Suppose a spring with spring constant 4 N/m is horizontal and has one end attach

Question

Suppose a spring with spring constant 4 N/m is horizontal and has one end attached to a wall and the other end attached to a 2kg mass. Suppose that the friction of the mass with the floor (i.e., the damping constant) is 3N * s/m.

a. Set up a differential equation that describes this system. Let x to denote the displacement, in meters, of the mass from its equilibrium position, and give your answer in terms of x, x', x''. Assume that positive displacement means the mass is farther from the wall than when the system is at equilibrium.

b. Find the general solution to you differential equation from the previous part. Use c1 and c2 to denote arbitrary constants. Use t for independent variable to represent the time elapsed in seconds. Enter c1 as c1 and c2 as c2. Your answer should be an equation of the form x=....

c. Is this system under damped, over damped, or critically damped? Enter a value for the damping constant that would make the system critically damped.

Explanation / Answer

k = 4 N/m, m = 2 , c = 3N * s/m.

mx'' + cx' + kx = 0

2x'' + 3x' + 4x= 0

x'' + 1.5 x' + 2 x = 0

b)

r^2 + 1.5 r + 2 = 0

r =( -1.5 +- i sqrt( 4*2 - 1.5^2) )/2

= -0.75 +- i sqrt(1.4375)

= -0.75 + - 1.19896 i

x(t) = e^(-0.75t) ( c1 sin (sqrt(1.4375 t) +   c2 cos (sqrt(1.4375 t))

c)

it is underdamped

as 1.5^2 -4 *2 < 0

c = 2sqrt(km)

= 2 *sqrt( 4 * 2)

= 4*sqrt(2)

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