Exploring b & n in Damped Harmonic Motion

[romd]

Homework Statement

Concerning damped harmonic motion (mass on a spring using cardboard discs as dampers); for the equation (below) of the graph describing the effect of different sized dampers on the time taken for amplitude of oscillations to halve, what do b (y-intercept) and n (gradient) represent? (A=area of damper; T=time taken for amplitude to halve)

Homework Equations

[tex]T=b.A^n[/tex]
(i.e. [tex]ln(T)=n.ln(A) + ln(b)[/tex] )

 

[Nate810]

The Attempt at a Solution b represents the time taken for the amplitude to halve when the area of the damper is equal to 1. n represents the rate at which the time taken for the amplitude to halve decreases as the area of the damper increases.

 

[Moetasim]

I would like to first clarify that the equation provided is not a standard equation for damped harmonic motion and may not accurately represent the behavior of the system. However, for the purpose of this question, let us assume that it is a suitable equation for the given scenario.

In this case, b and n represent the parameters of the equation, with b representing the y-intercept and n representing the gradient of the line on a logarithmic scale. The y-intercept, b, represents the value of T when A is equal to 1 (ln(A) = 0). This can be interpreted as the time taken for the amplitude to halve when there is no damper present (A=1).

On the other hand, the gradient, n, represents the rate of change between T and A on a logarithmic scale. A higher value of n would indicate a steeper slope, meaning that the time taken for the amplitude to halve decreases more rapidly as the damper size increases. This suggests that a larger damper has a greater effect on reducing the amplitude of oscillations.

It is important to note that the values of b and n may vary depending on the specific system and experimental conditions. Therefore, it is necessary to conduct further experiments and analyze the data to determine the most accurate values for these parameters.