Question

Two students are each given a spherical balloon that has a long string attached and is filled with an unknown gas whose density is referred to by the variable rhoG. The filled balloons each have a radius referred to by the variable R and a volume referred to by the variable V, and the balloons rise when released. A small valve is fitted to each balloon to allow the balloon to be deflated in a controlled manner. Part (a) The string and the balloon itself (not including the gas) have a combined known mass of MB. Derive an expression for the acceleration of the balloon when it is released in terms of the given quantities and physical constants, as appropriate. Air resistance is negligible, and the known density of air is rhoA. Part (b) The first student is given a stopwatch and a tape measure and asked to use only this equipment in an experiment to determine the density of the gas inside the balloon from a graph of the data. Describe an experimental procedure the student can use to accomplish this

Answer 1

The acceleration of the balloon when released can be calculated using the buoyant force and the weight of the balloon and string. The expression for the acceleration is a = (ρG * V * g) / (M + ρA * V).

Step-by-step explanation:

To derive an expression for the acceleration of the balloon when released, we need to consider the forces acting on it. The buoyant force, which is equal to the weight of the displaced air, is the force that causes the balloon to rise. The net vertical force on the balloon is the difference between the buoyant force and the weight of the balloon and string. Using Newton's second law of motion, we can write the expression for the acceleration as:

a = (ρG * V * g) / (M + ρA * V)

Where ρG is the density of the gas, V is the volume of the balloon, g is the acceleration due to gravity, M is the combined mass of the balloon and string, and ρA is the density of air.

Answer 2

Part (a) To derive an expression for the acceleration of the balloon when it is released, we can use Newton's second law of motion, which states that the net force on an object is equal to the product of its mass and acceleration (F = ma). In this case, the net force acting on the balloon is the difference between the buoyant force and the weight of the balloon.

The buoyant force can be calculated using Archimedes' principle, which states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. Since the balloon is filled with an unknown gas, the buoyant force can be expressed as FB = rhoG * V * g, where rhoG is the density of the gas, V is the volume of the balloon, and g is the acceleration due to gravity.

The weight of the balloon can be calculated as WB = (rhoG - rhoA) * V * g, where rhoA is the density of air.

Therefore, the net force acting on the balloon is given by F = FB - WB = (rhoG * V - (rhoG - rhoA) * V) * g = (rhoA * V) * g.

Applying Newton's second law, we have F = ma, where a is the acceleration of the balloon.

Thus, the expression for the acceleration of the balloon when it is released is a = (rhoA * V) * g / mB, where mB is the combined mass of the string and the balloon.

Part (b) To determine the density of the gas inside the balloon from a graph of the data, the student can follow this experimental procedure:

1. Inflate the balloon with the unknown gas and measure its radius, R, using the tape measure.

2. Release the balloon and start the stopwatch simultaneously.

3. Observe and record the time it takes for the balloon to rise a certain distance, such as one meter.

4. Repeat steps 1-3 multiple times to collect a set of data points.

5. Plot a graph of the time taken (y-axis) versus the square root of the radius (x-axis), sqrt(R).

6. The slope of the graph represents the square root of the acceleration of the balloon, sqrt(a).

7. Calculate the square root of the density of the gas, sqrt(rhoG), using the expression sqrt(a) = (rhoA * V) * g / mB derived in part (a).

8. Square the calculated value of sqrt(rhoG) to obtain the density of the gas, rhoG.

9. Record the density of the gas as the result of the experiment.

By following this experimental procedure and analyzing the data from the graph, the student can determine the density of the gas inside the balloon.