You're stuck in the exit row on a long flight, and you suddenly worry that your seatmate, who's next to the window, might pull the emergency window inward while you're in flight. The window measures \(50 \mathrm{cm}\) by \(90 \mathrm{cm} .\) Cabin pressure is \(0.75 \mathrm{atm},\) and atmospheric pressure at the plane's altitude is 0.25 atm. Should you worry?

Understanding how to convert pressure units is essential in physics, especially when dealing with problems involving forces and the behavior of gases. In our scenario concerning cabin pressure, we started with the measure of pressure in atmospheres (atm), a common unit used in everyday applications. However, since we often need to perform calculations using the SI (International System of Units), we must convert these pressures into Pascals (Pa).

The conversion is straightforward but pivotal. One atmosphere is equivalent to approximately \(1.013 \times 10^5\) Pascals. To convert from atmospheres to Pascals, you simply multiply the given value in atm by \(1.013 \times 10^5\). For instance, to convert 0.75 atm to Pascals, the calculation would be \(0.75 \times 1.013 \times 10^5\) Pascals. This conversion ensures that all subsequent calculations are consistent with SI units, which is crucial for accuracy in scientific measurements.

In physics problems involving forces acting over a surface, such as the example of cabin pressure on an airplane window, calculating the area of that surface is a crucial step. In this case, we have a rectangular window, and the area of a rectangle is found by multiplying its height by its width.

Before we perform this multiplication, we must ensure that the units are correct. Since SI units are standard in physics, we convert the dimensions from centimeters to meters. Consequently, a height of 90 cm becomes \(0.90\) m, and a width of 50 cm becomes \(0.50\) m. With both measurements in meters, we calculate the area by multiplying the two: the resulting area of the window is \(0.90 \, m \times 0.50 \, m\), or \(0.45\) square meters. It's often useful to visualize this area as it relates to the problem at hand—picture nearly half a square meter that must withstand the force due to the pressure difference across it.

After understanding pressure conversion and area calculation, we can move on to the force resulting from a pressure difference. This is where our physics problem connects directly with real-world concerns like the integrity of an airplane window during flight.

The force exerted by a pressure difference over an area is given by the equation: \(\text{Force} = \text{pressure difference} \times \text{area}\). The pressure difference is found by subtracting one pressure from another, such as atmospheric pressure outside the plane from the cabin pressure inside. In the airplane example, the inside is pressurized to a higher degree than the outside, creating a net force that pushes outwards.

Once we know the pressure difference and have calculated the window's area, we can compute the total force acting on the window. This force essentially tells us whether the window is at risk of being forced inward or not. If the force is significant, it's reasonable to have safety concerns, but airplane windows are designed to withstand much greater forces than what passengers could exert, ensuring that you can rest easy on your flight.