Question: Exercise Set 4:
The following chart shows loudness levels in decibels, some of which can damage the human ear.
Chart of sound intensity levels (loudness)for environmental noise
Weakest sound heard 0 dB
Rustling leaves 10 dB
A whisper in library at 2 m 30 dB
Conversation at home 50 dB
Conversation in restaurant 60 dB
Passenger car at 80 kph at 6 m 70 dB
Vacuum cleaner at 1m 70 dB
Freeway at 20 m 73 dB
Telephone dial tone 80 dB
At 90 - 95 dB sustained exposure may sustain hearing loss
Car wash at 6 m 90 dB
Train whistle at 150 m 90 dB
Hand drill 98 dB
Lawn mower at 1M 105 dB
Motorbike 100 dB
Jet take off at 300 m 100 dB
Sand blasting 115 dB
Threshold of discomfort Thunderclap 120 dB
Chain saw 120 dB
Oxygen torch 120 dB
Loud rock concert 115 dB
Pain threshold 130 dB Pneumatic riveter 125 dB
Aircraft carrier deck 140 dB
Eardrum rupture Jet take off at 25 m 150 dB
Decibels (dB) are a measurement configured to human sensitivity to sound. The formula for decibels (L) is:
L = 10 log(I/I0)
Where I = intensity, with units w/m^2 (watts per square metre).
I0 is the weakest sound that humans can hear, 10^-12 watts/m^2.
Therefore, 0dB is the quietest sound humans can hear, as:
L = 10 log(I0/I0)
= 10 log(1)
= 0
OFFICIAL
Question 1
By rearranging the given formula, determine the intensity in watts/m^2 of the following sounds
(a) (i) a chain saw
(ii) a vacuum cleaner at 1m
(iii) rustling leaves
(iv) a telephone dial tone
(v) a hand drill
(vi) an aircraft carrier deck
(b) Explain the reason for using sound intensity levels in decibels rather than intensity in watts/m^2.
top-center: chart/table of sound intensity levels with rows of sound names and dB values; above-middle area.
center: formula text L = 10 log(I/I0), definition of I and I0, and worked example showing L = 10 log(I0/I0) = 10 log(1) = 0.
bottom-center: title OFFICIAL and Question 1 with parts (a) and (b), listing sounds including chain saw, vacuum cleaner at 1m, rustling leaves, telephone dial tone, hand drill, and aircraft carrier deck.User: please solve step by step clearly!
1. **State the problem:**
We are given the formula for sound intensity level in decibels:
$$L = 10 \log\left(\frac{I}{I_0}\right)$$
where $L$ is the loudness level in decibels (dB), $I$ is the intensity in watts per square meter, and $I_0 = 10^{-12}$ watts/m$^2$ is the reference intensity (the quietest sound humans can hear).
We need to find the intensity $I$ for various sounds given their decibel levels $L$.
2. **Rearrange the formula to solve for $I$: **
Starting from
$$L = 10 \log\left(\frac{I}{I_0}\right)$$
Divide both sides by 10:
$$\frac{L}{10} = \log\left(\frac{I}{I_0}\right)$$
Rewrite the logarithm in exponential form:
$$10^{\frac{L}{10}} = \frac{I}{I_0}$$
Multiply both sides by $I_0$:
$$I = I_0 \times 10^{\frac{L}{10}}$$
3. **Calculate intensities for each sound:**
(a) (i) Chain saw: $L = 120$ dB
$$I = 10^{-12} \times 10^{\frac{120}{10}} = 10^{-12} \times 10^{12} = 1 \text{ watt/m}^2$$
(ii) Vacuum cleaner at 1m: $L = 70$ dB
$$I = 10^{-12} \times 10^{\frac{70}{10}} = 10^{-12} \times 10^{7} = 10^{-5} \text{ watts/m}^2$$
(iii) Rustling leaves: $L = 10$ dB
$$I = 10^{-12} \times 10^{\frac{10}{10}} = 10^{-12} \times 10^{1} = 10^{-11} \text{ watts/m}^2$$
(iv) Telephone dial tone: $L = 80$ dB
$$I = 10^{-12} \times 10^{\frac{80}{10}} = 10^{-12} \times 10^{8} = 10^{-4} \text{ watts/m}^2$$
(v) Hand drill: $L = 98$ dB
$$I = 10^{-12} \times 10^{\frac{98}{10}} = 10^{-12} \times 10^{9.8} = 10^{-12 + 9.8} = 10^{-2.2} \approx 6.31 \times 10^{-3} \text{ watts/m}^2$$
(vi) Aircraft carrier deck: $L = 140$ dB
$$I = 10^{-12} \times 10^{\frac{140}{10}} = 10^{-12} \times 10^{14} = 10^{2} = 100 \text{ watts/m}^2$$
4. **Explain why decibels are used instead of raw intensity:**
- The range of sound intensities humans can hear is extremely large, from $10^{-12}$ to over $10^{2}$ watts/m$^2$, spanning 14 orders of magnitude.
- Using decibels compresses this large range into a manageable scale (0 to about 150 dB).
- Decibels are logarithmic, which aligns better with human perception of loudness, which is also roughly logarithmic.
- This makes it easier to compare and communicate sound levels.
**Final answers:**
| Sound | $L$ (dB) | $I$ (watts/m$^2$) |
|---------------------|----------|-----------------------------|
| Chain saw | 120 | $1$ |
| Vacuum cleaner (1m) | 70 | $10^{-5}$ |
| Rustling leaves | 10 | $10^{-11}$ |
| Telephone dial tone | 80 | $10^{-4}$ |
| Hand drill | 98 | $6.31 \times 10^{-3}$ |
| Aircraft carrier deck| 140 | $100$ |