Damped harmonic motion sinusoid equation

[biles1234]

what is the equation? i have something written down in my notes but i really don't get it...

x=A(e^-kt)(cos omega t)

first of all, how is the amplitude calculated if it decreases over time?? is it averaged?

what is e?

second of all, to calculate k, you need hooke's law and you need displacement. is this displacement averaged over time as well?

what is omega??

and what cos of what angle is used??

sorry if I'm being really vague but I'm really confused about this. thank you.

 

[Tide]

1) The "amplitude" A is the initial amplitude.

2) e Euler's number - the base of natural logarithms = 2.718281728...

3) x IS the displacement.

4) omega is the circular frequency of the oscillation (in this case corrected for damping)

5) "cos" is the cosine (a trigonometric function) and you treat $\omega t$ as the angle (i.e. the phase of the oscillation).

 

[kyle8921]

The equation you have written down is the equation for damped harmonic motion, which describes the motion of an object that is oscillating back and forth with a decreasing amplitude over time. The amplitude, A, is the maximum displacement from the equilibrium position. It is not averaged, but rather calculated at any given time based on the initial conditions of the system.

The constant e is the base of the natural logarithm and is a mathematical constant that is approximately equal to 2.71828. It is used in this equation to model the exponential decay of the amplitude over time.

The value of k is determined by Hooke's law, which states that the force exerted by a spring is directly proportional to the displacement of the object from its equilibrium position. The displacement used in this equation is the instantaneous displacement at any given time, not an average over time.

Omega (ω) represents the angular frequency of the oscillation and is equal to 2π times the frequency (f) of the oscillation. The cosine function is used because it represents the periodic nature of the motion and the angle used is the angular displacement at any given time.

I hope this helps to clarify some of your confusion about the equation. It is a complex equation that requires a good understanding of mathematical concepts and physical principles, so don't worry if you still have questions. It is always helpful to consult with your teacher or a fellow classmate for further clarification.