The second-order diffrential equation m d 2 y/dt 2 + b dy/dt +ky =0 can be used
ID: 1943627 • Letter: T
Question
The second-order diffrential equation
m d2y/dt2 + b dy/dt +ky =0
can be used to model a harmonic oscillator, an undamped or underdamped harmonic oscillator can be used to make a clock. If we arrange for the clock to tick whenever the mass passes the rest position, then the time between ticks is equal to one-half of the natural period of the oscillator.
a) If dirt increases the coefficient of damping slightly for the harmonic oscillator, will the clock run fast or slow?
b) Suppose the spring provides slightly less force for a given compression or extension as it ages. Will the clock run fast or slow?
c) If grime collects on the harmonic oscillator and slightly increases the mass, will the clock run fast or slow?
Explanation / Answer
A) Increasing the dampening ratio would make the clock run slower because the spring would be more dampened (i.e. it would take longer to complete one full period).
B) Less force (smaller k value) implies the spring would be "bouncier" (easier to perturb), therefore the clock would run faster.
C) Increasing the mass would increase the force on the spring (higher k value), therefore the clock would run slower.