Write the differential equation that governs the motion of the damped mass-spring system, and find the solution that satisfies the initial conditions specified. Units are mks; γ is the damping coefficient, with units of kg/sec

m = 0.2, γ = 1.6 and k = 4

Initial displacement is 1 and initial velocity is -2

x" + _____ x' ____x = 0

x(t) =

Answer:

the solution that satisfies the initial conditions specified is;

x(t) = $c_1e^{-4t}cos(2t)$ + $c_2e^{-4t}sin(2t)$

Explanation:

Given the data in the question ;

m = 0.2, γ = 1.6, k = 4

x(0) = 1, x'(0) = -2

Now, the differential equation that governs the motions of spring mass system is;

mx" + γx' + kx = 0

so we substitute

0.2x" + 1.6x' + 4x = 0

divide through by 0.2

x" + 8x' + 20x = 0

hence, characteristics equation will be;

m² + 8m + 20 = 0

we find m using; x = [ -b±√(b² - 4ac) ] / 2a

m = [ -8 ± √((8)² - 4(1 × 20 )) ] / 2(1)

m = [ -8 ± √( 64 - 80 ) ] / 2

m = [ -8 ± √-16 ) ] / 2

m = ( -8 ± 4i ) / 2

m = -4 ± 2i

Hence, the general solution of the differential equation is;

x(t) = $c_1e^{-4t}cos(2t)$ + $c_2e^{-4t}sin(2t)$

From the initial conditions;

c₁ = 1, c₂ = 1

the solution that satisfies the initial conditions specified is;

x(t) = $c_1e^{-4t}cos(2t)$ + $c_2e^{-4t}sin(2t)$

6 0

3 years ago

Which workout schedule would be BEST suited for a weight-lifting workout that builds both strength and endurance

VLD [36.1K]

My workout schedule is day on - day off

This helps the body rest after the work out and let it gain/build muscle mass

Good luck

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4 years ago

Read 2 more answers

The system needs an ordinary friction-based brake to bring the train to a full stop. Explain why the magnetic brake is not very

BabaBlast [244]

Answer:

The slower the train is moving, the less are the changes of the magnetic flux, thus the eddy currents become weaker.

Explanation:

A magnetic brakes is not a very efficient way of braking when a train is moving slowly because at low speeds, the changes in the magnetic flux are very less and so it causes the eddy current to become weaker.

Let us find the drag force which is proportional to the velocity of two conducting plates.

The EMF that is induced in the eddy currents are : $E=v(B \times L)$