-Gravitational Field-Gravitational

Potential Energy

AP Physics CMrs. Coyle

Remember: Remember: Newton’s Law of Universal Gravitation

1 212 122

ˆmm

Gr

F r

-Universal Gravitation Constant G=6.67x 10-11 Nm2/kg2

-The gravitational force is a field force.

Review QuestionReview Question:

• Which exerts a greater force, the earth on the moon or the moon on the earth?

• Gravitational Field: tGravitational Field: the space around a mass. Here a test mass would feel a gravitational force.

• Gravitational Field Vector: Gravitational Field Vector: g

mF

g

Gravitational Field Vector, g at the surface of the Earth

2

2

E

E

E

E

M mmg G

R

Mg G

R

gg above the Earth’s surface above the Earth’s surface

• r = RE + h

Note:Note: • g decreases with increasing altitude• As r , the weight of the object

approaches zero

2E

E

GMg

R h

Variation of g with Height from the surface of the Earth

RememberRemember• The gravitational force is conservative

• The gravitational force is a central force (A central force has a direction towards the

center and its magnitude depends only on r)

• A central force can be represented by

ˆF F r r

Work done by the Gravitational Force Work done by the Gravitational Force • A particle moves from A

to B while acted on by a central force F

• We approximate the path along A to B with radial and arc zigzags

• The work done by F along the arcs is zero

• The work done by F along the radial direction is

( )dW d F r dr F r

Work done by the Gravitational Work done by the Gravitational ForceForce

• The work done is independent of the path and depends only on rf and ri

• This proves that the gravitational force is conservative.

( )dW d F r dr F r

( ) f

i

r

rW F r dr

Gravitational Potential EnergyGravitational Potential Energy• As a particle moves from A to B, its gravitational

potential energy changes by:

( ) f

i

f i

r

f i r

U U U W

U U F r dr

Gravitational Potential Energy of Gravitational Potential Energy of the Earth-particle systemthe Earth-particle system

• The reference point is chosen at infinity where the force on a particle would approach zero. Ui = 0 for ri =

• ∞This is valid only for rRE and not valid for r < RE

• U is negative because of the choice of Ui

( ) EGM mU r

r

Gravitational Gravitational Potential Potential

Energy of the Energy of the Earth-Earth-

particle particle systemsystem

Gravitational Potential Energy of any two particles

1 2GmmU

r

Gravitational Potential Energy of a system of any two particles

• U = -Gm1m2

rThe reference

point U=0 is at infinity.

Gravitational Potential Energy

Gravitational Potential Energy

• An outside force must do positive work to increase the separation between two objects

• This work gives the objects a greater potential energy (less negative).

Binding EnergyBinding Energy

• The absolute value of the potential energy is the binding energy

• An outside force must supply energy gretaer or equal to the binding energy to separate the particles to an infinite distance of separation.

• The excess energy will be in the form of kinetic energy of the particles when they are at infinite separation.

Systems with Three or More Systems with Three or More Particles (Configuration of Masses)Particles (Configuration of Masses)

• The total gravitational potential energy of the system is the sum over all pairs of particles

• Gravitational potential energy obeys the superposition principle

Systems with Three Particles

• The absolute value of Utotal represents the work needed to separate the particles by an infinite distance.

• Remember energy is a scalar quantity.

total 12 13 23

1 2 1 3 2 3

12 13 23

U U U U

mm mm m mG

r r r

Configurations of Masses

• Gravitational Forces are added using the vector component method.

• To find the Gravitational Potential Energyof the configuration of masses, the individual

energies are added as scalars. • A force would have to supply an amount of

energy equal to the individual energy in order to separate the masses by an infinite distance.

Ex #31

• A system consists of three particles, each of mass 5.00g, located at the corners of an equilateral triangle with sides of 30.0cm.

a)Calculate the potential energy of the system.b)If the particles are released simultaneously,

where will they collide?

Ans: a) -1.67x10-14 J