Problem Statement:

A mass m is attached to the end of a spring whose constant is k. After the mass reaches equilibrium, its support begins to oscillate vertically about a horizontal line L according to a formula $h(t)$. The value of h represents the distance in feet measured from L. Determine the DE of motion if the entire system moves through a medium offering a damping force that is numerically equal to (beta: B)(dx/dt).

I understand how to set up the equation and to solve when given numerical values for the above variables. I am stuck however with their solution. They are multiplying h(t) by the spring constant to get the DE: $$mx''(t) = -kx(t) - Bx'(t) + kh.$$
I thought it would be $$mx''(t) = -kx(t) - Bx'(t) + h(t)$$ and don't understand why we multiply by the spring constant for the $h(t)$ term.

Can anyone please help me understand this part?

Thank you!


  • $\begingroup$ Welcome to MSE! Please use the basic tutorial and quick reference guide and the Meta Read and enhance your question $\endgroup$
    – Jessie
    Feb 17, 2021 at 13:58
  • $\begingroup$ Units, for one thing. Your original idea isn't dimensionally consistent. You're adding Newtons to meters. The only scale in the problem available to multiply $h$ so it has units of $N$ is $m \ddot{x}$ is $k$. So you could easily "fix" your equation even without exactly understanding why $k$ has to be there. $\endgroup$ Feb 17, 2021 at 22:34

1 Answer 1


Because the length of the spring (better difference in length from the rest length) is $(x-h)$. $h$ is not an extra force, but the forced movement of the other end of the spring. So you could better write $$ m\ddot x + \beta\dot x+k(x-h)=0 $$

  • $\begingroup$ Thank you for the help! This clarifies my confusion :) $\endgroup$ Feb 23, 2021 at 19:00

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