Suppose a spring with spring constant 3 N/m 3 N m is horizontal and has one end attached to a wall and the other end attached to a 4 kg 4 k g mass. Suppose that the friction of the mass with the floor (i.e., the damping constant) is 3 N⋅s/m 3 N s m .
Set up a differential equation that describes this system. Let x x to denote the displacement, in meters, of the mass from its equilibrium position, and give your answer in terms of x,x′,x′′ x x ′ x ′ ′ . Assume that positive displacement means the mass is farther from the wall than when the system is at equilibrium.
Find the general solution to your differential equation from the previous part. Use c1c1 and c2c2 to denote arbitrary constants. Use tt for independent variable to represent the time elapsed in seconds. Enter c1c1 as c1 and c2c2 as c2. Your answer should be an equation of the form x=…x.
Enter a value for the damping constant that would make the system critically damped.

where m is the mass, c is the damping coefficient, k is the spring constant, x is the d²splacement, and the equation has been derived with respect to time t.

In the question, the

mass m = 3 kg,

spring constant k = 7 N/m,

damping constant c = 2N-s/m.

Differentiating x with time t, we take (d²x/dt²) = x'', and (dx/dt) = x'.

Substituting all the values in the standard differential equation, we get:

3x'' + 2x' + 7x = 0,

or, x'' + (2/3)x' + (7/3)x = 0 {Dividing by 3},

or, x'' + 0.67x' + 2.33x = 0, which is the required differential equation.

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The provided question is incorrect. The correct question is:

"Suppose a spring with a spring constant of 7 N/m is horizontal and has one end attached to a wall and the other end attached to a 3 kg mass. Suppose that the friction of the mass with the floor (i.e., the damping constant) is 2 N⋅s/m. Set up a differential equation that describes this system. Let x 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."