## 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 compress

Question

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?

in progress 0
1 year 2021-09-03T21:00:43+00:00 1 Answers 1 views 0

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 )

– 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