Download 3 AP Gravitational Field and Gravitational Potential Energy

-Gravitational Field
-Gravitational
Potential Energy
AP Physics C
Mrs. Coyle
Remember: Newton’s Law of Universal Gravitation
F12
m1m2
ˆ12
 G
r
2
r
or
F12
m1m2
 G
r2
-Universal Gravitation Constant
G=6.67x 10-11 Nm2/kg2
-The gravitational force is a field force.
The minus sign shows that the force is
opposite the unit vector r12 .
-
• Gravitational Field: the space around a mass.
• A test mass would feel a gravitational force
when placed in the field.
• Gravitational Field Vector:
g
Fg
m
Gravitational Field Vector, g at the
surface of the Earth
M Em
mg  G 2
RE
ME
g G 2
RE
g above the Earth’s surface
GM E
• r = RE + h
g
2
 RE  h 
Note:
• g decreases with increasing altitude
• As r , the weight of the object approaches
zero
Variation of g with Height from the
surface of the Earth
Remember
• The gravitational force is a conservative force.
• The gravitational force is also 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
F  F  r  rˆ
Work done by the Gravitational Force as the
particle moves from A to B
• The work done by F is:
dW

F

d
r

F
(
r
)
dr
 

• The path is approximated by
radial and arc zig zags.
• The work done by F along the
arcs is zero.
Work done by the Gravitational
Force
rf
W   F ( r ) dr
ri
• The work done is independent of the path and
depends only on rf and ri
• This proves that the gravitational force is
conservative because it is independent of the
path taken.
Gravitational Potential Energy
• As a particle moves from A to B, its gravitational
potential energy changes by:
U  U f  U i  W
rf
U f  U i    F (r ) dr
ri
Gravitational Potential Energy of the
Earth-particle system
• The reference point is chosen at infinity where the
force on a particle would approach zero.
• Ui = 0 for ri =

GM E m
U (r)  
r
• This is valid only for r > RE (outside the earth) and not
valid for r < RE
• U is negative because of the choice of Ui
Gravitational
Potential
Energy of the
Earthparticle
system
Gravitational Potential Energy of
any two particles
Gm1m2
U 
r
Gravitational Potential Energy of a
system of any two particles
• U = -Gm1m2
r
The 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 Energy
• The absolute value of the potential energy is the
binding energy
• An outside force must supply energy greater 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 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.
U total  U12  U13  U 23
 m1m2
m1m3
m2 m3 
 G 



r13
r23 
 r12
Systems with Three Particles
U total  U12  U13  U 23
 m1m2
m1m3
m2 m3 
 G 



r13
r23 
 r12
• The absolute value of Utotal represents the work needed to
separate the particles by an infinite distance.
• Remember energy is a scalar quantity.
Configurations of Masses
• Gravitational Forces are added using the vector
component method.
• To find the Gravitational Potential Energy
of 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, b) at the center of the triangle.