Review end of Chapter 7
Forces in Accelerating Reference
Frames
Newton’s Law of Universal Gravitation
Every particle in Universe attracts
every other particle with force:
directly proportional to product of
masses
inversely proportional to square of
distance between them.
Distinguish real forces from fictitious forces
“Centrifugal” force is a fictitious force
Real forces always represent interactions
between objects
Real forces go on left side of Newton’s Second
Law-Accelerations go on right side
mm
F = G 12 2
inverse square law r
F=ma
G = universal gravitational constant
= 6.673 x 10-11 N m² /kg²
Applications of Universal Gravitation
Gravitation Constant
Determined
experimentally
Henry Cavendish
gravitational force of uniform sphere on particle outside sphere
same as force exerted if entire mass of the sphere concentrated
at its center--Gauss’ Law
1798
The light beam
and mirror serve
to amplify the
motion
Acceleration due
to gravity
g will vary with
altitude
g=G
ME
r2
top 10 physics experiments
Applications of Universal Gravitation
gravitational force of uniform sphere on particle outside sphere
same as force exerted if entire mass of the sphere concentrated
at its center--Gauss’ Law
Acceleration due
to gravity
g will vary with
altitude
g=G
ME
r2
Gravitational Potential Energy
PE = mgy is valid only near
the earth’s surface
For objects high above the
earth’s surface, an alternate
expression is needed
PE = −G
MEm
r
Zero reference level is
infinitely far from the earth
Escape Speed
Escape Speed and radius
M
for sphere
Density ρ = M / V =
4 3
πR
4
3
M = π R3 ρ
speed needed for an
object “escape” from
planet
For the earth, vesc is
about 11.2 km/s
Note, v is independent
of
the mass of the object
RE
3
v esc =
2GME
RE
Determines Atmosphere
vesc =
v esc =
2GME
RE
2G 4 3 π R 3 E ρ
RE
vesc ∝ RE
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
2439
6052
6378
3397
71492
60268
25559
24764
1160
vesc RE/vesc
4.3
10.3
11.2
5
60
36
22
24
1.1
No Kepler’s Laws Here!!!
Ch. 8 General Torque Formula
Component of F ⊥ r
Torque and Equilibrium
First Condition of Equilibrium
OR
τ = r F sin θ
The lever arm, d, is the perpendicular
distance from the axis of rotation to a line
drawn along the direction of the force
d = r sin θ
so
τ = F d = F r sin θ
Example of a Free Body Diagram--Ladder
L
Θ
free body diagram shows normal force and
force of static friction acting on the ladder at
the ground
The last diagram shows the lever arms for the forces
from axis of rotation at ground
Pick axis and sum torques
The net external force must be zero
r
Σ F = 0 or
r
r
Σ Fx = 0 and Σ Fy = 0
•The Second Condition of Equilibrium states
The net external torque must be zero
r
Στ = 0
567
588
569
679
1192
1674
1162
1032
1055