[SOLVED] Physics 230 Homework Set 13

1. Once again, reconsider the oscillator studied extensively in Homework Sets 11 and 12
involving the mass m = 0.550 kg. The mass is attached to the spring and hung vertically,
where the oscillator is submerged in a resistive medium.The mass is subjected to a
harmonic force of the form F(t) = F0 cos(!t) where F0 = 2.50 N. It is found that the
damping parameter, , is one-eighth the critical value.Find
a) the frequency that gives rise to the maximum amplitude,
b) the maximum amplitude,
c) the quality factor.Now reconsider this damped, driven harmonic oscillator subject to a driving frequency
that diers from that found in a). Find
d) the amplitude, A(!), at this frequency.
e) the phase angle, , when the system is driven at a frequency ! = 2 rad/s, and2. For a damped, driven harmonic oscillator, the displacement of the mass from equilibrium
is described by the solution to the equation of motion, given by Eqn. (3.10), where both
the steady-state solution and the transient response need to be included.As presented in
class, the general solution to Eqn. (3.10) is of the form
x(t) = x1(t) + x2(t) = A(!) cos(!t ) + Bet/2 cos(!1t + ), (1)
where x1 corresponds to the steady-state solution whereas x2 is the transient solution,
where !1 (!2
0 (/2)2)1/2 is the frequency of the transient response.Reconsider the oscillator of the previous problem where the system is driven at frequency
! = 2 rad/s. At time t = 0, the mass is displaced from the equilibrium position by
5.50 cm in the downward direction and is given an initial shove, imparting a speed of
0.450 m/s on the mass in the downward direction toward the floor.a) Construct an expression for the time-dependent velocity of the mass, which is attached
to the spring.
b) Using the aforementioned initial conditions imposed on the oscillator, construct a
system of algebraic equations involving B and .
c) Show that B = 0.0448 m and = 32.2 = 0.562 rad are solutions to the above
system of algebraic equations.3. Reconsider the steady state solution and the transient response of the oscillator of the
previous problem. Using a spreadsheet program (i.e. Excel),
a) generate a plot of x1(t) = A(!) cos(!t ) vs t,
b) generate a plot of x2(t) = Bet/2 cos(!1t + ) vs t, and
c) generate a plot of x1 vs t, x2 vs t, and x(t) = x1 + x2 vs. t, all on the the same plot.For all three of these distinct plots, the functions should be plotted over the time period
t = 0 to 5.00s, with a time step no greater than 0.0250s. Each plot needs to include a
title and the axes should be labeled and include units.4. A transverse wave traveling along a string is described by the function
z(x, t) = (0.21 m) sin
(0.36m1
)x + (0.52s1
)t
(2)a) Calculate the amplitude, wavelength, frequency, and velocity of the wave.
b) In what direction is the wave traveling?c) Show that the aforementioned expression is a solution to the 1D wave equation.
d) Calculate the maximum speed and acceleration of the transverse displacement of a
point on the string.