Problem 52 - Pearson Physics

Question

In a pig-calling contest a caller produces a sound with an intensity level of \(100 \mathrm{~dB}\). How many such callers would be required to reach the pain level of \(120 \mathrm{~dB}\) ?

Step-by-Step Solution

Answer

100 callers are needed to reach 120 dB.

Step 1: Understand the Decibel Scale

Decibels (dB) are a logarithmic unit used to measure sound intensity. Every increase of 10 dB represents a tenfold increase in sound intensity. Therefore, going from 100 dB to 120 dB involves changes in orders of magnitude.

Step 2: Calculate the Intensity Ratio

We need to calculate how much more intense 120 dB is compared to 100 dB. Since each 10 dB increase is a tenfold increase, 120 dB is 10 times louder than 110 dB, and 110 dB is 10 times louder than 100 dB. Thus, the ratio is: 10 times 10 = 100.

Step 3: Determine the Number of Callers Needed

Since each caller produces 100 dB, and we need a combined intensity level of 120 dB, we need 100 callers.The equation is derived from: 10\times \log_{10}(N) = 10\times 2 \ (since 120 - 100 = 20 \ and 20/10 = 2) Solving for \(N\), we find \(N = 10^2 = 100\). Thus, 100 callers are required to achieve 120 dB.

Key Concepts

Sound Intensity

Sound intensity refers to the power per unit area carried by a sound wave. It is often measured in watts per square meter (W/m²). Sound waves are vibrations that travel through a medium like air, water, or solids. When you speak or make noise, these vibrations travel and reach people's ears as sound.
The intensity of sound helps us understand how much energy the wave is transporting. More energy means a louder or more intense sound. Imagine a drum: a gentle tap emits less intense sound than a hard strike. In simple terms, higher sound intensity translates to more powerful and louder sounds.

• Measured in decibels (dB)
• Relates to the energy carried by sound
• Higher dB means a louder sound

Sound intensity plays a crucial role in understanding how we perceive different volumes and in contexts like noise control, broadcasting, and even in competitions!

Logarithmic Scale

The decibel scale used to measure sound intensity is a logarithmic scale. Unlike a linear scale where numbers increase directly (e.g., 1 to 2 to 3), a logarithmic scale increases by orders of magnitude. Each step on a logarithmic scale represents a multiplication of the previous step by a given number.
In the context of sound, every 10 dB increase signifies a tenfold increase in sound intensity. So, 20 dB is ten times more intense than 10 dB, 30 dB is 100 times more intense, and so on. The power of a logarithmic scale lies in its ability to represent how humans perceive sound and light. Our ears and brains interpret volume changes on a logarithmic scale, which allows us to hear both quiet whispers and loud noises without much strain.

• Decibels use a base-10 logarithm
• Conveys massive range differences compactly
• Mirrors human auditory perception

Understanding this can help when calculating the combined noise levels of multiple sources, like determining the combined effort in a call of many voices!

Sound Pressure Levels

Sound pressure levels (SPL) express the effectiveness of sound, typically using the decibel scale. They describe how loud a sound is at a particular location, based on pressure variations in the air. Think of SPL as the 'oomph' behind the sound you hear.
Sound pressure is caused by the fluctuations in air pressure due to sound waves. Higher pressure variations mean higher SPL, which can be uncomfortable or even painful for our ears. Different levels of SPL have different labels:

• Quiet whisper: around 20 dB
• Normal conversation: about 60 dB
• Rock concert: roughly 110 dB
• Pain threshold: approximately 120 dB

This helps us judge the appropriate number of callers needed in a situation like a pig-calling contest. If every caller adds their share of sound to reach 100 dB, combining energies from enough callers can reach the 120 dB pain threshold.