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Elena-2011 [213]
3 years ago
14

A mass with mass 4 is attached to a spring with spring constant 24 and a dashpot giving a damping 20. The mass is set in motion

with initial position 4 and initial velocity 2. (All values are given in consistent units.) Find the position function
Physics
1 answer:
aleksandrvk [35]3 years ago
7 0

Answer: x(t) = 14e^{-2t} - 10e^{-3t}

Explanation: In a mass-spring-damper system, the differential equation that rules the motion of the mass is: mx" + cx' + kx = 0

Using m = 4, k = 24 and c = 20, we have

4x" + 20x' + 24x = 0

Simplifying, we have

x" + 5x'+ 6x = 0

The characteristic equation of this differential is

r^{2} + 5r + 6 = 0

The solutions for the quadratic equation are: r_{1} = -2 and r_{2} = -3

Hence:

x(t) = C_{1}e^{-2t} + C_{2}e^{-3t}

x'(t) = -2C_{1}E^{-2t} - 3C_{2}e^{-3t}

To determine the constants, we have the initial conditions x(0) = 4 and

x'(0) = 2, then:

x(0) = C_{1} + C_{2} = 4\\          C_{1} = 4 - C_{2}

x'(0) = -2C_{1} -3C_{2} = 2\\-2(4-C_{2}) -3C_{2} = 2\\C_{2} = -10\\C_{1} = 4 - C_{2}\\C_{1} = 14

Substituing the constants:

x(t) = 14e^{-2t} - 10e^{-3t}

The position function for this system is: x(t) = 14e^{-2t} - 10e^{-3t}

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