Laws

∆U = −W

U(∞)− U(R) = −∫ ∞

~F(r) · d~r

= −GMm∫ ∞

1r2 dr

= −[

− U(R) = −0 +GMm

U(R) = −GMmR

Kepler’s Laws

∆U = −W

U(∞)− U(R) = −∫ ∞

~F(r) · d~r

= −GMm∫ ∞

1r2 dr

= −[

− U(R) = −0 +GMm

U(R) = −GMmR

Kepler’s Laws

∆U = −W

U(∞)− U(R) = −∫ ∞

~F(r) · d~r

= −GMm∫ ∞

1r2 dr

= −[

− U(R) = −0 +GMm

U(R) = −GMmR

Kepler’s Laws

∆U = −W

U(∞)− U(R) = −∫ ∞

~F(r) · d~r

= −GMm∫ ∞

1r2 dr

= −[

− U(R) = −0 +GMm

U(R) = −GMmR

Kepler’s Laws

∆U = −W

U(∞)− U(R) = −∫ ∞

~F(r) · d~r

= −GMm∫ ∞

1r2 dr

= −[

− U(R) = −0 +GMm

U(R) = −GMmR

Kepler’s Laws

The change in gravitational potential energy ∆U

is path independent.

∆U = −W

Kepler’s Laws

The force from this potential energy is just the

derivative (since we used an integral to derive it).

F(r) = −dUdr

= − ddr

(−GMm

)= −GMm

The minus sign indicates the force points radially inward.

Kepler’s Laws

The force from this potential energy is just the

derivative (since we used an integral to derive it).

F(r) = −dUdr

= − ddr

(−GMm

)= −GMm

The minus sign indicates the force points radially inward.

Kepler’s Laws

Lecture Question 13.2A large asteroid collides with a planet of mass m orbiting astar of mass M at a distance r. As a result of the collision,the planet is knocked out of its orbit, such that it leaves thesolar system. Which of the following expressions gives theminimum amount of energy that the planet must receive inthe collision to be removed from the solar system?

(a) GMm/r

(b) GMm/r2

(c) GMm/√

(d) Gm/r

(e) Gm/r2

Kepler’s Laws

Johannes Kepler was a 17th century

mathematician who developed three laws of

planetary motion.

1. The Law of Orbits: all planets move in elliptical orbitswith the Sun at one focus.

Kepler’s Laws

2. The Law of Areas: a line that connects a planet to theSun sweeps out equals areas in equal time intervals (i.e.,dA/dt = constant).

∆A =12

r2(∆θ)→ dA =12

r2 dθdt

L = rp⊥ = rmv⊥ = rmrω → dAdt

Kepler’s Laws

∆A =12

r2(∆θ)→ dA =12

r2 dθdt

Kepler’s Laws

∆A =12

r2(∆θ)→ dA =12

r2 dθdt

Kepler’s Laws

3. The Law of Periods: the square of the period of anyplanet is proportional to the cube of the semimajor axis ofits orbit.

F = ma

GMmr2 = m(rω2) = mr2

Kepler’s Laws

F = maGMm

r2 = m(rω2) = mr2(

Kepler’s Laws

F = maGMm

r2 = m(rω2) = mr2(

Kepler’s Laws

When one object orbits a much larger object,

mechanical energy is conserved.

For a circular orbit,

F = ma→ GMmr2 = m

→ GMm2r

mv2 = K

E = K + U

2r− GMm

E = −GMm2r

(for an elliptical orbit, E = −GMm/2a)

Kepler’s Laws

F = ma→ GMmr2 = m

r→ GMm

mv2 = K

E = K + U

2r− GMm

E = −GMm2r

Kepler’s Laws

F = ma→ GMmr2 = m

r→ GMm

mv2 = K

E = K + U

2r− GMm

E = −GMm2r

Kepler’s Laws

F = ma→ GMmr2 = m

r→ GMm

mv2 = K

E = K + U

2r− GMm

E = −GMm2r

Kepler’s Laws

The total energy of an orbiting body is negative.

E = −GMm2r

Chapter 13 - Gravity

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