A system with mass = 150g_and a spring constant of k=100N/m has a damping constant y=0.9.Assume the mass was pulled to the right 30 cm at t=0 and released.d) Estimate the time at which the amplitude has decayed to 4 of its initial value.e) Assume the system is connected to a forcing function given by (in Newtons)F(t)=2coswtf) Estimate the value of the amplitude at resonance.

(a) Given: The mass of the system is 150g. The value of the spring constant is 100 Nm. The value of the damping constant is 0.9. The mass is pulled to the right by 30 cm. Introduction: Damped harmonic oscillators are vibrating systems for which the amplitude of vibration decreases over time. Since nearly all physical systems involve considerations such as air resistance, friction, and intermolecular forces where energy in the system is lost to heat or sound, accounting for damping is important in realistic oscillatory systems.Calculation: Write the expression of the angular velocity. ω0=km Here, k is the spring constant, m is the mass of the particle. Substitute 100 Nm for k and 0.150 kg for m in the above expression. ω0=100 Nm0.150 kgω0=25.8 rads Write the expression of the amplitude. xt=xmeγt2m Substitute xm4 for xt, 0.9 for γ, 0.150 kg for m in the above expression. xm4=xme-0.9t20.150 kg14=e-0.9t20.150 kgln0.25=-0.9t20.150 kg-1.39=-0.9t0.300 kgt=0.5 s Thus, the time at which the amplitude decreases to 14 of its initial value is 0.5 s.

Answered: 150g_and a spring constant of k=100N/m…

150g_and a spring constant of k=100N/m has a damping constant y=0.9. A system with mass = Assume the mass was pulled to the right 30 cm at t=0 and released. d) Estimate the time at which the amplitude has decayed to 4 of its initial value.