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!
Alex
Best Answer
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 $$ m\ddot x + \beta\dot x+k(x-h)=0 $$