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Physics 103: Lecture 13
Newtonian Gravity
Extended Objects : Center of Gravity

03/05/2003
Today’s lecture will cover
Newton’s Law of Gravitation
Kepler’s Laws
Center of Gravity
Physics 103, Spring 2003, U. Wisconsin
1
Lecture 13 - Preflight 1, 2 & 3

You are driving a car with constant speed around a
horizontal circular track. On a piece of paper draw a Free
Body diagram for the car.
How many forces are acting on the car?
2%
9%
61%
24%
1 2 3 4 5
a)
gravity
b)
normal force of road
c)
centripetal force
3%
Fc
0%
20%
40%
60%
80%
Pretty Sure
Not Quite Sure
Just Guessing
03/05/2003
N
mg
N
Physics 103, Spring 2003, U. Wisconsin
2
Lecture 13 Preflight 4 & 5

The net force on the car is
Zero
Pointing radially inward, toward the center of the circle
Pointing radially outward, away from the center of the circle
32%
55%
To travel in a circle
something must supply the
centripetal acceleration.
13%
0%
03/05/2003
20%
40%
60%
Pretty Sure
Not Quite Sure
Just Guessing
Physics 103, Spring 2003, U. Wisconsin
3
Lecture 13 - Preflight 6 & 7

Suppose you are driving through a valley whose bottom has a circular
shape. If your car has mass M, what is the normal force exerted on you
by the car seat as you drive past the bottom of the hill?
Fn < Mg
V
Fn = Mg
down
Fn > Mg
Normal force must also supply
centripetal acceleration
20%
51%
29%
Pretty Sure
Not Quite Sure
Just Guessing
0%
03/05/2003
20%
40%
60%
Physics 103, Spring 2003, U. Wisconsin
4
A SPECIAL POINT
Center of mass (or center of gravity)
If line of force passes thru c.m. No rotation
m1r1  m2 r2 m3 r3 m4 r4 .......
rcm 
m1 m2  m3  m4  ........
Center of mass is the same as the center of the volume
Provided: uniform density -- material!

03/05/2003
Symmetry
Physics 103, Spring 2003, U. Wisconsin
5
Question 1
Can a body’s center of gravity be outside its volume?
a) yes
b) no
Center of gravity (CG) is defined as:
In three dimensions, the coordinates of CG
xcm 
m1 x1  m 2 x2 
m1  m 2 
ycm 
m1 y1  m 2 y2 
m1  m 2 
zcm 
m1z1  m2 z2 
m1  m2 
where m1 is the mass of element at
coordinate (x1,y1,z1) …
CG can be outside the volume.
03/05/2003
Physics 103, Spring 2003, U. Wisconsin
6
Question 2
Where is the center of gravity of a “yummy” donut?
It is at the origin of the circular ring, half way from the
bottom of the donut - where there is no dough.
03/05/2003
Physics 103, Spring 2003, U. Wisconsin
7
A moment later……

M  R
New c.g.  old c.g. m   

8 M
R
shift 
8
C G has shifted along the line of symmetry away from the bite.
03/05/2003

Physics 103, Spring 2003, U. Wisconsin
8
Newton’s Law of Gravitation

Every particle in the universe attracts every other particle with a force
along the line joining them. The force is directly proportional to the
product of their masses and inversely proportional to the square of the
distance between them.
Note: “particle”!
If an extended object you must treat the vector
sum of all the forces. This is done automatically
by considering the object as if it were of the
same mass concentrated at the “center of mass”
(or the center of gravity!?)
If a system of extended objects you must still
consider the center of mass)
03/05/2003
Physics 103, Spring 2003, U. Wisconsin
9
Newton’s Law of Gravity
m
Magnitude:
r
Fg = G
mM
r2
M
G = 6.67 x 10-11 N m2/kg2
Direction: attractive (pulls them together)
force on M due to m is away from M center and
force on m due to M is away from m center
Mm
PE  G
r
PE(r  )  0
Work done to bring mass m from infinity to the proximity of mass M
Only differences in potential energy matter
Zero point is arbitrary.
03/05/2003

Physics 103, Spring 2003, U. Wisconsin
10

Close to the Surface of the Earth

Consider an object of mass m near the surface of the earth.
Fg = ma = G
mg
M m
PE1  G E
RE
M m
PE 2  G E
RE  h
 1
1 
PE  GME m 

R
R

h
 E

E
03/05/2003
a=g = G
mM
M
RE2
r2
=9.8 m/s2
r~RE






h
h

PE  GME m
 GME m
 2  h 
RE RE  h

RE 1
R
 
E 
GM
h
PE  m 2 E h  mgh for small
RE
RE
Physics 103, Spring 2003, U. Wisconsin
11
Escaping Gravity


Kinetic energy of the rocket must be greater than the gravitational
potential energy
Defines minimum velocity to escape from gravitational attraction
KE  PE  constant
Consider case when speed is just sufficient to escape to infinity
with vanishing final velocity
At infinity, KE + PE = 0, therefore, on Earth,
1 2
M m
mvesc  G E  0 
2
RE
v esc 

03/05/2003
2GME
 11.2 km/s  25000 mph
RE
Physics 103, Spring 2003, U. Wisconsin
12
Lecture 13 Preflight 8 & 9

Two satellites A and B of the same mass are going around Earth in
concentric orbits. the distance of satellite B from Earth’s center is twice
that of satellite A. What is the ratio of the centripetal acceleration of B to
that of A?
Since the only force is the
gravitational force, it must
scale as the inverse square
of their distances from the
center of the Earth.
1/8
1/4
1/2
0.707
1.0
4%
GmME
FB
rA2 1
rB2

 2
GmM
FA
rB 4
E
rA2
03/05/2003
47%
44%
1%
Pretty Sure
Not Quite Sure
Just Guessing
4%
0%
10%
20%
30%
40%
50%
Physics 103, Spring 2003, U. Wisconsin
13
Lecture 13 Preflight 10 & 11

Suppose Earth had no atmosphere and a ball were fired from the top of
Mt. Everest in a direction tangent to the ground. If the initial speed were
high enough to cause the ball to travel in a circular trajectory around the
earth, the ball’s acceleration would
• be much less than g (because the ball
doesn’t fall to the ground)
• be approximately g
• depends on the velocity of the ball
23%
49%
Pretty Sure
Not Quite Sure
Just Guessing
28%
0%
10%
03/05/2003
20%
30%
40%
50%
Physics 103, Spring 2003, U. Wisconsin
14
Orbits

Acceleration is provided by gravity
mM
mac  Fg  G 2
R
03/05/2003
Physics 103, Spring 2003, U. Wisconsin
15
Kepler’s Emperical Laws




Based on Tycho Brahe’s astronomical measurements
Orbit of a planet is an ellipse with the Sun at one focus
Equal areas swept out in equal times.
T2
R3
= constant
Newton:
V2
mM
m
 mR  2  G 2
R
R
2

T
R 3 GM

2
T
4 2
03/05/2003
Physics 103, Spring 2003, U. Wisconsin
Do read Examples
7.14-7.16 if you
have not
already done so.
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