If the amplitude of loudness of sound becomes twice, then loudness increases by a factor of :

Question

Accepted Answer

When discussing the relationship between the amplitude of sound and its perceived loudness, it's important to understand the underlying principles of sound physics. The loudness of a sound is related to the amplitude of the sound wave, but it is not a direct linear relationship. Instead, loudness is often measured in decibels (dB), which is a logarithmic scale.

1. **Explanation for Option 2 (Correct Answer):**
   - The loudness of a sound is proportional to the square of its amplitude. This means that if the amplitude of a sound wave is doubled, the loudness increases by a factor of the square of 2, which is 4. This is because loudness is related to the intensity of the sound wave, and intensity is proportional to the square of the amplitude. Therefore, when the amplitude is doubled, the intensity becomes four times greater, leading to an increase in loudness by a factor of 4.

2. **Explanation for why other options are incorrect:**
   - **Option 1 (Factor of 2):** This option is incorrect because it suggests a linear relationship between amplitude and loudness. However, as explained, loudness is proportional to the square of the amplitude, not the amplitude itself. Therefore, doubling the amplitude results in a fourfold increase in loudness, not a twofold increase.
   
   - **Option 3 (Factor of 8):** This option is incorrect because it overestimates the increase in loudness. A factor of 8 would imply a more complex relationship than the square of the amplitude. Since loudness is proportional to the square of the amplitude, doubling the amplitude results in a fourfold increase, not eightfold.
   
   - **Option 4 (Factor of 16):** This option is incorrect because it significantly overestimates the increase in loudness. A factor of 16 would imply that the loudness increases by the fourth power of the amplitude, which is not the case. The correct relationship is that loudness increases by the square of the amplitude, leading to a fourfold increase when the amplitude is doubled.

In conclusion, the correct answer is Option 2, as it accurately reflects the relationship between amplitude and loudness, where loudness increases by a factor of 4 when the amplitude is doubled.