Description
1. Reconsider the simple harmonic oscillator studied extensively in Homework Set 11 involving
the mass m = 0.550 kg, which is attached to the spring and hung vertically. This
time, however, the oscillator is submerged in a resistive medium.
After oscillating for
6.00s, the maximum amplitude decreases to one-fourth of the initial value. Calculate
a) the damping parameter, , and the damping constant, b,
b) the angular frequency of oscillation, !, and
c) the quality factor, Q.
2. Reconsider the oscillator of the previous problem. 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) Obtain an expression for the displacement of the mass in the form
x(t) = Aet/2 cos(!t + ), giving numerical values for A, !, , and . Notice that
the oscillator is experiencing very light damping and should be treated as such.
b) Use a spreadsheet program (i.e. Excel) to plot this function over the time period
t = 0 to 4.00s, with a time step no greater than 0.0200s. Your columns of input data
must be labeled with the correct SI units. Your plot needs to include a title and the
axes should be labeled and include units.
3. Again reconsider the oscillator studied extensively in Homework Set 11 involving the mass
m = 0.550 kg, which is attached to the spring and hung vertically. This time, the
oscillator is placed in a resistive medium where the oscillator is found to be critically
damped.
a) Determine the value of the damping constant b, and therefore the damping parameter
, that is required to produce this critical damping.
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.
b) Obtain an expression for the displacement of the mass in the form
x(t)=(A + Bt)et/2, giving numerical values for A, B, and .
c) Find the maximum displacement from equilibrium and the time, t
0
, when this occurs.
4. Reconsider the oscillator of the previous problem. Using a spreadsheet program
(i.e. Excel), plot this function over the time period t = 0 to 0.500s, with a time step
no greater than 0.0100s. Your columns of input data must be labeled with the correct SI
units. Your plot needs to include a title and the axes should be labeled and include units.
From the plot, determine the maximum displacement from equilibrium and the time, t
0
,
when this occurs. How does this compare to the result from problem 3?
5. Again reconsider the oscillator studied extensively in Homework Set 11 involving the mass
m = 0.550 kg, which is attached to the spring and hung vertically. This time, the
oscillator is placed in a resistive medium where the oscillator is found to be heavily
damped. The resistive medium is described by a damping parameter that is triple the
value found in problem 3.
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) Obtain an expression for the displacement of the mass in the form
x(t) = et/2 [Ae+↵t + Be↵t
], where ↵ ⌘ p(/2)2 !2
0, giving numerical values for
A, B, , and ↵.
b) Using a spreadsheet program (i.e. Excel), plot this function over the time period
t = 0 to 0.500s, with a time step no greater than 0.0100s. Your columns of input
data must be labeled with the correct SI units. Your plot needs to include a title
and the axes should be labeled and include units. Include the plot generated in
problem 4 for comparison.
