Download Potential and Kinetic Energy

Energy
Preliminary example
First a simple example: a mass under uniform gravity, acceleration
dv
dv v dt
dy
Integrating
g
g
v dv g dy
gives
1 2
v
gy const
2
This constant is determined by the initial conditions v 0 and y 0 . (Alternatively, without the constant
1 2 1 2
g y2 y1 .)
2 v2
2 v1
Obviously a handy quantity. We can use this formula to see how speed depends on height (but not
how anything depends on time). It is called ‘energy per unit mass’, E m. (The ‘per mass’ part of this
definition is most interesting for things which change mass, but for these we gave to change Newton’s
second law anyway).
The quantity 12 v2 gy is called a ‘first integral’ of the system.
Work
If a particle with a constant force F acting on it is moved in a straight line from position vector r 1 to r2
the work done is defined as F r2 r1 . That is the Force times the distance traveled in the direction
of the force.
If the force depends on position the work done is an integral, taking a one dimensional example F x work is
x2
F dx
x1
We can also use time as a variable if convenient
t2
F
dx
dt
dt
F
dr
dt
dt
t1
In the vector case this becomes
t2
t1
For example the work done moving an mass m in a uniform gravitational field
(only depends on height y).
gj is mg y 2
y1 Further out in space where the force between two bodies is GMm r 2 when they are a distance r apart,
the work done moving them further apart is
r2
r1
GMm dr
r2
GMm 1
1
r2
1
r1 Kinetic Energy
Kinetic energy (energy of motion) is defined as the work done accelerating a mass from rest to its
speed v. It is always
1
K mv2
2
Potential Energy
Potential energy is the work done on moving a mass from rest at some reference position to rest at
another reference position in space, ignoring any friction or drag forces. So for a uniform gravitational
field
V mg y y1 or for planetary orbit, rather perversely taking the reference to be at infinity
V
GMm r2
Conservation of Energy
In a frictionless system where no work is done by external forces the total energy (potential energy
plus kinetic energy) stays constant. (It is a first integral of the system).
E V K is a constant
or
d
V K 0
dt
Loss of Energy
In a system with friction of drag, frictional forces always act to oppose the motion removing some of
the energy:
d
V K
dt
0
Potential Energy of Spring-mass system
Spring with natural length l, displaced a distance r has a restoring force F Law) where k is the spring constant.
k
l
r
l (this is Hooke’s
The potential energy
V
F dr k
r
l
l dr k
r
l
2
2l
const
Taking the reference position of the spring to be r 0 we have that the constant of integration is zero.
The equations of motion are mr̈ k
l
r
l so (Simple Harmonic Motion)
r
l A cos
k
t B sin
ml
and we obtain constants A and B from r and ṙ at t 0.
2
k
t
ml
Energy E K V is
1 2 1k
mv r
2
2l
l
2
Circular Pendulum
E
1 2
mv mgl 1
2
cos α The interesting cases are when the total energy is less than, or greater than 2mgl.
The phase space plot(α and v) of lines of constant energy for the circular pendulum.
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