For Exam #1:
(Assume a mass m, and a spring constant k)

There once were a mass and a spring,
To each other they firmly did cling.
But one day in the lab
Walter gave them a jab
At what frequency did they to swing?
(Acceptable answers: omega-naught, Sqrt[k/m])

For Exam #2:

There once were three beads on a string,
But one of them thought it was king.
It sat in the middle
Its thumbs it did twiddle
In which normal mode did they swing?
(Answer: Mode n=2)

On Phase in Complex Numbers:

There once was a number named i,
Whose beloved was moniker'd pi
From atop their friend e
And drinking green tea,
They twisted each term that came nigh!

On the pendulum mode:

On a magical string hung a bob
Hooked up with a spring to a blob
When displaced right together
Even without the tether
United in motion they throb!

On Matrix Multiplication:

There once was a matrix named Fred,
With a vector he wished to be wed.
But Fred was too wide
He had three on a side
And the two column vector, she fled!

On Intensity:

When it seems that you have the propensity,
To go off defining intensity,
Choose an area wide
As you wish to inscribe
And on it just find power density!

On Hilbert Space:

David Hilbert once had an ambition
To describe perfectly his condition
He traced out a vector
To the mind, quite a specter
In dimensions of his own volition!

On Group Velocity

There’s no need to respond with ferocity
If you’re asked to find group velocity
Take the d’rivative in k
Of omega – hooray!
At k-average, you’ll find that monstrosity!
(Note: Group velocity is not really a monstrosity.)