Problem 34 - College Physics

Question

A large balloon of mass \(226 \mathrm{~kg}\) is filled with helium gas until its volume is \(325 \mathrm{~m}^{3}\). Assume the density of air is \(1.29 \mathrm{~kg} / \mathrm{m}^{3}\) and the density of helium is \(0.179 \mathrm{~kg} / \mathrm{m}^{3}\). (a) Draw a force diagram for the balloon. (b) Calculate the buoyant force acting on the balloon. (c) Find the net force on the balloon and determine whether the balloon will rise or fall after it is released. (d) What maximum additional mass can the balloon support in equilibrium? (e) What happens to the balloon if the mass of the load is less than the value calculated in part (d)? (f) What limits the height to which the balloon can rise?

Step-by-Step Solution

Answer

The force diagram of the balloon includes its weight going downwards and the buoyant force upwards. The buoyant force acting on the balloon is calculated using the volume, density of air, and gravity. The net force is the difference between the buoyant force and the weight of the balloon. When this force is positive, the balloon will rise; otherwise, it will fall. The maximum additional mass the balloon can support can be calculated by equating the buoyant force with the total weight of the balloon and the added mass. If a load weighs less than the maximum, the balloon will rise. The height to which the balloon can rise is limited by the decrease in air density with altitude.

Step 1: Draw the Force Diagram

The only two forces acting on the balloon are gravity and the buoyant force. The weight of the balloon, acting downward, can be represented by the mass of the balloon multiplied by gravity. The buoyant force, acting upwards, is the weight of the air displaced by the balloon.

Step 2: Calculate the Buoyant Force

The buoyant force can be calculated by multiplying the volume of the balloon by the density of air and gravity, i.e., \(F_{buoyant} = V_{balloon} \times density_{air} \times g\). Substitute the given values into this equation to calculate the buoyant force.

Step 3: Find the Net Force

The net force can be calculated by subtracting the weight of the balloon from the buoyant force. This will tell if the balloon will rise or fall.

Step 4: Calculate the Maximum Additional Mass

When the balloon is in equilibrium, the buoyant force equals the weight of the balloon plus the weight of the additional mass. Therefore, the maximum additional mass is \( (F_{buoyant} - weight_{balloon}) / g\)).

Step 5: Predicting Balloon Behaviour

If the load mass is less than the maximum mass calculated in step 4, the balloon will rise. It will continue to rise until it reaches the height where the density of the air equals the average density of the balloon.

Step 6: Discussing the Limits

The height to which the balloon can rise is limited by the decreasing air density with height. As the balloon rises, the air becomes less dense and provides less buoyancy. The balloon will stop rising when its average density equals the air density.

Key Concepts

Force Diagram

When analyzing how forces act on an object such as a balloon, it's crucial to visualize a force diagram. This diagram helps illustrate all the forces at play.
For our helium-filled balloon, two key forces are involved:

• Weight (gravity): This force pulls the balloon downward and is determined by multiplying the mass of the balloon by gravitational acceleration, symbolized as \( F_{gravity} = m_{balloon} \times g \).
• Buoyant Force: An upward force exerted by the displaced air, countering gravity, based on the density and volume of displaced fluid.

By sketching these forces, students can visualize how they work in opposition, providing a clearer understanding of buoyancy dynamics.

Buoyant Force Calculation

To understand why objects float or sink, the concept of buoyant force is essential. It quantifies how much lighter an object "feels" in a fluid compared to air.
The buoyant force (\( F_{buoyant} \) is calculated using the principle of displaced fluid, which states:

• \( F_{buoyant} = V_{balloon} \times \rho_{air} \times g \)
• Where:
  • \( V_{balloon} \) = Volume of the displaced fluid (here, air)
  • \( \rho_{air} \) = Density of the air, typically \(1.29 \, \text{kg/m}^3 \)
  • \( g \) = Acceleration due to gravity, approx. \(9.81 \, \text{m/s}^2 \)

Inserting the values for our balloon into this formula reveals the force needed to counteract the balloon's weight.

Net Force

Net force determines if the balloon will ascend or descend. It is the cumulative force acting on an object after accounting for all individual forces.
Here's how to find it:

• Calculate the weight of the balloon: \( F_{gravity} = m_{balloon} \times g \)
• The net force is: \( F_{net} = F_{buoyant} - F_{gravity} \)

• If \( F_{net} > 0 \) (buoyant force exceeds weight), the balloon rises.
• If \( F_{net} < 0 \) (weight exceeds buoyant force), the balloon descends.

Understanding this concept helps students predict the motion of the balloon.

Equilibrium

In physics, an object is in equilibrium when all forces acting on it are balanced, resulting in no net force.This means the object remains stationary or moves at a constant velocity.
In the context of our balloon, equilibrium is achieved when:

• The upward buoyant force equals the combined downward forces of the balloon’s weight and any additional mass.
• Maximum additional mass can be calculated from:\(\text{Max. Mass} = \frac{F_{buoyant} - F_{gravity}}{g} \)

Reaching equilibrium means the balloon stays at the same height without rising or falling, useful in determining lifting capacity.

Air Density

Air density plays a significant role in determining buoyancy. It's essentially the mass of air per unit of volume and influences how much lift a balloon can achieve.
Factors affecting air density include:

• Altitude: Higher altitudes mean lower air density, affecting buoyant force negatively.
• Temperature and humidity: Warmer temperatures and higher humidity levels can decrease air density.

When a balloon ascends, the air density reduces, meaning its buoyant force decreases until it equals the weight of the balloon and its load. This variation limits the maximum height, as eventually, the balloon will reach an altitude where it can no longer rise.