Question

A low-pressure weather system can decrease the atmospheric pressure from 1 atm to 0.99 atm. By what percent will this decrease the force on a rectangular window from the outside? (Note: Assume the window is \(6 \mathrm{m}\) by \(3 \mathrm{m}\) and the glass is \(3 \mathrm{cm}\) thick.) A. \(1 \%\) B. \(10 \%\) C. \(\frac{1}{3 \%}\) D. \(30 \%\)

Short answer

The force decreases by 1%. The correct answer is A.

Step-by-step solution

Determine the change in atmospheric pressure

The initial pressure is 1 atm, and the final pressure is 0.99 atm. Calculate the change by subtracting the final pressure from the initial pressure.Change in pressure = 1 atm - 0.99 atm = 0.01 atm.

Calculate the force for initial and final pressures

Force is given by the formula \[ F = P \times A \].Where, - \( P \) is the pressure, - \( A \) is the area of the window.First, find the area of the window: \[ A = 6 \text{ m} \times 3 \text{ m} = 18 \text{ m}^2 \]Initial Force: \[ F_{initial} = 1 \text{ atm} \times 18 \text{ m}^2 = 18 \text{ atm} \text{ m}^2 \]Final Force: \[ F_{final} = 0.99 \text{ atm} \times 18 \text{ m}^2 = 17.82 \text{ atm} \text{ m}^2 \]

Determine the change in force

Subtract the final force from the initial force to find the change in force.\[ \text{Change in force} = 18 \text{ atm} \text{ m}^2 - 17.82 \text{ atm} \text{ m}^2 = 0.18 \text{ atm} \text{ m}^2 \]

Calculate the percent decrease in force

To find the percent decrease, use the formula \[ \text{Percent decrease} = \frac{\text{Change in force}}{\text{Initial force}} \times 100 \]Substitute the values: \[ \text{Percent decrease} = \frac{0.18 \text{ atm} \text{ m}^2}{18 \text{ atm} \text{ m}^2} \times 100 = 1 \text{ \text{Percent}} \]

Key concepts

Pressure Change

Atmospheric pressure is essentially the weight of the air above us in the atmosphere. Standard atmospheric pressure at sea level is typically about 1 atm (atmosphere). When there is a change in the weather, such as a low-pressure system, this pressure can decrease.

In the problem, the pressure drops from 1 atm to 0.99 atm. This small change might seem negligible, but it can have effects on large surfaces like windows.

To find the change in pressure, you simply subtract the new pressure from the old pressure: \[ \text{Change in pressure} = 1 \text{ atm} - 0.99 \text{ atm} = 0.01 \text{ atm} \]

This 0.01 atm change equates to a reduction in atmospheric force upon the window. This understanding is crucial as we dive into force calculations.

Force Calculation

Force exerted by the atmosphere on a surface is calculated using the formula given by physics:

\[ F = P \times A \]

Where:

• P is the pressure (in atm in this exercise)
• A is the area of the surface in contact with the atmosphere (in square meters for this exercise)

First, you need to determine the area of the window, which is given as 6 meters by 3 meters.
The area would be:
\[ A = 6 \text{ m} \times 3 \text{ m} = 18 \text{ m}^2 \]

Next, we calculate the initial and final forces:

• Initial force with 1 atm: \[ F_{initial} = 1 \text{ atm} \times 18 \text{ m}^2 = 18 \text{ atm m}^2 \]
• Final force with 0.99 atm: \[ F_{final} = 0.99 \text{ atm} \times 18 \text{ m}^2 = 17.82 \text{ atm m}^2 \]

These calculations show how a change in pressure translates to a change in the force exerted on the window.

Percent Decrease

To express the effect of the pressure decrease in a way that's easy to understand, you can convert the change in force to a percentage.

The formula used here is:

\[ \text{Percent decrease} = \frac{\text{Change in force}}{\text{Initial force}} \times 100 \]

First, find the change in force by subtracting the final force from the initial force:
\[ \text{Change in force} = 18 \text{ atm m}^2 - 17.82 \text{ atm m}^2 = 0.18 \text{ atm m}^2 \]

Then, plug this result into the percent decrease formula:
\[ \text{Percent decrease} = \frac{0.18 \text{ atm m}^2}{18 \text{ atm m}^2} \times 100 = 1 \text{\text{Percent}} \]

This result tells us that the force on the window decreased by 1 percent due to the drop in atmospheric pressure.

Physics Problem Solving

Solving physics problems often involves multiple steps. By breaking them down into manageable chunks, understanding becomes easier.

Here’s a summary of the steps taken to solve the given exercise:

• Identify given values: Initial pressure, final pressure, the dimensions of the window.
• Calculate the change in pressure: Subtract the final pressure from the initial pressure.
• Find the area of the window: Multiply its height by its width.
• Use the pressure and area to calculate forces: Plug these values into the force formula for both initial and final scenarios.
• Calculate the change in force: Subtract the final force from the initial force.
• Determine the percent decrease: Divide the change in force by the initial force and multiply by 100.

Approaching problems strategically like this helps ensure you don’t miss critical steps and makes complex problems more approachable. It's also a good practice to double-check each step to ensure accuracy.