A horizontal spring-mass system has low friction, spring stiffness 205 N/m, and mass 0.6 kg. The system is released with an initial compression of the spring of 13 cm and an initial speed of the mass of 3 m/s.
(a) What is the maximum stretch during the motion? m
(b) What is the maximum speed during the motion? m/s
(c) Now suppose that there is energy dissipation of 0.02 J per cycle of the spring-mass system. What is the average power input in watts required to maintain a steady oscillation?

a) x_max = 0.20794 m

b)  v_max = 3.8436 m/s

c) P = 0.05883 W

Explanation:

Given:

- The stiffness k = 205 N / m

- The mass m = 0.6 kg

- initial compression of the spring xi = 13 cm

- initial speed of the mass vi = 3 m/s

Find:

(a) What is the maximum stretch during the motion? m

(b) What is the maximum speed during the motion? m/s

(c) Now suppose that there is energy dissipation of 0.02 J per cycle of the spring-mass system. What is the average power input in watts required to maintain a steady oscillation?

Solution:

- Conservation of energy principle can be applied that the total energy U of the system remains constant. So the Total energy is:

                          U = K.E + P.E

                          U = 0.5*m*v^2 + 0.5*k*x^2

- We will take initial point with given values and maximum compression x_max when v = 0.

                          0.5*m*vi^2 + 0.5*k*xi^2 = 0.5*k*x_max^2

                          (m/k)*vi^2 + xi^2 = x_max^2

                          x_max = sqrt ( (m/k)*vi^2 + xi^2 ) = sqrt ( (.6/205)*3^2 + .13^2  

                          x_max = 0.20794 m

- The angular speed w of the harmonic oscillation is given by:

                          w = sqrt ( k / m )

                          w = sqrt ( 205 / 0.6 )

                          w = 18.48422 rad/s

- The maximum velocity v_max is given by:

                          v_max = - w*x_max

                          v_max = - (18.48422)*(0.20794)

                          v_max = 3.8436 m/s

- The amount of power required to stabilize each oscillation is given by:

                         P = E_cycle / T

Where, E = Energy per cycle  = 0.02 J

             T = Time period of oscillation

                         T = 2π/w

                         P = E_cycle*w / 2π

                         P = (0.02*18.48422) / 2π

                         P = 0.05883 W