Note: This is based on material from Chapter 13, which is not required for the Test 3 this year (the date of Thanksgiving this year is the reason the schedule is a bit different than 1999...) However, it could easily find itself on the final exam...

A simple harmonic oscillator consists of a mass (0.3 kg) attached to a spring (spring constant = 0.5 N/m). There is no damping. The maximum velocity of the mass is measured to be v_max = 2.0 m/s.

a) What is the angular frequency of the motion?

b) What is the amplitude of motion?

c) If you were to shake the spring by hand, what frequency of shaking would lead to the maximum motion of the mass?

a) For a simple harmonic oscillator consisting of a mass on a spring, the angular frequency is related to the mass and the spring constant via

₀ = (k/m)^1/2

= (0.5 N/m / 0.3 kg)^1/2 =

= 1.29 rad/s

b) The only additional information we are given is the maximum velocity. For simple harmonic motion, the velocity is given by

v(t) = A sin(t + )

this the maximum velocity is related to the amplitude by

v_max = A

Thus the amplitude is easily found from

A = v_max/ = (2.0 m/s)/(1.29 rad/s)

= 1.55 m

c) Now the situation is changed - if we are shaking the system, we now have a driven oscillator. The maximum amplitude will occur, for a given external driving force, when the driving force is at the resonant frequency. If there is no damping in the system, the resonant frequency is exactly the natural frequency of the system (it is slightly less than that if there is damping, but there isn't here), thus the maximum amplitude will occur if it is shaken with

_resonance = ₀ = 1.29 rad/s

To convert this from an angular frequency to an ordinary frequency we use

f = /2 = (1.29 rad/s)/(2)

= 0.205 Hz