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.