A horizontal platform vibrates with simple harmonic motion

A horizontal platform vibrates with simple harmonic motion in the horizontal direction with a period of 0.97 s. A body on the platform starts to slide when the amplitude of vibration reaches 0.285 m.
Find the coefficient of static friction between body and platform. The acceleration of gravity is 9.8 m/s^2 .

Thanks for the possible help and explanation.

it may look at first impossible to solve this since you are not given a mass, but we will see that you do not need the mass

first, we write the frequency of a harmonic oscillator:

f = (1/2pi) Sqrt[k/m] where k is the spring constant and m is the mass

solving for k, we get

k = 4 pi^2 f^2 m

now, we know that the force of a spring on an object is

F = k x where x is the displacement; the greater the displacement the greater the force

as long as F is less than the frictional force between the mass and the platform, the mass will nto slip, however, if F exceeds friction, the mass will slip

so we want to find the condition where F exactly equals friction; for this we need an expression for the friction force, which we know is u mg where u is the coefficient of friction, then we have

k x = u m g

but we know k from before, so using that expression for k we get

(4 pi^2 f m) x = u m g

the m's cancel leaving us with an expression for u

u = 4 pi^2 f^2 x/g = 4 (3.14)^2 (1/0.97s)^2 (0.285m)/9.8m/s/s

solve for u