Question

Atmospheric pressure is about ${10}^5\mathrm{N}/{\mathrm{m}}^2$, meaning that a $100,000\mathrm{N}$ weight of air-corresponding to a mass of $10,000\mathrm{kg}$ -sits atop very square meter of the ground (at or near sea level). If the air density were constant at $1.25\mathrm{kg}/{\mathrm{m}}^3$ - rather than decreasing with height as it actually does - how high would the atmosphere extend to result in this weight (mass)?

Short answer

Answer: The atmosphere would extend to 8000 meters.

Step-by-step solution

Calculate the force

We are given the atmospheric pressure (P) as ${10}^{5}N/m^2$ and the area (A) as $1m^2$. We can find the force by using the formula: Force (F) = Pressure (P) * Area (A) F = $({10}^{5}N/m^2)\ast (1m^2)={10}^{5}N$

Calculate the mass

Now that we have the force (F), we can find the mass by using the formula: Mass (m) = Force (F) / Gravity (g) Given g is approximately $9.81m/s^2$, thus: m = $({10}^{5}N)/(9.81m/s^2)\approx {10}^{4}kg$

Calculate the height

We have found the mass (m) in a specific area. Now, we need to find the height (h) of the atmosphere. We are given the constant air density as $1.25kg/m^3$. We can find the height using the formula: Height (h) = Mass (m) / (Area (A) * Air Density) h = $({10}^{4}kg)/(1m^2\ast 1.25kg/m^3)=8000m$ Therefore, the height of the atmosphere with a constant air density of $1.25kg/m^3$ would extend to 8000 meters.

Key concepts

Air Density

Air density is a key factor in understanding atmospheric pressure and atmospheric physics in general. It represents the mass of air contained within a certain volume. On Earth's surface, air density can be approximated to be around $1.25{\text{ kg/m}}^3$.
Air density is not constant and usually decreases with altitude. At sea level, air is compressed by the weight of the air above, making it denser. As you ascend in altitude, the air becomes thinner, resulting in reduced air density.

• Conceptual Importance: Understanding air density is vital for calculations involving buoyancy, aerodynamics, and meteorology.
• Applications: Air density knowledge is applied in designing aircraft, weather forecasting, and calculating the height of the atmosphere based on constant pressure assumptions.

In scenarios like our problem, where air density is considered constant, it's much simpler to compute the hypothetical height of the atmosphere, given known mass and pressure.

Force Calculation

Force calculation is indispensable in understanding pressures and forces exerted by the atmosphere. The force exerted by air on a surface is calculated using pressure and area. The basic formula to find force $F$ is:
$$F=P\times A$$where $P$ is the pressure, and $A$ is the area.

• Atmospheric Pressure: Given as ${10}^5{\text{ N/m}}^2$ at sea level, illustrating how much force is exerted on a square meter of the Earth’s surface by the atmosphere.
• Force Example: In the exercise, with a surface area of $1{\text{ m}}^2$, the atmospheric force is calculated as ${10}^5\text{ N}$.

Understanding how to calculate force aids in visualizing how much air pressure affects objects and living beings on Earth. It is also pivotal in further calculations involving mass and gravitational force applications.

Gravitational Acceleration

Gravitational acceleration $g$ plays a fundamental role in determining the weight force exerted by an object. On Earth, $g$ is approximately $9.81{\text{ m/s}}^2$.
This constant value is used to calculate the weight or gravitational force acting on an object. By applying Newton's second law, weight $W$ can be found with the formula:$$W=m\cdot g$$where $m$ is the mass.

• Importance: Understanding gravitational acceleration allows us to compute how much "pull" the Earth exerts on an object, which directly links to how pressure is experienced due to mass (like air mass).
• Example Context: In our scenario, dividing atmospheric force by $g$ gives us the mass of air exerting pressure on the surface, approximately ${10}^4\text{ kg}$.

This concept is essential not only for physics calculations but also for real-world applications such as engineering and flight dynamics.