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EXAM I, PHYSICS 4304
FEBRUARY 25, 2002
Dr. Charles W. Myles
INSTRUCTIONS: Please read ALL of these before doing anything else!!!
PLEASE write on one side of the paper only!!
PLEASE do not write on the exam sheets, there will not be room! If you do not have paper,
I will give you some.
PLEASE show all of your work, writing down at least the essential steps in the solution of a
problem. Partial credit will be liberal, provided that the essential work is shown. Organized
work, in a logical, easy to follow order will receive more credit than disorganized work.
NOTE!!! the setup (THE PHYSICS) of a problem will count more heavily in the grading
than the detailed mathematics of working it out.
PLEASE clearly mark your final answers and write neatly. If I cannot read your answer, you
can't expect me to give it the credit it deserves and you are apt to lose credit.
PLEASE FOLLOW THESE SIMPLE DIRECTIONS!!!! THANK YOU!!!
NOTE!!!! Work any four (4) of the six problems. Each problem is equally weighted and worth
25 points for a total of 100 points on this exam.
1. A block of mass m is moving on a horizontal surface where measurements have found that
the retarding force is proportional to the 1/6 (one sixth) power of the velocity. That is
F = -mkv1/6, where k is a constant. At t = 0, the velocity is v0 and the mass is at the origin.
Find (in any order)
a. The velocity as a function of time (v(t)). (5 points)
b. The time it takes the mass to stop. (5 points)
c. The position as a function of velocity (x(v)). (5 points)
d. The position as a function of time (x(t)). (5 points)
e. The distance the mass travels before it stops. (5 points)
The following integral might be useful. The constant of integration is not shown. The lower limit
is important in this problem! vp dv = (vp+1)/(p+1), where p is any power (p  -1)
2. Consider a mass m, confined along the x-axis by a conservative force which is derivable from
a potential energy function given by U(x) = (½)kx2 – ()x3, ( k &  are positive constants)
a. If the mass has total mechanical energy E, find a general expression for the velocity v at
position x. (That is, find the functional form of v(x). This expression will not have the
time in it!) (5 points)
b. If the mass has total mechanical energy E, derive an equation whose solution will give the
turning points of the particle. (Don’t try to solve this equation for the case of general E!).
What is the velocity at these turning points? Solve the equation for the turning points in
the special case when E = 0 (5 points)
c. Find the equilibrium points for this potential. Determine whether these are points of
stable or unstable equilibrium. Also find the values of x where U(x) = 0. Use all of these
results to sketch the potential U(x). (5 points)
d. Calculate the restoring force F(x). (5 points)
e. Derive the equation of motion for this particle (don’t try to solve it!). (5 points)
NOTE!!!! Work any four (4) of the six problems.
3. A block of mass m moves along the x-axis. At time t = 0, it is at the origin and its velocity is
v0. At time t = 0, a time-dependent force given by F = F0e-kt begins to act, where F0 and k are
constants. Find (in any order)
a. The velocity as a function of time (v(t)). (5 points)
b. The position as a function of time (x(t)). (5 points)
c. What are v(t) and x(t) when t is very small? In a few complete and grammatically
correct English sentences, explain what this small time means physically. Do v(t) and
x(t) resemble any familiar results in this limit? (5 points)
d. What are v(t) and x(t) when t is very large? In a few complete and grammatically correct
English sentences, explain what this long time means physically. Do v(t) and x(t)
resemble any familiar results in this limit? (5 points)
e. When first formulating this problem, I originally was going to ask you to compute the
time it takes the mass to stop. However, I then realized that computing this in the usual
way (finding the time t when v(t) = 0) leads to unphysical results. Despite this, compute
this time by this method anyway. In a brief, complete and grammatically correct English
sentence, explain why is this an unphysical result. (5 points)
The following integral might be useful. The constant of integration is not shown. The lower
limit is important in this problem!
e-kt dt = (e-kt)/(-k)
The following expansion might be useful: For |kt| <<1, e-kt  1-kt + (½)(kt)2 - …
4. A mass m = 1.5 kg is attached to an ideal spring of constant k = 7.0 N/m. At t =0 it is
released from the origin (x = 0) with initial speed v0 = 8.0 m/s. Consider this oscillator under
the following two DIFFERENT conditions:
a. WITH NO FRICTION. Find numerical values for the frequency, the period, the
maximum velocity, the maximum amplitude, the maximum acceleration, and the total
mechanical energy. Using (some of) these values, write an expression for the time
dependence of the displacement x(t) (that is, use the computed information to evaluate
the integration constants in the general solution for x(t).) (12 points)
b. WITH FRICTION. A damping force proportional to the velocity is present. The damping
force constant is b = 1.5 N s/m. Find numerical values for the frequency, the period, and
the decrement of the motion. Calculate the time it takes for the amplitude to reduce to
half of its maximum value. Using (some of) these values, write an expression for the time
dependence of the displacement x(t) (that is, use the computed information to evaluate
the integration constants in the general solution for x(t).) Is this oscillator underdamped,
overdamped, or critically damped? (13 points)
NOTE!!!! Work any four (4) of the six problems.
5. Consider a one-dimensional, driven, damped harmonic oscillator with mass m, spring
constant k and damping constant b. It is underdamped. The driving force is F0 sin(t), where
F0 is a constant and  is a frequency.
a. Find the steady state (particular) solution for the displacement x(t). This means to derive
the results for D() and (). It does not mean to copy them directly from your sheet of
paper! (15 points)
b. Consider the steady state solution. If  = ()0, compute the amplitude D() and the
phase angle () exactly at resonance. Compute the quality factor Q of this oscillator.
(10 points)
The following trig identities might be useful:
cos(  ) = coscos -/+ sinsin; sin(  ) = sincos  cossin
6. Consider a one-dimensional periodic force, F(t). Over one time period T, it has the form
= - F0,
(-½)T  t < 0
= F0,
0 < t  (½)T
where F0 is a constant. Note: T is a time period for which one can define a frequency as
= 2/T. Using these facts in your solution below will be useful!
F(t)
a. Find the Fourier series representation of F(t). (15 points)
b. Consider a one-dimensional, driven harmonic oscillator with NO DAMPING (b =  = 0).
The driving force has the form F(t), above. Use the known solution for the steady state
displacement xs(t) of an undamped oscillator with a sinusoidal driving force, along with
the results of part a), to express the solution for x(t) for this driven oscillator (with F(t)
above) as a series solution. (10 points)
The following integrals might be useful. The constants of integration are not shown.
The lower limits are important in this problem!
cos(ax)dx = (1/a)sin(ax)
sin(ax)dx = -(1/a)cos(ax)