Download More on energy plus gravitation

More on energy plus gravitation
•  Tutorial homework due Thursday/Friday
•  LA applications due Monday: lacentral.colorado.edu
1
Energy diagram
A particle affected just by conservative forces has a constant
total energy (E) while the potential (U) and kinetic energy (K)
change. If the conservative force is gravity, the change
depends on y.
Energy
E = K +U
When the potential energy
equals the total energy the
particle is at its max height.
Any higher would require a
negative kinetic energy.
E
K2
K1
U g1 U g 2
U
y1
y2
y
Potential energy depends on the forces. Total energy depends on
initial conditions (for example, how high or how hard a ball is thrown).
2
Graphs of potential energy
Spring potential energy is given by
a parabola equation: Us ( x) = 12 k (Δx)2
The force associated with the
potential energy is in the direction
which reduces the potential energy.
At the minimum the force is zero and the
object is in a stable equilibrium position
Example is a ball in the bottom of a bowl
This is just a way to visualize what is going on; the
potential energy graph is for a spring, not gravity!
3
Graphs of potential energy
Can also have more complicated
potential energy curves
Any maxima are places where
the forces are zero but is, in
fact, an unstable equilibrium
Example is a ball on top of an inverted bowl
Can get potential energy for any
position using graph. From E=K+U
can determine K or E given the other.
4
Clicker question 1
Set frequency to BA
A cart rolls without friction along a track. The graph of U vs. the
x-position is shown. The total mechanical energy is 45 kJ.
To within 2 kJ, what is the maximum kinetic energy
over the stretch of track shown?
50
E
40
A.  10 kJ
30
U(kJ)
B.  15 kJ
20
C.  30 kJ
10
D.  35 kJ
0 0 40
80
120 160
x(m)
E.  45 kJ
Since E=K+U, the maximum K is when U is a minimum which is
for 120<x<140 where U=10 kJ so K = E − U = 45 kJ − 10 kJ = 35 kJ
Can use Energy Skate Park PhET simulation to explore further.
5
Getting force from potential
We previously found that the work done by a conservative
force is the opposite of the change in potential: Wc = −ΔU
Let us consider this equation in one dimension
Note that force and potential energy may depend on position
So
Wc = −ΔU in 1-D becomes Fx ( x) ⋅ Δx = −ΔU ( x)
and therefore Fx ( x) = −
ΔU ( x)
(note this looks like a slope)
Δx
dU ( x)
If we let Δx → 0 then Fx ( x) = −
dx
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Force–potential energy relationship
dU ( x) Force acts to reduce
In one dimension : Fx ( x) = −
potential energy
dx
Check if it works for
spring potential:
Check if it works
for gravity:
Us = 12 kx 2 so Fx ( x) = − d (12 kx 2 ) = −kx
dx
Ug = mgy so Fy = − d (mgy ) = −mg
dy
The slope of the potential energy
versus position graph gives the
magnitude and position of the force
associated with that potential energy
7
Clicker question 2
Set frequency to BA
A cart rolls without friction along a track. The graph of U vs. the
x-position is shown. The total mechanical energy is 45 kJ.
At which of the following points does the force have the largest
50
value in the positive x direction?
E
40
A.  25 m
30
U(kJ)
B.  50 m
dU(x)
20
Fx (x) = −
C.  60 m
10
dx
D.  80 m
0 0 40
80
120 160
x(m)
E.  150 m
The slope of U vs x is 0 at 25 m and 60 m
The slope of U vs x is positive at 50 m and 150 m,
leading to a negative force.
The slope of U vs x at 80 m is negative, giving a positive force.
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Newton’s Law of Gravity
Newton and Einstein are generally thought
to be the two greatest physicists ever.
Not only did Newton come up with the three laws
of motion and invent calculus, he was the first to
realize that the force associated with things falling
was also responsible for astronomical phenomena.
Gm1m2
Newton’s Law of Gravitation can be written as FG = r 2
Between any two masses (here m1 & m2) there is an attractive
force proportional to the product of the masses and inversely
proportional to the square of the distance between them.
Side note: this force manifestly obeys Newton’s 3rd law
9
Gravitational Force
Gm1m2 is the force of gravity which is felt by each mass
FG =
r2
and directed towards the other mass.
Newton figured out the 1/r2 dependence, assuming that the
celestial objects and the Earth were point particles.
By inventing integral calculus he could prove that for a mass
m2, outside a spherical mass m1, the force of gravity was as
if all of the mass m1 was at the center of the sphere.
Therefore for any two spherically symmetric objects, the
distance r that enters into the force of gravity is the distance
between the centers of the spheres.
10
Force rules
Gm1m2
−11
2
2
FG =
G
=
6
.
67
×
10
N
⋅
m
/
kg
is
the
force
of
gravity
with
r2
Newton’s 2nd law still works. The net force
! on !an object
determines the object’s acceleration: Fnet = ma
Remarkably, the mass in Newton’s 2nd law (called the inertial
mass) is the same as the mass in the law of gravitation (called
the gravitational mass). Einstein figured out (230 years later)
that this “coincidence” could be explained by assuming space
and time were curved (in the theory of general relativity).
Remember, force is still a vector and the law of superposition
still works. To find the net gravitational force on an object,
determine the magnitude and direction of the force from all
other masses and then add these forces together.
11
Force of gravity on Earth
Gm1m2
F
=
mg
How does g
correspond to our new force FG =
?
2
r
If we consider mass 2 to be the Earth (ME) and r to be the radius
of the Earth (RE) then we can write F = m GM E
G
1
RE2
Using known values we can find that
GM E (6.67 ×10−11 N ⋅ m2 / kg 2 )(5.97 ×1024 kg )
=
= 9.8 m/s2
2
2
RE
(6.38 ×106 m)
So, on the surface of the Earth, the force of
GM E
F
=
m
= m1g
1
gravity between the Earth and an object m1 is G
2
RE
We can only use FG = mg if the distance above the
surface is very small compared to the radius.
12