Question

A mass of 5 kg stretches a spring 10 cm. The mass is acted on by an external force of 10 sin(θ) N (newtons) and moves in a medium that imparts a viscous force of 2 N when the speed of the mass is 4 cm/s. If the mass is set in motion from its equilibrium position with an initial velocity of 3 cm/s, formulate the initial value problem describing the motion of the mass. Then (a) Find the solution of the initial value problem in Problem 6. (b) Identify the transient and steady state parts of the solution. (c) Plot the graph of the steady state solution. If the given external force is replaced by a force of 2 cos(ωt) of frequency ω, find the value of ω for which the amplitude of the forced response is maximum. 

          A mass of 5 kg stretches a spring 10 cm. The mass is acted on by an external force of 10 sin(θ) N (newtons) and moves in a medium that imparts a viscous force of 2 N when the speed of the mass is 4 cm/s. If the mass is set in motion from its equilibrium position with an initial velocity of 3 cm/s, formulate the initial value problem describing the motion of the mass. Then (a) Find the solution of the initial value problem in Problem 6. (b) Identify the transient and steady state parts of the solution. (c) Plot the graph of the steady state solution. If the given external force is replaced by a force of 2 cos(ωt) of frequency ω, find the value of ω for which the amplitude of the forced response is maximum.
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Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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A mass of 5 kg stretches a spring 10 cm. The mass is acted on by an external force of 10 sin(θ) N (newtons) and moves in a medium that imparts a viscous force of 2 N when the speed of the mass is 4 cm/s. If the mass is set in motion from its equilibrium position with an initial velocity of 3 cm/s, formulate the initial value problem describing the motion of the mass. Then (a) Find the solution of the initial value problem in Problem 6. (b) Identify the transient and steady state parts of the solution. (c) Plot the graph of the steady state solution. If the given external force is replaced by a force of 2 cos(ωt) of frequency ω, find the value of ω for which the amplitude of the forced response is maximum.
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Transcript

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00:01 Hi in the given problem we will be determining the position y position y of the mass at time t so we are given with m divided up prime plus gamma y prime plus k y is equal to g the function of g t so here m is equal to 5 gt is equal to 10 sine t over 2 and l is equal to 10 centimeter or we can say this is 0 .1 meter and we have g is equal to 9 .8 meter per second square which is acceleration due to gravity k is equal to m g over l that is equal to 490 and fd is equal to minus gamma v which is equal to minus 2 newton and v is called to 4 centimeters meter per second which is equal to 0 .04 meter per second and gamma is equal to 50 so the ordinary differential equation the initial condition is initial condition of differential equation is y0 y0 is equal to 0 and y prime 0 is to 3 or 0 .03 meter per second.
01:41 So we have the particular solution as yp for the given differential equation as equal to a, e...
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