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.
m1m2
FG 2
r
inverse square law
G = universal gravitational constant
= 6.673 x 10-11 N m² /kg²
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
ME
gG 2
r
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
MEm
PE  G
r
– Zero reference level is infinitely
far from the earth
Escape Speed
• speed needed for an
object to “escape” from
planet
• For the earth, vesc is about
11.2 km/s
• Note, v is independent of
the mass of the object
v esc 
2GME
RE
Black holes, escape speed and the
speed of light
GM
GM
2
R 2
vesc 
vesc
R
RBH
GM
 2
c
R ~ 1 cm for Earth
R ~ 3 km for Sun
Ch. 8 General Torque Formula
Component of F  r
OR
 r F sin 
• 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 
so
 = F d = F r sin 
Torque and Equilibrium
• First Condition of Equilibrium
• The net external force must be zero

 F  0 or


 Fx  0 and  Fy  0
•The Second Condition of Equilibrium states
The net external torque must be zero

  0