The amplitude of a lightly damped oscillator decreases by 3.0 %during each cycle. What percentage of the mechanical energy of the oscillator is lost in each cycle?

Short Answer

Expert verified

The percentage of mechanical energy lost in each cycle is 6%.

Step by step solution

01

The given data:

The amplitude of the oscillator decreases by 3% in each cycle.

02

Understanding the concept of damping:

Damping is the restrain of the vibratory motion of a body affecting its oscillations due to the dissipation of energy. Here, the oscillator is damped resulting in a constant decrease of the energy that results in restricting its motion that is a decrease in its amplitude.

Formula:

The mechanical energy of the body undergoing oscillations,

E=12kxm2 ….. (i)

Where, k is the spring constant and xm is the amplitude or the maximum displacement of the body.

03

Calculate percentage of the mechanical energy of the oscillator is lost in each cycle:

Let the initial mechanical energy of the oscillator according to equation (i) be,

E=12kxm2 ….. (I)

Here, xm is the initial amplitude of the oscillator.

You have the amplitude of the oscillations decreased by 3% .

So, the amplitude of next oscillation will become,

data-custom-editor="chemistry" xm'=xm-0.03xm=0.97xm

Thus, the decreased mechanical energy of the next cycle of oscillator is given using the above data in equation (i) as follows:

E'=12kxm'2=12k0.97xm2=0.9412kxm'2=0.94E

So, the loss in the amount of mechanical energy of the oscillator is given as follows:

E=E'-E=0.94E-E=-0.06EEE=-0.06=6%

The negative sign shows that energy is lost.

Hence, the percentage of mechanical energy lost in each cycle is 6%.

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