what is the formula to find..If the driving frequency is reduced slightly (but the driving amplitude remains the same), at what frequency will the amplitude of the mass's oscillation be half of the maximum amplitude?

To find the frequency at which the amplitude of the mass's oscillation is half of the maximum amplitude, we need to consider the equation for the amplitude of a driven harmonic oscillator.

The equation for the amplitude of a driven harmonic oscillator is given by the following formula:

A = (F0 / sqrt((m * (w0^2 - w^2)^2) + (b^2 * w^2))) / sqrt((w0^2 - w^2)^2 + (b^2 * w^2) / m^2)

Where:
A is the amplitude of the oscillation
F0 is the driving force amplitude
m is the mass of the object
w0 is the natural angular frequency of the oscillator
w is the angular frequency of the driving force
b is the damping coefficient

In this case, we want to find the frequency at which the amplitude is half of the maximum amplitude. So we can set A = A_max / 2 and solve for w.

Let's assume that we know the values of F0, m, w0, and b. Plug these values into the equation and solve for w when A = A_max / 2.

By rearranging the equation and substituting the given values into it, we can find the frequency at which the amplitude of the mass's oscillation is half of the maximum amplitude.