Question

Equations of Simple Harmonic Motion (basic)

PLEASE! show work and only answer if you know how to do it. People keeps giving me the wrong answer.

Analyzing Newton's 2^nd Law for a mass spring system, we found a_x = -k/m X. Comparing this to the x-component of uniform circular motion, we found as a possible solution for the above equation: x = Acos(omega t) v_x = - omega Asin(omega t) a_x = - omega^2 Acos(omega t) with omega = square root k/m, and A the amplitude (maximum displacement from equilibrium). Consider an oscillator with mass 0.217 kg and spring constant 21.6 N/m. If the amplitude of the oscillation is 4.53 cm, what is the maximum speed? Consider an oscillator with frequency 31 Hz. If at t=0 the oscillator is at a maximum displacement from the equilibrium of +5.8 cm. what is the displacement 3.46 seconds later? Consider an oscillator with mass 0.31 kg and spring constant 14.2 N/kg. The oscillator is displaced from the equilibrium position by +5.9 cm and released. After what time, is the oscillator for the first time at a displacement of -2.25 cm?

Answer 1

Q_x = -k/m x

solving the above condition we get,

        x = A cos wt

        v = - wA sin wt

        a = - w²A cos wt

a)    k =21.6 N/m ,    m= 0.217 kg           A = 4.53 cm

              v = -wA sin wt

              for maximum speed, sin wt = -1

                             in the third quadrant, wt= 3pi/2      sin 3pi/2 = - sin pi/2 =-1

                       v_max = wA = sqrt(k/m) A = sqrt( 21.6/0.217) 4.53 x10^-2

                       v_max = 0.45 m/s

b) f = 31 Hz,   at t =0 ,    x = +5.8 cm

         w = 2pi f = 2pi x 31 = 62 pi

               x = A coswt

           at t =0    5.8 = A cos 0

                     A = 5.8 cm

             at   t = 3.46 s,      x = 5.8 cos ( 2pi x f x 3.46)

                                          x = + 4.02 cm

c)   m = 0.31 kg ,      k = 14.2 N/m

             w = sqrt (m/k) = sqrt(0.31 /14.2) = 0.148

t =0 ,   x = +5.9 cm

                    x = A coswt  

                       A = 5.9 cm

             x = A cos wt

               cos wt = x/A

                      cos wt = -2.25/5.9

                                  = - 0.38

                         cos wt = cos 112     =   cos (0.622 pi)

                                 w t = 0.622 pi

                                       t = 0.622 pi /0.148

                                       t = 4.2 x3.14 s

                                       t = 13.18 s