1
HW 5 Solutions
Energy of Harmonic Oscillators
As you know, a common example of a
harmonic oscillator is a mass attached to a
spring. In this problem, we will consider a
horizontally
moving block attached to a
spring. Note that, since the gravitational
potential energy is not changing in this case,
it can be excluded from the calculations.
For such a system, the potential energy is
stored in the spring and is given by
2
1
2
Uk
x
=
,
where
k
is the force constant of the spring and
x
is the distance from the equilibrium position.
The kinetic energy of the system is, as always,
2
1
2
Km
v
=
,
where
m
is the mass of the block and
v
is the speed of the block.
We will also assume that there are no resistive forces; that is,
E=const
.
Consider a harmonic oscillator at four different moments, labeled A, B, C, and D, as shown in
the figure . Assume that the force constant
k
, the mass of the block,
m
, and the amplitude of
vibrations,
A
, are given. Answer the following questions.
a)
Which moment corresponds to the maximum potential energy of the system?
A
B
C
D
Note that the sign of
x
does not matter, just its magnitude, because
U
is a
quadratic
form of
x
.
b) Which moment corresponds to the minimum kinetic energy of the system?
Again, A.