What is the significance of the base e in the damped oscillations equation?

[maccha]

My textbook gives the equation A=Ao(e^-bt/2m) for the changing amplitude of damped oscillations. What I don't understand is where this equation comes from. Why make it to the base e? Why not make the equation A=Ao(f^T/t) where f is the factor by which it is decay and T is the period.

 

[ehild]

Yes, but b is the damping constant and m is the oscillating body of mass m. The formula is the result when you solve the equation of motion of the oscillator. There are two forces, acting on the oscillating body: Hook's force -Dx and a damping force which is proportional to the velocity. So ma=-Dx-bv, where x is the change of length of the spring and a is the acceleration, v is the velocity of the body, D is the spring constant and b is the damping factor. This equation can be solved for x, and it results in a sine function of time, where the amplitude decreases exponentially. You are right, one can use other base for the exponential, but it is simpler in maths to use the base e. e^x is a nice function, both its differential quotient and integral is the same as itself.

ehild