How Long Until the Oscillator's Energy Halves?

[NAkid]

Homework Statement

A mass M is suspended from a spring and oscillates with a period of 0.820s. Each complete oscillation results in an amplitude reduction of a factor of 0.985 due to a small velocity dependent frictional effect. Calculate the time it takes for the total energy of the oscillator to decrease to 0.500 of its initial value.

Homework Equations

The Attempt at a Solution

Tried .985^t = .5, then t = log(.5)/log(.985) but this didn't work... any other suggestions?

 

[Shooting Star]

... any other suggestions?

Try a direct and simple approach, without calculus. The total E of the spring is (1/2)kA^2, where A is the amplitude. For the energy to become E/2, you can find A.

After each oscillation, it becomes cA, where c = 0.985. If it starts out with amplitude A0 and energy E0, then you can calculate after how many swings the energy becomes E0, by using the above mentioned process. Then find the time.

 

[Moetasim]

That is a good approach, but you need to take into account the fact that the amplitude decreases with each oscillation. This means that the energy also decreases with each oscillation. You can use the equation for the energy of an oscillator, E = 1/2*k*A^2, where k is the spring constant and A is the amplitude. Since the amplitude decreases by a factor of 0.985 with each oscillation, you can use this to find the relationship between the energy and the number of oscillations. Once you have this relationship, you can solve for the number of oscillations it takes for the energy to decrease to 0.500 of its initial value. From there, you can use the period and the number of oscillations to find the total time it takes for the energy to decrease to 0.500 of its initial value. I hope this helps!