Phys 325 HW 6A due Thursday March 12, 2015 by 1pm Name:________________1.] 20 points This upside-down pendulum has atorsional spring κ at its base. It has governingnonlinear ODE ML2&&θ= −κθ+ MgL sinθ. Asdiscussed in class and in discussion motion near thetop, at θ small, has linearized ODE ML2&&θ+ (κ− MgL)θ= 0 and is unstable if κ < MgL.Let us ask how this system behaves if κ is too small.Take κ = MgL/2 for illustration.The system has three equilibrium points ( one is θ = 0 which we know is unstable) Theother two are at θ = ±θeq. Find θeq. (give it in both radians and degrees.) You will needto numerically solve the transcendental equation θeq for the point at which total torque iszero ( or equivalently where total PE is minimum.Let θ(t) = θeq + ε(t) where ε(t) is a small angle measuring deviation away from thisequilibrium but θeq is not necessarily small. Find the linear differential equationgoverning small ε in the form meff d2ε/dt2 + keffε=0. You will need the Taylor Seriesexpansion for sin(θeq + ε(t)) Show that the motion near this equilibrium is stable. Whatis its natural frequency?You may find it useful to set g = L = m = 1 and then solve the resulting entirelynumerical problem. There is no loss of generality is doing so, as we can alwayschoose units to make it so. If, however, we wish to recover the parametricdependence on these quantities, we can insert factors with the desired dimensions.For example, if you want a frequency, you take your numerical answer and multiplyit by (g/L)1/2 – which has the right dimensions.2] 20 points (you will need a calculator) Alightly damped system is driven nearresonance. Consider parameters ζ = 0.02, k= 2, and m = 2/49 acted on from quiescentinitial conditions: xo = vo = 0 by the force F= sin 6.9t; Find G(6.9) and φ(6.9). Writethe deviation from static equilibrium to be of the form x = xparticular + xcomplementary and findthe two coefficients in xcomplementary needed to match initial conditions for total x. Make anumerical plot of x(t) from t = 0 to t = 20. Use a time resolution dt that is finer than 1/ωn.Check your plot: are the initial conditions met? (you may need to zoom in near the plotorigin) From the plot, judge whether steady-state has been achieved by time 20. Whatwas the duration of the transient?Phys 325 HW 6B due Thursday March 12, 2015 by 1pm Name:________________1. 15 points The pictured pendulum (with a massless rod oflength L and point mass M on its tip) oscillates in an oil bath;the mass on the end suffers a drag force F =-cv proportional toits speed v and directed opposed to its velocity vector. Ignorebuoyancy. a) Derive the differential equation for arbitraryamplitude theta (but so that the mass remains submerged).b) Linearize it & Check it for obvious errors: is the dampingpositive? is the effective stiffness positive? What is itsdamped frequency of vibration? (you may assume the systemis underdamped) c) For initial conditions θ = Θ, dθ/dt = Ω,find the subsequent θ(t).2. 25 points Consider the pictured mass-spring systemexcited by a moving base y(t) and with a dragforce = -c times the velocity of the mass relative tothe lab. We wish to find the motion x(t)representing the displacement from equilibrium(Xeq=natural spring length) relative to the movingbase. We are interested in the steady state motion,i.e, the motion x(t) after the effects of initialconditions have died away. There are two stagesto this problem: a) derivation of the differential equation. It should come out the form indicated here: Meff&&x + Ceff&x + Keffx = D&&y + E&yYou are advised to use F= ma to derive it. The sum of the forces has two terms, withmagnitudes: k times the stretch of the spring, and c times the velocity of the mass relativeto the lab. But be careful in your derivation: x and X represent motion relative to themoving base; the mass's acceleration is therefore NOT d2x/dt2. Nor is its velocity relativeto the lab equal to dx/dt. Check the differential equation you derived. What are the fivecoefficients M,C,K,D and E?: Does the effective force (the right hand side) scale sensiblywith dy/dt and d2y/dt2? If the base y is accelerating to the right, for example, is there anapparent force acting on m to its left? Is your equation dimensionally consistent? Are theeffective stiffness and damping positive? Does the equation reduce to what you shouldhave if y(t) = 0? b) obtaining the particular solution. Substitute y(t) = Yo sin { ωt } and use thestandard formulas for harmonic responses in terms of the quantities G(ω) and φ(ω). Youneedn’t re-derive them, but you will have to substitute for all the forms of thecoefficients. ( You need not consider initial conditions; they only affect the constants inthe homogeneous part of the general solution xh, which dies away after enough time; thequestion only asks for the steady state part of the solution. ) You may save a lot ofalgebra if you remember that the particular solution associated with the sum of two forcesis the sum of the particular