Question

Consider a simple harmonic oscillator function that corresponds to some vibration in the vertical direction: y(t) = 10 cos [ 20 π t − π/2 ]

(a) what is the amplitude of vibration (in arbitrary units)?

(b) what is the vertical displacement at t = 0?

(c) what is the period of the vibration?

(d) what is the sign (positive, negative, or zero) of the vibration’s velocity at t = 0?

(e) Use the information gathered in steps (a) through (d) to sketch a plot of disturbance y versus time t, showing at least two peaks and at least two troughs of the function. Make sure to label the tick marks on the horizontal and vertical scales.

Answer 1

Given

   simple harmonic oscillator equation is

   y(t) = 10 cos(20 pi t - pi/2)

which is in the form of y(t) = A cos[wt - phi]

   A- amplitude, W - angular frequency , phi - phase angle

a) the amplitude of vibration is A = 10 m

b) the vertical displacement at t = 0 is
  
   y(t) = 10 cos(20 pi t - pi/2)

   y(0) = 10 cos(20 pi 0 - pi/2)
   y(0) = 10 cos(-pi/2)
   y(0) = 0 m

c) the period of the vibration is W = 2pi/T ==> T = 2 pi/W = 2pi/20 pi = 0.1 Hz

d) he sign (positive, negative, or zero) of the vibration’s velocity at t = 0 is

  

   y(t) = 10 cos(20 pi t - pi/2)

   v(t) = dy(t)/dt= -10 (20 pi)sin(20 pit-pi/2)
           = - 200 pi sin (20 pi t - pi/2)

   v(0) = -200pi sin(20 pi*0 - pi/2)
= -200 pi(-1)
= 200 pi
so the sign is +ve