# Spring/Mass Systems: Driven Motion

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.

• 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. Feb 17, 2021 at 22:34
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 $$m\ddot x + \beta\dot x+k(x-h)=0$$