# Damped Oscillator Problem - Very Hard

#### Dillio

1. Homework Statement

I have read the chapter twice and I have read through the notes several times to help me with the homework assignment. It deals with damped Harmonic Oscillations.

Problem:
You have a mass submerged horizontally in oil and a spring with a k of 85 N/m pulls on a mass of 250g in oil with a b = 0.07 Kg/s

1. What is the period of oscillation?
I found the angular frequency of the system and then used the 2(pi) / omega to find the period. I found this to be around 0.3407 seconds. Is this correct?

2. How long does it take for the amplitude to die down to 0.5 amplitude of the max? There seems to be nothing in the book or the notes that helps with solving this unless I am missing something. I do not know a distance (or position), Amplitude, or phase angle to use the equation found in the book.

I found an answer of 4.95 seconds but I am not sure if that is correct since no equation in the book solves something like this. I took the Amplitude term of the damped harmonic oscillator equation and set it equal to 0.5A and solved for t.

3. How long until the total energy is 0.5 the initial value? The book just gives the rate of energy loss in terms of a velocity value and a b value, which was not given.

Absolutely no clue here....

I appreciate ANY help! Thanks.

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#### alphysicist

Homework Helper
Hi Dillio,

1. Homework Statement

I have read the chapter twice and I have read through the notes several times to help me with the homework assignment. It deals with damped Harmonic Oscillations.

Problem:
You have a mass submerged horizontally in oil and a spring with a k of 85 N/m pulls on a mass of 250g in oil with a b = 0.07 Kg/s

1. What is the period of oscillation?
I found the angular frequency of the system and then used the 2(pi) / omega to find the period. I found this to be around 0.3407 seconds. Is this correct?
That looks right to me.

2. How long does it take for the amplitude to die down to 0.5 amplitude of the max? There seems to be nothing in the book or the notes that helps with solving this unless I am missing something. I do not know a distance (or position), Amplitude, or phase angle to use the equation found in the book.

I found an answer of 4.95 seconds but I am not sure if that is correct since no equation in the book solves something like this. I took the Amplitude term of the damped harmonic oscillator equation and set it equal to 0.5A and solved for t.
That looks right to me.

3. How long until the total energy is 0.5 the initial value? The book just gives the rate of energy loss in terms of a velocity value and a b value, which was not given.
In #2 you found the time for the amplitude to reach half of its starting value.

For #3, when the energy is half of its value, what is the amplitude (compared to the original amplitude)? Once you answer that you can follow the same procedure you used in #2.

#### Dillio

For #3, when the energy is half of its value, what is the amplitude (compared to the original amplitude)? Once you answer that you can follow the same procedure you used in #2.
I used E = 0.5kA^2 and found A to be equal to sqrt([2E]/k). To solve for the energy when it is one half of its original value. I made the second energy equation: 05E = 0.5kA^2. I solved for this amplitude and found sqrt(E/k). That means the second amplitude is related to the initial energy by: E/sqrt(2)

I solved this for t in the amplitude equation and actually found 2.4 seconds, which is about half the time value I found in part 2. Does this make sense?

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