need help with differential equations question ASAP!!?

Question

here's the question, it sounds like physics, but i'm in diff. eq and i don't have a clue how to start it...(we are currently working on nonhomogeneous diff eqs)

A front-loading washing machine is mounted on a thick rubber pad that acts like a spring; the weight W = mg (with g = 9.8 m/s2) of the machine depresses the pad exactly 0.13 cm. When its rotor spins at ω radians per second, the rotor exerts a vertical force
F0 cos(ωt)
Newtons on the machine. Neglecting friction, determine at what speed (in revolutions per minute) resonance vibrations will occur?
thanks!!

Answer 1

Yup, it's physics.

The problem says that the rubber pad acts like a spring. So we need a spring constant k. The value of k is implied in the statement that the weight of the machine depresses the pad by 0.13 cm. A spring supplies a force of magnitude kx where x is the amount of compression or extension. At equilibrium (i.e., when the machine isn't moving) this force equals the weight of the machine. So

W = mg = k(.0013)

Thus, k = m(9.8)/.0013 where m is the mass of the machine.

So your ODE would be

m d²x/dt² + kx = F0 cos(ωt)

As you know, the homogeneous solution of this gives the "natural" frequency of the system. Resonance occurs when the external force has that same frequency.

Note that ω has units of radians per second, and your answer is supposed to be in units of revolutions per minute.