Linear MethodsPlease answer question number 10 Problems 6 though 10 involve modelling the motion of an object attached toa spring. The function, y(t), denotes the displacement of the mass from theequilibrium position, and is a function of time, t. The velocity and accelerationaredyv(t) = y/(t) =dtand a(t) = y"(t) =-respectively.dt2All work, solutions and answers should be written in terms of t.Problem 6. The simple harmonic motion of a 2 kg mass attached to a spring with springconstant, k = 32, is governed by the differential equation.dy+ 32y(t) = 0dt2which has general solution y(t) = c, cos(4t) + c2 sin(4t).%3DShow that y(t) = 5 cos(4t +) is a solution to this differential equation by erpressingthis function as a linear combination of y1 (t) = cos(4t) and y2(t) = sin(4t).• Identify the amplitude, frequency and phase for this simple harmonic motion and sketcha graph of y(t).The spring-mass system in Problem 6 is modified by introduction of a dampingforce proportional to the velocity. The resulting differential equation for themotion isdydy+c+ 32y(t) = 0.dt2Problem 7. Find y(t) for e = 20, y(0) = 2, y(0) = -1. Is the system overdamped,underdamped, or critically damped.Problem 8. Determine the impact of changing the dampening force.• The damping constant is decreased to c = 16, find the general formula for the motionof the mass. What sort of damping results?If the damping constant is decreased to c = 8, find the general formula for the motionof the mass. What sort of damping results?The spring-mass system in Problem 6 is modified by introduction of an externaldriving force, F(t). The resulting differential equation for the motion isdy+ 32y(t) = F(t).%3Ddt²Problem 9. For an external force, F(t) = 28 cos(3t), find the function y(t) that describesthe motion of the mass. Does the motion remain bounded as t → +0? Explain your answer.Problem 10. For an external force, F(t) = 28 cos(4t), find the function y(t) that describesthe motion of the mass. In this case, the frequency of the system and the external forceare the same resulting in a phenomenon known as resonance. Describe what happens ast + +o0?

Question

Linear MethodsPlease answer question number 10 Problems 6 though 10 involve modelling the motion of an object attached toa spring. The function, y(t), denotes the displacement of the mass from theequilibrium position, and is a function of time, t. The velocity and accelerationaredyv(t) = y/(t) =dtand a(t) = y&#34;(t) =-respectively.dt2All work, solutions and answers should be written in terms of t.Problem 6. The simple harmonic motion of a 2 kg mass attached to a spring with springconstant, k = 32, is governed by the differential equation.dy+ 32y(t) = 0dt2which has general solution y(t) = c, cos(4t) + c2 sin(4t).%3DShow that y(t) = 5 cos(4t +) is a solution to this differential equation by erpressingthis function as a linear combination of y1 (t) = cos(4t) and y2(t) = sin(4t).• Identify the amplitude, frequency and phase for this simple harmonic motion and sketcha graph of y(t).The spring-mass system in Problem 6 is modified by introduction of a dampingforce proportional to the velocity. The resulting differential equation for themotion isdydy+c+ 32y(t) = 0.dt2Problem 7. Find y(t) for e = 20, y(0) = 2, y(0) = -1. Is the system overdamped,underdamped, or critically damped.Problem 8. Determine the impact of changing the dampening force.• The damping constant is decreased to c = 16, find the general formula for the motionof the mass. What sort of damping results?If the damping constant is decreased to c = 8, find the general formula for the motionof the mass. What sort of damping results?The spring-mass system in Problem 6 is modified by introduction of an externaldriving force, F(t). The resulting differential equation for the motion isdy+ 32y(t) = F(t).%3Ddt²Problem 9. For an external force, F(t) = 28 cos(3t), find the function y(t) that describesthe motion of the mass. Does the motion remain bounded as t → +0? Explain your answer.Problem 10. For an external force, F(t) = 28 cos(4t), find the function y(t) that describesthe motion of the mass. In this case, the frequency of the system and the external forceare the same resulting in a phenomenon known as resonance. Describe what happens ast + +o0?

Accepted Answer

Resonance and oscillation.

Problem 10. For an external force, F(t) = 28 cos(4t), find the function y(t) that describes the motion of the mass. In this case, the frequency of the system and the external force are the same resulting in a phenomenon known as resonance. Describe what happens as t + +o0?