Problem 36 - College Physics

Question

(a) Calculate the buoyant force of air (density 1.20 \(\mathrm{kg} / \mathrm{m}^{3} )\) on a spherical party balloon that has a radius of 15.0 \(\mathrm{cm}\) . (b) If the rubber of the balloon itself has a mass of 2.00 \(\mathrm{g}\) and the balloon is filled with helium (density 0.166 \(\mathrm{kg} / \mathrm{m}^{3}\) ), calculate the net upward force (the "lift") that acts on it in air.

Step-by-Step Solution

Answer

(a) The buoyant force is 0.166 N. (b) The net upward force is approximately 0.1234 N.

Step 1: Convert Radius to Meters

The radius of the balloon is given as 15.0 cm. First, convert this to meters by dividing by 100: \[ 15.0 \text{ cm} = 0.15 \text{ m} \]

Step 2: Calculate Volume of the Balloon

With the radius converted to meters, calculate the volume of the balloon using the formula for the volume of a sphere: \[ V = \frac{4}{3} \pi r^3 \] Substituting the radius, \[ V = \frac{4}{3} \pi (0.15)^3 \approx 0.0141 \text{ m}^3 \]

Step 3: Calculate Buoyant Force

The buoyant force can be calculated using the formula: \[ F_{b} = \rho_{\text{air}} \cdot V \cdot g \] where \( \rho_{\text{air}} = 1.20 \text{ kg/m}^3 \), \( V = 0.0141 \text{ m}^3 \) and \( g = 9.81 \text{ m/s}^2 \). Substituting the values,\[ F_{b} = 1.20 \times 0.0141 \times 9.81 \approx 0.166 \text{ N} \]

Step 4: Calculate Mass of Helium

Determine the mass of helium in the balloon using the formula: \[ m = \rho_{\text{He}} \cdot V \] where \( \rho_{\text{He}} = 0.166 \text{ kg/m}^3 \). So,\[ m = 0.166 \times 0.0141 \approx 0.00234 \text{ kg} \]

Step 5: Total Mass of Balloon

Add the mass of the rubber to the mass of the helium to find the total mass:\[ m_{\text{total}} = m_{\text{He}} + m_{\text{rubber}} \]Converting the mass of the rubber from grams to kilograms: \[ m_{\text{rubber}} = 2.00 \text{ g} = 0.00200 \text{ kg} \]Thus,\[ m_{\text{total}} = 0.00234 + 0.00200 = 0.00434 \text{ kg} \]

Step 6: Calculate Weight of the Balloon

The weight of the balloon is found using:\[ W = m_{\text{total}} \cdot g \]Substituting the total mass and gravitational acceleration,\[ W = 0.00434 \times 9.81 \approx 0.0426 \text{ N} \]

Step 7: Calculate Net Upward Force

Net force ("lift") is the difference between the buoyant force and the weight of the balloon. \[ \text{Net force} = F_{b} - W \]Substitute the values:\[ \text{Net force} = 0.166 - 0.0426 \approx 0.1234 \text{ N} \]

Key Concepts

Density

Density is a measure of how much mass is packed into a given volume. It's a crucial concept in understanding buoyant forces and the behavior of different substances in fluids. Mathematically, density (\( \rho \)) is expressed as mass divided by volume, or \( \rho = \frac{m}{V} \).
For example, in our balloon problem, the densities of air and helium are given as 1.20 kg/m³ and 0.166 kg/m³, respectively. This means that air is denser than helium. This difference in density is key to the balloon's ability to float when filled with helium.

• Each substance has its own characteristic density.
• Floating or sinking depends on comparing the density of the object to the fluid surrounding it.
• Lower density substances float on higher density fluids.

A helium-filled balloon rises because helium's density is less than that of the surrounding air. Understanding density helps us predict and calculate how different substances will behave when immersed in a fluid.

Net Upward Force

The net upward force, often referred to as "lift," is the force that determines whether an object will rise, float, or sink in a fluid. It's the difference between the buoyant force exerted by the fluid and the object's weight.
In our balloon scenario, the net upward force is calculated as follows:

• The buoyant force is the force with which the air pushes the balloon upwards.
• The weight of the balloon includes both the mass of the helium inside and the balloon's material.
• Subtract the weight from the buoyant force to get the net upward force.

In formula form, this can be seen as: \[ \text{Net force} = F_{b} - W \] Where \( F_{b} \) is the buoyant force and \( W \) is weight. If the net force is positive, the balloon will rise; if negative, it will sink; if zero, it will float suspended. In our example, the net upward force came out to approximately 0.1234 N, allowing the balloon to rise in the air.

Volume of a Sphere

The volume of an object tells us how much space it occupies. For spherical objects like balloons, calculating the volume involves a specific formula due to their round 3D shape: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere.
In our problem, the radius of the balloon is 0.15 meters. Plugging this radius into the formula, we found the volume to be approximately 0.0141 m³.

• The volume helps determine how much fluid is displaced when the sphere is submerged, crucial for calculating buoyant forces.
• Larger volumes displace more fluid, leading to larger buoyant forces.

The volume is a fundamental part of finding the buoyant force using Archimedes' principle, which states that the buoyant force is equal to the weight of the displaced fluid. This is why knowing the volume is essential in buoyancy calculations for balloons.