Calculating Decay Constant of Damped Oscillations

[h6872]

Homework Statement

Marie observes damped oscillations of a glider on an air track. She observed that the amplitude decreased to 50% of its original value after 10 seconds. What is the decay constant for the motion of the glider?

Homework Equations

The Attempt at a Solution

It seems the equation including the decay constant (involving the displacement, amplitude, angular frequency, and time) seems a little overcomplicated, considering the information given for this particular example.

However, I do know that if the motion of the damped glider is plotted, then the decay constant τ can be derived from the equation τ = -log e/slope. From the information given, I know that the graph will be of an exponential function, but how do I know what the slope of it would be given a 50% reduction in 10 seconds? Please help!

 

[LowlyPion]

Homework Statement

Marie observes damped oscillations of a glider on an air track. She observed that the amplitude decreased to 50% of its original value after 10 seconds. What is the decay constant for the motion of the glider?

Homework Equations

The Attempt at a Solution

It seems the equation including the decay constant (involving the displacement, amplitude, angular frequency, and time) seems a little overcomplicated, considering the information given for this particular example.

However, I do know that if the motion of the damped glider is plotted, then the decay constant τ can be derived from the equation τ = -log e/slope. From the information given, I know that the graph will be of an exponential function, but how do I know what the slope of it would be given a 50% reduction in 10 seconds? Please help!

Won't the equation be proportional then to e^(-kt) ?

Doesn't that mean then that e^(-k*10) = 1/2 ?

What happens if you then take the natural log of both sides?

 

[baba89]

Your approach is correct. The slope of the graph can be determined by taking two points on the curve, one at the initial amplitude and one at the amplitude after 10 seconds, and using the formula for slope (change in y/change in x). In this case, the change in y would be the change in amplitude (from 100% to 50%) and the change in x would be 10 seconds. Once you have the slope, you can plug it into the equation τ = -log e/slope to calculate the decay constant.