University of Michigan
Physics 340, Fall 2003 5 Sept. 2003
Assignment 1: The Harmonic Oscillator, Complex Representations, Superposition
Required reading: French, Chap. 1
Chap. 2 through p 26
Chap. 3 through p 62
We start with a version of the basic physics problem for the harmonic oscillator:
1. A block of mass m=0.5 kg is free to oscillate at the end of a spring. At time t = 0, the block is
launched from its rest position at x = 0, by a hammer strike which imposes an instantaneous
velocity v0 = 40 cm/s. The block then executes simple harmonic motion with a maximum
excursion of A = 10 cm.
a) Find the oscillation frequency, the period, and the spring constant. (Hint: Write an
expression for the position of the block as a function of time. Differentiate to find the
velocity. Solve for the initial conditions.)
b) What is the velocity at t = π/4 s? Explain.
c) Plot the kinetic and potential energy of the block vs. time. (Recall that the potential
energy of a linear spring is given by
2
1
2
Ukx
=.) What is the total energy vs. time?
d) An identical block-spring system starts with a different set of initial conditions: it is
extended to x = 10 cm, and released at t = 0. Write an expression for the position of
this block as a function of time.
e) What is the phase difference between the two systems?
f) What is the total energy of the system vs time? Compare to the first system.
One purpose of this problem is to show that the general solution requires both sines and
cosines, but that the mix of the two is just determining the overall phase angle, which is
simply adjusting to the initial conditions. It is obviously the same “motion” in either case,
with the same total energy.
But then we end up with some math refreshment:
2. French 1-1. Do the problem using both the Cartesian and the exponential representation of
the complex numbers. Note that French discusses these numbers as vectors, with “length”
and “angle/direction”, where we would talk about “complex numbers” with “modulus” and
the “phase”.
3. a. French 1-6
b. Display the above relationships geometrically using the Argand diagram.
4. French 1-12
5. French 2-3
6. Use phasor addition to calculate the amplitude and phase (A and φ) of the vibration given by
the sum
()cos()(10)cos()(17)sin()
xtAtcmtcmt
ωφωω
=+=+
