Question

A 11.9-kg object oscillates at the end of a vertical spring that has a spring constant of 1.80 ✕ 10⁴ N/m. The effect of air resistance is represented by the damping coefficient

b = 3.00 N · s/m.

(a) Calculate the frequency of the damped oscillation.
Hz

(b) By what percentage does the amplitude of the oscillation decrease in each cycle?
%

(c) Find the time interval that elapses while the energy of the system drops to 3.00% of its initial value.
s

Answer 1

Given:

• Mass of an object :
• Spring constant :
• Damping coefficient :

​​​​​​(a) Frequency :-

Angular frequency of damped oscillation is given by,

The frequency is given by,

This is the frequency of the damped oscillation.

(b) Percentage of amplitude :-

Position of an object under damped oscillation is described as,

In this displacement over one cycle of time , the amplitude changes from, ​​​​​​​.

For a fractional decrease of amplitude is given by,

Therefore, the amplitude of the oscillation decrease in each cycle by 2.02% .

(c) Time interval :-

we know that the energy is proportional to the square of the amplitude of the oscillation as,

For damped oscillation we know that,

Therefore,

We have given that the energy of the system drops to 3% of its initial value, hence,

​​​​​​​This is the time interval that elapses while the energy of the system drops to 3% of its initial value.

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