Problem 12 - College Physics

Question

Landing on Venus. One of the great difficulties in landing on Venus is dealing with the crushing pressure of the atmosphere, which is 92 times the earth's atmospheric pressure. (a) If you are designing a lander for Venus in the shape of a hemisphere 2.5 \(\mathrm{m}\) in diameter, how many newtons of inward force must it be prepared to withstand due to the Venusian atmosphere? (Don't forget about the bottom!) (b) How much force would the lander have to withstand on the earth?

Step-by-Step Solution

Answer

(a) 137826 kN on Venus; (b) 1.49 MN on Earth.

Step 1: Calculate Area for Venus

The lander's shape is a hemisphere, so calculate the total surface area including the base. The radius \( r \) is half the diameter: \( r = \frac{2.5}{2} = 1.25 \, \mathrm{m} \). The surface area of the curved part of the hemisphere is \( 2\pi r^2 \), and the base area is \( \pi r^2 \). Thus, the total surface area \( A \) is:\[A = 2\pi r^2 + \pi r^2 = 3\pi r^2\]Substitute \( r = 1.25 \, \mathrm{m} \):\[A = 3\pi (1.25)^2 = 3\pi (1.5625) = 4.6875\pi \, \mathrm{m}^2\]Convert \( \pi \) to its approximate value \( 3.1416 \) for computation:\[A \approx 4.6875 \times 3.1416 = 14.725 \, \mathrm{m}^2\]

Step 2: Calculate Force for Venus

Now, use the area to calculate the force. The atmospheric pressure on Venus is 92 times that on Earth, with Earth having a pressure of approximately \( 101325 \, \mathrm{Pa} \). Thus, Venus has a pressure \( P_v = 92 \times 101325 \, \mathrm{Pa} \). Calculate the force \( F_v \) using \( F = PA \):\[F_v = 92 \times 101325 \times 14.725 \]Compute this to find the inward force:\[F_v \approx 92 \times 101325 \times 14.725 = 137825811.75 \, \mathrm{N} \]The lander must withstand approximately \( 137826 imes 10^3 \, \mathrm{N} \).

Step 3: Calculate Area for Earth

The surface area of the hemisphere remains the same when calculating for Earth, which we have already computed as roughly \( 14.725 \, \mathrm{m}^2 \).

Step 4: Calculate Force for Earth

The atmospheric pressure on Earth, \( P_e \), is approximately \( 101325 \, \mathrm{Pa} \). Use this to calculate the force \( F_e \) the lander must withstand on Earth:\[F_e = 101325 \times 14.725\]Compute this to find the inward force:\[F_e \approx 101325 \times 14.725 = 1491456.25 \, \mathrm{N}\]The lander must withstand approximately \( 1.49 \times 10^6 \, \mathrm{N} \) on Earth.

Key Concepts

Venus Atmospheric Pressure

Venus is known for its extremely harsh atmospheric conditions, especially in terms of pressure. The pressure on Venus is a staggering 92 times greater than the atmospheric pressure on Earth. This means that any lander or spacecraft attempting to operate on Venus must be designed to withstand this intense pressure.

Pressure is essentially a measure of force applied per unit area, and on Earth, the atmospheric pressure is about 101,325 Pascals (Pa). On Venus, this pressure can be calculated as follows:

• The pressure on Venus, denoted as \( P_v \), is \( 92 \times 101,325 \, \text{Pa} \).
• This equals approximately \( 9,321,900 \, \text{Pa} \).

Understanding Venusian pressure is crucial for designing structures that can survive the conditions found on this planet. Thus, when planning a Venus mission, these calculations ensure that the equipment can endure such high pressures without failing.

Force Calculation

Calculating the force a lander needs to withstand on Venus involves understanding the relationship between pressure, force, and area.

The formula used is:

• Force \( F \) is equal to Pressure \( P \) multiplied by Area \( A \).
• Mathematically, this is expressed as \( F = PA \).

For the Venus lander:

• The total surface area \( A \) of the lander, a hemisphere including its base, is already worked out to be approximately \( 14.725 \, \text{m}^2 \).
• The pressure on Venus \( P_v \) is \( 92 \times 101,325 \, \text{Pa} \).
• Substitute these values into the formula to find the force \( F_v \):
• \( F_v = 92 \times 101,325 \times 14.725 \approx 137,826,000 \, \text{N} \).

This substantial force is due to the significantly higher atmospheric pressure on Venus compared to Earth. The calculation ensures the lander is reinforced to handle up to around 137,826,000 Newtons of force.

Hemisphere Surface Area

Knowing how to calculate the surface area of a hemisphere is vital in determining how much atmospheric force it must endure. A hemisphere is half of a sphere, and understanding this geometry aids in solving pressure-related problems, like the lander on Venus.

For a hemisphere with a diameter of 2.5 meters:

• The radius \( r \) becomes half the diameter: \( r = 1.25 \, \text{m} \).
• The curved surface area is \( 2\pi r^2 \).
• The base, a circle, has an area of \( \pi r^2 \).
• Total surface area \( A \) of the hemisphere, therefore, is \( 3\pi r^2 \).
• Here, substituting \( r = 1.25 \), we find \( A = 3\pi (1.25)^2 \approx 14.725 \, \text{m}^2 \).

By understanding and computing the surface area accurately, one can easily determine the needed reinforcement for the lander against the incredible atmospheric pressures both on Venus and Earth.