Student is first given an oral
task based on the prerequisites and goals the module developers have.
Here we have a spring hanging from a ring stand support. We then
place a mass on the spring so that the mass is able to oscillate.
1. Draw a free body diagram for a mass on a spring for the following
i. when it is moving with a maximum velocity
ii. when it is moving with no velocity (at
the bottom of its motion)
2. Write Newton's 2nd Law mathematically (i.e. write
the differential equation for the motion of the mass) and explain how you
came up with your answer.
3. Why is the sign in front of the kx negative? What does
it imply about the motion of the mass and how would the motion of the mass
be different if the sign were positive?
4. Does the equation hold
during the entire motion? Explain.
5. (PRE-REQ) What is the solution to the equation ?
i. What determines the coefficients A,
ii. Are there any other ways to write the solution?
6. Consider a mass held where the spring is in its unstretched
position. The mass is then released. Write down all the initial conditions
needed to obtain A,B.
i. in words
7. The mass starts from where the spring is stretched a distance
d from equilibrium and a a clock is started when the velocity of
the mass is maximum. Make a plot of the masses' position vs. time.
8. Make a similar plot if the mass starts where the spring is
stretched the same distance, d as above, but it is given a starting
push. Again the clock is started when the velocity of the mass is maximum.
9. How would the motion of the mass change if the spring were
stretched more and the mass was released (without the push)? Make a plot
of the motion.
10. How would the motion change if a heavier mass were used but
the spring was stretched the same distance d?Again the mass is given
no starting push. Make a plot of the motion.
11. If damping had to be considered how would your differential
equation change? Explain.
Student now uses material on the
web at: http://www.math.rpi.edu/www/diffeq/links/springmass/index.html.
The student then moves on to the
at the right shows two identical, massless, frictionless, springs, each
with spring constant k, hanging from a bar. Attached to one spring is a
mass m1. Attached to the other spring is a mass m2,
where m2 > m1. At t = 0, the two masses are connected
to the springs and released from rest.
(Note: When the masses are at their starting height, the springs
are at their unstretched lengths.)
A. On a single set of y vs. t axes (where y is
position and t is time), sketch a graph of the motion of each of
the masses. Label your axes clearly and identify which curve corresponds
to which mass.
B. i) Determine the equation which gives y
as a function of t for m1.
ii) Determine the equation which gives y as a function of
t for m2.
iii) Describe what each symbol in each of the equations represents.
C. Describe how you would determine when during the oscillations, if
at all, the masses would be at the same height. (Do not work out the details,
just clearly describe the steps.) Explain your reasoning.
Please address questions and comments about this protocol
to Mel Sabella.
|University of Maryland