Problem 67 - Physics

Question

A portable radio is sitting at the edge of a balcony \(5.1 \mathrm{~m}\) above the ground. The unit is emitting sound uniformly in all directions. By accident, it falls from rest off the balcony and continues to play on the way down. A gardener is working in a flower bed directly below the falling unit. From the instant the unit begins to fall, how much time is required for the sound intensity level heard by the gardener to increase by \(10.0 \mathrm{~dB} ?\)

Step-by-Step Solution

Answer

It takes approximately 0.843 seconds for the intensity level to increase by 10 dB.

Step 1: Understanding the Problem

The portable radio is dropping from a height of 5.1 m and we need to determine how much time it takes for the sound intensity to increase by 10 dB for a gardener standing directly below. We start by examining how the intensity of sound changes with distance and how the drop time relates to this change.

Step 2: Formula for Sound Intensity Increase

The intensity level in decibels (dB) is calculated with the formula \[ L = 10 \times \log\left(\frac{I}{I_0}\right) \]where \(L\) is the intensity level, \(I\) is the intensity at a given point, and \(I_0\) is the reference intensity. A change of 10 dB corresponds to the intensity being multiplied by 10.

Step 3: Relationship Between Intensity and Distance

Sound intensity \(I\) is inversely proportional to the square of the distance \(r\) from the source, given as \[ I \propto \frac{1}{r^2} \]Thus, if the distance decreases by a factor of \(\sqrt{10}\), the intensity becomes 10 times greater.

Step 4: Calculate the Factor and New Distance

Since the intensity needs to increase by a factor of 10 (because a 10 dB increase), the new distance from the gardener to the radio should be \(5.1 / \sqrt{10}\). Calculate this distance: \[ r_\text{new} = \frac{5.1}{\sqrt{10}} \approx 1.613 \text{ m} \]

Step 5: Use Kinematics to Find Time

Use the kinematic equation for an object falling under gravity from rest: \[ h = \frac{1}{2} g t^2 \]Where \(h\) is the distance fallen (5.1 m - 1.613 m = 3.487 m) and \(g\) is the acceleration due to gravity (9.8 m/s²). Solving for \(t\): \[ 3.487 = \frac{1}{2} \times 9.8 \times t^2 \]\[ t^2 = \frac{3.487 \times 2}{9.8} \approx 0.711 \]\[ t \approx \sqrt{0.711} \approx 0.843 \text{ seconds} \]

Key Concepts

Decibel (dB)

The decibel (dB) is a unit that measures the intensity of sound. It’s a logarithmic unit, which means it scales in a non-linear way. This can be a bit tricky to understand at first, but let's break it down.
Firstly, the formula for sound intensity level in decibels is given by:\[ L = 10 \times \log\left(\frac{I}{I_0}\right) \]where:

• \(L\) is the sound intensity level in decibels.
• \(I\) is the intensity of the sound being measured.
• \(I_0\) is the reference sound intensity, typically set to the threshold of hearing, which is about \(10^{-12} \text{ W/m}^2\).

This formula reflects how humans perceive sound. A 10 dB increase usually indicates that the sound intensity is 10 times more powerful. This logarithmic nature reflects our hearing perception: each 10 dB increment represents sound intensity increasing by a power of 10. Whether you're listening to music or measuring ambient noise, understanding decibels helps us communicate the sound's strength effectively.

Kinematics in physics

Kinematics is a branch of physics that deals with the motion of objects. It’s concerned with the concepts of velocity, acceleration, and displacement, without considering the forces that cause the motion. In our exercise, understanding kinematics is crucial to solve how long it takes for the radio to fall.Consider an object in free fall, such as our fallen radio. We use the basic kinematic equation:\[ h = \frac{1}{2} g t^2 \]where:

• \(h\) is the height from which the object falls.
• \(g\) is the acceleration due to gravity, approximately \(9.8 \text{ m/s}^2\) on Earth.
• \(t\) is the time it takes for the object to fall that distance.

This equation helps us understand how the time of fall varies depending on the height and gravity. By calculating the displacement and time using kinematic equations, we developed a clear understanding of the object's motion during its fall. This allows us to predict how long the radio takes to reach a certain point.

Inverse-square law

The inverse-square law is a principle that applies to various physical phenomena, including sound and light. It states that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity.In our scenario, when the radio falls closer to the gardener, sound intensity increases because of the inverse-square relationship expressed as:\[ I \propto \frac{1}{r^2} \]where:

• \(I\) is the intensity.
• \(r\) is the distance from the sound source.

So, if the distance from the radio to the gardener changes, the intensity of the sound increases or decreases with the square of that distance. For the intensity level to increase by 10 dB, the distance must decrease by a factor of \(\sqrt{10}\), meaning relatively smaller changes in distance significantly impact intensity. This principle is essential for understanding phenomena in various fields, from acoustics to electromagnetism.