Anonymous asked in Science & MathematicsMathematics · 2 months ago

Suppose a mass of m = 2 kg is suspended from a spring with spring constant k = 10 N/m.  ?

Suppose a mass of m = 2 kg is suspended from a spring with spring constant k = 10

N/m. Furthermore, suppose the spring is submerged in a medium that provides a damping force that is 4 times the velocity. The mass is released from a position 1 m below equilibrium with no initial velocity. Find the equation of motion and explain whether the system underdamped, overdamped or critically damped?

1 Answer

  • 2 months ago

    I'll get you started in the right direction.

    We can set up a differential equation simply by using Newton's law ΣF = ma. First, we identify all of the forces acting on the spring.

    Naturally, we have gravity acting on this suspended spring. The magnitude of the force due to gravity is given by

    F_g = mg

    The damping force also acts on the spring. We are given that this force is four times the velocity at any given time. Thus, the magnitude of the damping force is given by

    F_d =  4v, where v is the velocity at any time t.

    Finally, there is a "restorative" force which tends the mass-spring system to it's equilibrium position. According to Hooke's law, the magnitude of this force is given by

    F_s =  kx, where k is the spring constant and x is the displacement from the equilibrium position.

    Now, if we take the down direction to be negative, we can assign signs to the forces F_g, F_d, and F_s according to which direction they will act on the mass. 

    Note that the gravitational force acts in the downward direction, thus we assign a negative to F_g.

    Since the damping force will always oppose the direction of travel of the spring system, we may assign a negative to F_d as well( whatever sign the velocity has at time t, we negate it).

    Finally, the restorative force too may be assigned a negative since it opposes the direction of travel as well.

    Summing up the forces we get the following:

    F_g + F_d + F_s = ma

    -mg -  4v -  kx =  ma

    Note that v = dx/dt and a = dv/dt = dx²/dt²

    Thus, we may rewrite as

    -mg - 4(dx/dt) - kx =  mdx²/dt²

    We can rearrange as

    mdx²/dt² + 4dx/dt + kx = -mg

    Note that this a second order ODE. I'll let you take it from here.

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