please show all work :) 1.A) Write down Newton's second law for a damped harmonic oscillator and identify theterms.B) Assume a general solution: x(t) = exp{-at} [ASin(@t) +BCos(@t)], and describe inone sentence how we solve for a and o. Simply state the answer for these two variables.C) Assume a spring constant of 4.00-N/m and a mass of 4.00-kg and determine what isrequired for CRITICALLY damped motion. Graph x(t) if x(0) = 10 cm and it is at restfor this critically damped case.

(A) In Simple harmonic motion, the frictional forces such as air resistance do include. But now when the air resistance is considered the motion will slow down or dampen. This is known as a damped harmonic oscillator. And the motion is known as damped harmonic motion. And this frictional force is proportional to the velocity (v) of the object. Therefore, Fresistance∝-vFresistance=-bv Where b be defined as the vicious damping constant.Let m be defined as the mass of an object, and k be defined as the spring constant. Then according to Newton’s second law Fnet=-kx-bvma=-kx-bvmd2xdt2=-kx-bv a=d2xdt2,v=dxdtmd2xdt2+kx+bv=0d2xdt2+kmx+bmv=0d2xdt2+γdxdt+ω2x=0 ω2=km,γ=bm Where, a be defined as the acceleration. v be defined as the velocity ω be defined as the angular frequency of oscillation. γ be defined as the damping ratio.(B) The general solution is defined as, xt=e-αtAsinωt+BcosωtPlugging the given solution in the differential equation, there will be three results found for α, which will describe the motion of the system. Therefore, the quadratic equation can be written as, α2x+γαx+ω2x=0α2+γα+ω2x=0α2+γα+ω2=0 ∵x≠0On Solving:α=γ±γ2-4ω22Andω=km(C) Given the mass of the object is 4 kg, and the spring constant is defined as 4 N.m-1. The condition for the critically damping be defined as, γ2=4ω2γ2=4kmbm2=44 N.m-14 kgb2=4m2 s-2b2=44b2=64 kg2.s-2b=8 kg.s-1 In critical damping, the system returns to equilibrium very quickly without oscillation.The initial position be defined as x (0) = 10 cm. And given for critical damping, it is at rest. That means velocity is zero. Therefore, xt=e-αtAsinωt+Bcosωtx0=B10=Bddx0=0Aω-αB=0A=αBωA=0 α=0Therefore,xt=10cost ω=1Therefore, the graph be plotted as,