A 500g hardboiled egg moves on the end of a spring with forc

A 50.0-g hardboiled egg moves on the end of a spring with force constant k = 20.0 N/m. It is released with amplitude 0.60 m at t = 0 s. A damping force F_x = -bv acts on the egg. After it oscillates for 4.00 s, the amplitude of the motion has decreased to 0.150 m. a) Calculate the magnitude of the damping coefficient b. b) Calculate the angular frequency w\'. c) Verify that the phase angle phi = 0. d) Find the displacement x at time t = 4.00 s. e) How long it takes to reduce the amplitude less than 1%.

Solution

frequency w0 = sqrt(k/m) = 20
period T = 2pi/w0 = 0.314
N (no. of cycles) = elapsed time/T = 4/0.314 = 12.74
is the \"logarithmic decrement\", the proportionate damping loss over one oscillation cycle. AR is the amplitude ratio = 4.
= ln(AR)/N = 0.1088
= /sqrt(4pi^2+^2) = 0.0173
B = 2*w0*m = 2*k/w0 = 2/sqrt(km) = 0. 0346 N-s/m

a) damping coefficient = 0.0346 N-s/m

b) Angular frequency = 20

c) Phase angle = 0

d) displacement = 0.15*cos(20*4) = 0.026 m = 2.6 cm