A heavy table of mass M is vibrationally isolated by being hung from the ceiling by springs, so that its frequency of vertical oscillation is \omega_0 (take \omega_0 to be 2\pi/sec, a typical

Question

value). Assume now that the ceiling vibrates vertically with amplitude A at frequency \omega\neq\omega_0, i.e. y_c(t)=A\cos(\omega t).

a. Write down the dynamical equation that relates a(t), the acceleration of the table, to its position y(t), and the position of the ceiling. Although M and k will appear in this
equation, you should be able to replace them with \omega_0. Show that the equation you get this way is the same as if a force proportional to y_c(t)=A\cos(\omega t) were acting on the mass – spring system.

This system is referred to as a driven harmonic oscillator. Its steady state solution is y_c(t)=A\cos(\omega t). NOTE that it responds solely at the drive frequency \omega\neq\omega_0, not at the
natural frequency of the oscillator \omega\neq\omega_0. (Actually there is also a transient at \omega\neq\omega_0 that fades away with time in a real system due to damping).

b. By substituting the above expression for y(t) (and the a(t) that results from this) in your equation from part (a), you should be able to obtain and solve a simple equation for C(\omega).

c. With what amplitude, y_t, will the table oscillate if the building (i.e. ceiling) oscillates with amplitude 0.01 cm at a (typical) frequency of 15 Hz? This ratio is called the isolation factor at \omega\neq\omega_0.