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Chapter 9: Problem 276

The amplitude of a damped oscillator decreases to \(0.9\) times its original magnitude in \(5 \mathrm{~s}\). In another \(10 \mathrm{~s}\) it will decrease to \(\alpha\) times its original magnitude, where \(\alpha\) equals (A) \(0.81\) (B) \(0.729\) (C) \(0.6\) (D) \(0.7\)

Short Answer

Expert verified
The amplitude of the damped oscillator decreases to 0.9 times its original magnitude in 5 seconds. After another 10 seconds, it will decrease to \(\alpha\) times its original magnitude, where \(\alpha\) equals 0.729. This is because after 5 seconds, the amplitude is 0.9 times the original value, and after another 10 seconds, the amplitude decreases to \(0.81 \times 0.9 = 0.729\) times its original magnitude. Hence, the correct answer is (B) 0.729.

Step by step solution

01

Understand the concept of a damped oscillator.

In a damped oscillator, the amplitude of oscillation decreases exponentially over time. The rate of decrease depends on the damping constant. In this case, we are given the information that the amplitude decreased 0.9 times its magnitude after 5 seconds.
02

Calculate the decrease in amplitude after another 10 seconds.

We are given that after the initial 5 seconds, the amplitude decreases by 0.9 times its original value. To find the decrease factor after another 10 seconds, we just need to multiply this factor again after 5 seconds and then after the total 10 seconds. After 5 seconds decrease: 0.9 After another 5 seconds (total 10 seconds): \(0.9 \times 0.9 = 0.81\) After another 5 seconds (total 15 seconds): \(0.81 \times 0.9= 0.729\)
03

Select the correct answer.

After going through the given choices, we can see that the correct value for 伪 is 0.729, which is option (B). Thus, the amplitude of the damped oscillator decreases to 0.729 times its original magnitude after a total of 15 seconds.

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