The decibel level of a jackhammer is 130 dB relative to the threshold of hearing. Determine the sound intensity produced by the jack hammer
The Correct Answer and Explanation is :
To determine the sound intensity produced by the jackhammer, we can use the formula that relates sound intensity and decibel level:
[
\text{dB} = 10 \times \log \left( \frac{I}{I_0} \right)
]
Where:
- (\text{dB}) is the decibel level (130 dB in this case),
- (I) is the sound intensity we need to find,
- (I_0) is the reference intensity, which is the threshold of hearing and equals (1 \times 10^{-12} \, \text{W/m}^2).
Step-by-step calculation:
- Rearrange the formula to solve for the sound intensity (I):
[
\frac{I}{I_0} = 10^{\frac{\text{dB}}{10}}
]
[
I = I_0 \times 10^{\frac{\text{dB}}{10}}
]
- Substitute the known values:
- ( \text{dB} = 130 \, \text{dB} )
- ( I_0 = 1 \times 10^{-12} \, \text{W/m}^2 )
[
I = 1 \times 10^{-12} \times 10^{\frac{130}{10}}
]
- Simplify the exponent:
[
\frac{130}{10} = 13
]
[
I = 1 \times 10^{-12} \times 10^{13}
]
[
I = 10^{1} = 10 \, \text{W/m}^2
]
Final answer:
The sound intensity produced by the jackhammer is (10 \, \text{W/m}^2).
Explanation:
Sound intensity is a measure of the energy transported by sound waves through a medium (such as air). The decibel scale is logarithmic, meaning a small increase in decibels represents a large increase in sound intensity. A 130 dB level, as in the case of the jackhammer, indicates an extremely loud sound.
In this case, a jackhammer produces a sound that is 130 dB relative to the threshold of hearing, which means its sound intensity is (10 \, \text{W/m}^2). For context, this is a very high sound intensity. For example, normal conversation occurs at about 60 dB, which corresponds to a sound intensity of (10^{-12} \, \text{W/m}^2), while a rock concert might reach 120 dB.
Because the scale is logarithmic, every increase of 10 dB represents a tenfold increase in sound intensity. Thus, the jackhammer’s sound intensity is 10 times higher than a sound at 120 dB and 100 times higher than a sound at 110 dB.
This makes jackhammers one of the loudest sources of sound in everyday life, often requiring ear protection to avoid hearing damage.