Question

Ballooning on Mars. It has been proposed that we could explore Mars using inflated balloons to hover just above the surface. The buoyancy of the atmosphere would keep the balloon aloft. The density of the Martian atmosphere is 0.0154 kg/m\(^3\) (although this varies with temperature). Suppose we construct these balloons of a thin but tough plastic having a density such that each square meter has a mass of 5.00 g. We inflate them with a very light gas whose mass we can ignore. (a) What should be the radius and mass of these balloons so they just hover above the surface of Mars? (b) If we released one of the balloons from part (a) on earth, where the atmospheric density is 1.20 kg/m\(^3\), what would be its initial acceleration assuming it was the same size as on Mars? Would it go up or down? (c) If on Mars these balloons have five times the radius found in part (a), how heavy an instrument package could they carry?

Short answer

(a) Calculate radius and mass for balance on Mars. (b) Balloon rises initially on Earth. (c) Larger balloons can carry heavier loads.

Step-by-step solution

Calculate Volume of Balloon

To solve these questions, we need to first understand the buoyancy concept. For the balloon to hover, the buoyant force must equal the gravitational force on the plastic. The buoyant force is the weight of the air displaced, which can be calculated using the volume of the sphere derived from the balloon. For part (a), the formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius.

Buoyant Force Equals Gravitational Force

Assuming the density of Martian air as \( \rho_m = 0.0154 \text{ kg/m}^3 \), the buoyant force \( F_b \) is given by the expression \( F_b = \rho_m V g \). For the balloon to hover, \( F_b \) must equal the gravitational force on the balloon, \( F_g = m g \), where \( m \) is the mass of the balloon. Convert the mass per area of the balloon's skin into SI units: \( 5.00 \text{ g/m}^2 = 0.005 \text{ kg/m}^2 \). The surface area of the balloon is \( A = 4 \pi r^2 \). Thus, the gravitational force is \( F_g = 0.005 \times A \times g \).

Solve for Radius and Mass on Mars

Set \( \rho_m V g = 0.005 \times A \times g \) and solve for \( r \). The volume \( V = \frac{4}{3} \pi r^3 \) and the area \( A = 4 \pi r^2 \). By canceling \( g \) and \( \pi \), we derive \( \rho_m \frac{4}{3} r = 0.005 \times 4 \). Solving for certain terms: \( r = \left( \frac{0.005 \times 4}{4/3 \times \rho_m} \right)^{1/2} \). Simplifying gives us the radius \( r \), then substitute back to find the mass of the balloon.

Initial Acceleration on Earth

To determine the initial acceleration on Earth, calculate the buoyant force \( F_b = \rho_e V g \) where \( \rho_e \) is Earth's atmospheric density \( 1.20 \text{ kg/m}^3 \). Use Newton's second law: \( a = \frac{F_{ ext{net}}}{m} \). Since the buoyant force will be greater than the weight, \( F_{ ext{net}} = F_b - F_g = V(\rho_e - \rho_m)g \), the balloon will rise, giving positive acceleration.

Carrying Capacity on Mars

For five times the radius found in part (a), we use the formula for volume \( V' = \frac{4}{3} \pi (5r)^3 \) and calculate the new net buoyant force as \( F_b' = \rho_m V' g \). This force now must overcome the weight of the balloon's skin plus the new payload, \( F_g' = (m' + m_{ ext{package}}) g \). Solving \( F_b' = F_g' \) will give \( m_{ ext{package}} \), indicating how heavy an instrument it can carry.

Key concepts

Mars Exploration

Exploring Mars presents unique challenges due to its thin atmosphere and lower gravity compared to Earth. Mars' atmosphere primarily consists of carbon dioxide and has a density of just 0.0154 kg/m\(^3\). This low density means that for objects like balloons to float, they need to be very light and large in volume to displace enough Martian air and create sufficient buoyancy. By utilizing balloons to hover, scientists can gather data on Martian weather, terrain, and surface properties from above.

The idea of using balloons for Martian exploration is driven by the need to cover vast areas quickly and efficiently without ground-level obstacles. Balloons provide the versatility needed in a hostile environment where powered flight may be impractical or too expensive. Through Mars exploration, these techniques could revolutionize how we study other planets, offering a balance between cost and data-gathering effectiveness.

Balloon Dynamics

The fundamental principle behind balloons' dynamics is buoyancy, which allows them to float or hover in an atmosphere. On both Mars and Earth, this principle operates under the same basic physics laws — but the environments are different, which impacts how these dynamics play out.

For a balloon to hover:

• The buoyant force must equal the gravitational force acting on the balloon.
• The buoyant force is the weight of the displaced air by the balloon's volume, while the gravitational force is the weight of the balloon's material.

In the case of Mars, the balloon must be large enough to displace a volume of air that equals the weight of the balloon due to the low air density. Similarly, if released on Earth, the same balloon needs to adjust for the higher atmospheric density, meaning it would experience a different buoyant force, leading it to rise faster initially.

Density and Buoyancy Concepts

Density, a crucial concept in buoyancy, refers to the mass per unit volume of a substance. The lower the density of the surrounding atmosphere, the more challenging it becomes for a balloon to remain afloat. On Mars, with its sparse atmosphere, achieving buoyancy requires a significant volume for even a lightweight balloon.

Buoyancy occurs due to differences in density. When the density of the air inside the balloon is less than the air it displaces, a buoyant force is generated, enabling it to float. In practical terms:

• If the densities of the gas in the balloon and the surrounding air are equivalent, the balloon won't rise or fall.
• An imbalance, with the balloon less dense than the surrounding air, causes it to rise due to buoyancy.

These principles are foundational in designing balloons for Martian exploration, where achieving the necessary buoyancy with minimal weight is key.

Gravitational Force Calculations

Gravitational force, often referred to as weight, is the force by which a planet's gravity pulls objects towards its center. This force can be calculated using the formula \( F_g = mg \), where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity. On Mars, \( g \) is about 3.71 m/s\(^2\), significantly less than Earth's \( 9.81 m/s^2 \).

When designing balloons for hovering on Mars:

• The gravitational force must be precisely balanced with the buoyant force to achieve stable hovering.
• By understanding \( g \), it becomes easier to calculate how much air needs to be displaced to counteract the balloon's weight.

This understanding is crucial since any changes in the balloon's mass or atmospheric conditions directly affect its ability to float. Calculations must be carefully performed to ensure that these balloons can not only hover but also carry scientific instruments as per the mission requirements.