91Ó°ÊÓ

To calculate the potential energy of the pin before it is dropped, we rely on the fundamental potential energy formula: \( PE = mgh \). This represents the energy stored in an object due to its position relative to the ground. In our exercise, the mass of the pin \( m \) is \( 0.55 \, \text{g} = 0.00055 \, \text{kg} \), the acceleration due to gravity \( g \) is \( 9.8 \, \text{m/s}^2 \), and the height \( h \) from which the pin is dropped is \( 0.28 \, \text{m} \).

Let's break that down into steps:

• Mass conversion: First, convert the mass from grams to kilograms because the formula requires SI units.
• Substitute values: Replace \( m \), \( g \), and \( h \) with their appropriate values into the formula, and simplify to find the potential energy.

For this scenario, potential energy amounting to \( 1.5124 \times 10^{-3} \) joules represents the total mechanical energy available for conversion to sound when the pin impacts the floor.

When the pin lands, all its mechanical energy stored as potential energy transforms into sound energy. This scenario embodies the law of conservation of energy, which states that energy cannot be created or destroyed—only converted from one form to another.

This conversion assumes that the sound energy generated spreads out and radiates equally in all directions. Throughout the 1.5 seconds duration of sound emission, the calculated power of sound accounts for how much energy per second turns into sound. In our solution:

• Energy transformation: The potential energy is transformed into sound energy, calculated using the relation \( P = \frac{E}{t} \), where \( P \) is the power, \( E \) the energy, and \( t \) the time.

By dividing the total energy \( 1.5124 \times 10^{-3} \) joules by the time \( 1.5 \) seconds, we realize the energy conversion results in a power of approximately \( 1.0083 \times 10^{-3} \, \text{W} \). This power quantifies the rate at which energy is transferred to sound.

The total mechanical energy of the pin as it falls encompasses its initial potential energy. In physical dynamics, mechanical energy is the sum of potential energy (due to position) and kinetic energy (due to motion). For the pin dropping exercise, it's fundamental to recognize that initially, all its mechanical energy is in the form of potential energy.

Once released, as the pin begins its fall, this potential energy starts converting into kinetic energy. However, since our exercise assumes full conversion to sound, we focus only on the potential energy converted entirely into sound upon landing. This simplifies the problem by ignoring air resistance and other forces that might convert some energy into different states.

Specific points to remember in this context include:

• Energy conservation: The amount of mechanical energy is constant, merely transitioning between forms until it converts fully to sound.
• No losses considered: Our simplified model disregards non-conservative forces to maintain the conversion clarity.

Sound intensity relates to how much sound power passes through a certain area. Here, it's crucial in determining the farthest distance a sound can be audibly perceived. The formula for sound intensity \( I \) in this scenario is \( I = \frac{P}{4\pi r^2} \), requiring sound to be distributed uniformly across a sphere's surface with radius \( r \). This formula forms a mathematical relationship between power, intensity, and distance.

Steps taken to find the distance include:

• Threshold intensity: The minimum intensity detectable by the human ear is approximately \( 1 \times 10^{-12} \, \text{W/m}^2 \).
• Solving the equation: Setting \( I = 1 \times 10^{-12} \, \text{W/m}^2 \), and using the known power, solve for \( r \), giving the maximum distance.
• Maximum distance: Upon solving, the maximum theoretical distance is close to \( 8963 \, \text{m} \), though real-world factors may diminish it.

This formula highlights how surface area growth affects intensity and why energy dissipates as sound spreads from the source.