Post on 17-Jan-2016
Gravity
Unanswered questionsGalileo describes falling objects by rolling objects down a ramp.
But why does everything accelerate the same rate regardless of its mass?
Kepler describes planetary motions.
What force can account for the elliptical paths of the planets?
Two seemingly unrelated observations, but…
Newton unites the two in one master stroke at the age of 24.
Inverse Square Law
€
ra c =
2π( )2R
T 2
T 2 = kR3
r a c =
2π( )2R
kR3 ~1
R2
€
FG = maC
~ m1
R2
~ Mm1
R2
FG =GMm
R2
Circular motion result
Kepler’s observational law
Centripetal acceleration behaves like inverse square
Newton’s 2nd Law for centripetal direction
Constant of proportionality determined by Cavendish
Newton’s 3rd Law suggests both masses are important
€
G = 6.67 ×10−11 Nm 2
kg2
The Force Law
Direction?Along a line connecting the center of the two masses.
Action at a distance. How does the force GET from the Earth to the moon?
“I feign no hypothesis regarding action at a distance.”
€
FG =GMm
R2
m
M
R
Potential Energy Revisited
€
FG = mg
Remember how we used to write the force of gravity.
From this, we derived an expression for gravitational potential energy:
€
UG = mgh
This only applies near surface of earth! (When GME/RE2=9.8 m/s2)
More generally, we have
€
UG = −GMm
R
As usual, we have a choice for where we set UG = 0. When using this equation, the choice is made for us.
Two ApplicationsTerrestrial (free fall near the surface of a planet or star)
Celestial (circular orbit around a planet or star)
This most general expression is always true, but sometimes the first expression is simpler to implement (it has limited application, however, so be careful!).
€
FG = mg
€
FG =GMm
R2
Terrestrial ApplicationExample:
The radius of the Earth is 6.4 x 106 m and the value of g is 9.8 m/sec2. What is the mass of the Earth?
€
FG =GME m
R2
FG = gm
GME m
R2 = gm
ME =gR2
G
=9.8 6.4 ×106
( )2
6.67 ×10−11 = 6.0 ×1024 kg
Celestial Application
2
2
22
2411
2
6
or,
6 106.67 10
7680
6.79 10
c G
s es
e e
F F
m mvm G
R Rm m
v G R GR v
xR x
x m
Radius of the earth is about 6.38 x 106 m at the equator. That gives the altitude above the surface to be:6.79x106 – 6.38x106 = 0.41x 106 m, or 410 km.
“g” ~ 8.68 m/s2
The ISS orbits the earth with a speed of approximately 7680 m/s.What is the orbital radius of the station, and what is its altitude?
Torque
Dynamics
Which applied force results in the largest angular acceleration of the bolt?
€
rF 1
€
rF 2
€
rF 3
Dynamics
€
rF 1
€
rF 2
€
rF 3
Which applied force results in the largest angular acceleration of the bolt?
Dynamics
€
rF 1
€
rF 2
Which applied force results in a clockwise angular acceleration? A counter-clockwise angular acceleration?
O
€
rr
€
rF Perp
€
rF Par
Dynamics quantified
Consider a force acting on a rigid body, some distance away from a fixed pivot point.
Only the perpendicular part contributes to rotation!Where does the parallel part go?
€
rτ = r
r r F Perp
Can split the force into components.
Dynamics quantified
Which applied force results in the largest angular acceleration of the bolt?
€
rF 1
€
rF 2
€
rF 3
Dynamics quantified
€
rF 1
€
rF 2
€
rF 3
Which applied force results in the largest angular acceleration of the bolt?
Direction of torque
€
rF 1
€
rF 2
But torque is a vector, and vectors only point in a single direction.
Direction of torque is given by right hand rule.Draw and tip to tailPoint fingers of right hand in the direction of Curl fingers in direction ofThumb points in direction of torque: either into (clockwise, negative) or out of
(counter-clockwise, positive) page
€
rr
€
rF
€
rr
€
rF
€
rr
€
rr
Direction of torque is the direction the applied force tends to cause the object to rotate.
F1 provides a clockwise torque.F2 provides a counter-clockwise torque.
Newton’s LawsNewton’s 1st Law – If there is no net torque on an object, then the object rotates at a
constant angular velocity (could be zero angular velocity).Newton’s 2nd Law – If the net torque on an object about a point is not zero, then the
net torque produces an angular acceleration about that point.
€
rF
€
rr
€
τ =rF
= r ma( )
= r m rα( )( )
= mr2α
The quantity mr2 is the rotational equivalent of mass, and is called moment of inertia.
Newton’s 3rd Law – For every action, there is an equal and opposite reaction.
O
Example
20m 40m
A B
€
rτ net, A =
r τ T, A +
r τ B, A
0 = (±)FT rT + (±)FB rB
0 = −FT rT + FB rB
FB =FT rT
rB
=8000(20)
(60)= 2667 N
A truck crosses a massless bridge supported by two piers. What force much each pier exert when the truck is at the indicated position?
€
rF A
€
rF B
€
rF T = 8000 N
€
rτ net, B =
r τ T, B +
r τ A, B
0 = (±)FT rT + (±)FA rA
0 = +FT rT − FA rA
FA =FT rT
rA
=8000(40)
(60)= 5333 N
€
2667 + 5333 = 8000 N
Sample Torque Problems
X = __________
X = __________