1. We consider an object of mass m attatched to a spring which exerts
a restoring force given by F=-kx, where k is the spring constant.
a. we know that frequency f of the system is expressed by
f=(1/2*pi)*square root(k/m). Show that this f is the only
possible frequency, that is, no other frequency other than this is
possible.
b. the total mechanichal energy E, defined by the sum of the kinetic
energy (K) and potential energy (V) is given by
E= K + V = 1/2mv^2 + 1/2 kx^2
where is the reference point of the potential energy chosen?
c. show that E is a constant dependent of time, that is, E is a
conserved quantity.
d. We consider the trajectories of the harmonic oscillator in the two
dimentional phasespace (p,x). We show that any trajectory must circle
in the clock-wise direction. We will achieve this in the following
step: suppose a phase space point at t=0 is located at (Po,0), where
Po>0. show that, at a small positive t, the point will be located in
the first quadrant of the phase space.
e. For given m and k, a phase space trajectory follows vastly
different paths depending on the initial conditions (Po,Xo) at t=0.
Explain why two trajectories corresponding to two different initial
conditions will never cross.
f. If one chooses the reference point for the potential energey at
the maximum value of x represented by x-max, the total mechanichal
energy can be expressed by E= K + V = 1/2mv^2 + 1/2Kx^2 -
1/2kx^2max.
show that E is a conserved quantity and find the area encircled by the
phase trajectory for a given value of E. |
