A particle of mass 5 gm moves along the x-axis under the influence of two forces:...

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A particle of mass 5 gm moves along the x-axis under the influence of two forces: (i) a force of attraction toward the origin, O, which is numerically equal to the 60 times the distance, and (ii) a damping force proportional to the instantaneous speed such that when the speed is 10 cm/s, the force is 300 g cm s?. The particle starts from rest at a distance 30 cm from O. a) Show that the DE for the motion is d?x d x +6 ++12x = 0 dt? dt b) Derive the expression for the position of the particle at any time. c) Find the amplitude, period and frequency of oscillation. d) Find the logarithmic decrement (the rate at which oscillation amplitude decreases). e) In general for what ranges of damping constant will the motion be (i) overdamped, (ii) underdamped, or (iii) critically damped? Given everything remains constant. A particle of mass 5 gm moves along the x-axis under the influence of two forces: (i) a force of attraction toward the origin, O, which is numerically equal to the 60 times the distance, and (ii) a damping force proportional to the instantaneous speed such that when the speed is 10 cm/s, the force is 300 g cm s?. The particle starts from rest at a distance 30 cm from O. a) Show that the DE for the motion is d?x d x +6 ++12x = 0 dt? dt b) Derive the expression for the position of the particle at any time. c) Find the amplitude, period and frequency of oscillation. d) Find the logarithmic decrement (the rate at which oscillation amplitude decreases). e) In general for what ranges of damping constant will the motion be (i) overdamped, (ii) underdamped, or (iii) critically damped? Given everything remains constant.