Question

10. A balloon is filled with helium (density =0.178 kg/m 3) to a volume that has a radius of 1.50 m and is then connected to a string as shown in the figure. If the mass of the skin of the balloon is 2.00 kg, determine the tension in the string when the balloon is at equilibrium. The density of air is 1.28 kg/m 3

Answer 1

The tension in the string when the helium balloon is at equilibrium can be found by calculating the buoyant force and the weight of the balloon. The buoyant force is the product of the density of air, the volume of the balloon, and gravity. Tension equals the buoyant force minus the weight, which is the sum of the helium mass and the balloon skin's mass.

Step-by-step explanation:

To determine the tension in the string when the balloon is at equilibrium, we must consider the forces acting on the balloon. The balloon experiences an upward buoyant force due to the displaced air, and it also experiences a downward force due to gravity, which is the sum of the mass of the helium gas and the mass of the balloon's skin.

To calculate the buoyant force, we use the equation:

• Buoyant Force = (Density of air) × (Volume of the balloon) × (Acceleration due to gravity)

To find the volume of the balloon, we use the formula for the volume of a sphere:

• Volume = 4/3 π × radius³

The downward force due to gravity is the weight of the balloon, which is the sum of the weight of the helium and the skin of the balloon:

• Weight = (Mass of helium + Mass of balloon skin) × (Acceleration due to gravity)

To calculate the mass of helium, we use:

• Mass of helium = Density of helium × Volume of helium

Once we have the buoyant force and the gravity force, the tension in the string at equilibrium is simply the difference between these two forces:

• Tension = Buoyant Force - Weight

Make sure to keep all units consistent throughout the calculations (e.g., use meters for volume calculations and kilograms for mass).

Answer 2

The tension in the string when the balloon is at equilibrium is approximately 133.737 Newtons.

To determine the tension in the string when the balloon is at equilibrium, first, we need to understand the forces acting on the balloon. There are two main forces at play here:

1. The buoyant force (upward force) exerted by the air due to the displacement of the air by the helium inside the balloon.
2. The gravitational force (downward force) acting on the mass of the balloon's skin.

At equilibrium, these two forces will equate to the tension in the string.

Let's calculate these forces step by step:

Calculate the volume of the balloon.
The volume V of a sphere can be calculated using the formula:

`\[ V = (4)/(3) \pi r^3 \]`
where r is the radius of the balloon.

Given:

`\[ r = 1.50 \text{ m} \]`


`\[ V = (4)/(3) \pi (1.50)^3 \]`

`\[ V = (4)/(3) \pi (3.375) \]`

`\[ V = 4.5 \pi \]`

`\[ V \approx 4.5 * 3.14159 \]`

`\[ V \approx 14.137 \text{ m}^3 \]`

Calculate the buoyant force exerted on the balloon by the air.
The buoyant force
`\( F_b \)` can be calculated using the Archimedes' principle:

`\[ F_b = (\text{density of air} - \text{density of helium}) * V * g \]`
where g is the acceleration due to gravity, approximately
`\( 9.81 \text{ m/s}^2 \)`.

Given:

`\[ \text{density of air} = 1.28 \text{ kg/m}^3 \]`

`\[ \text{density of helium} = 0.178 \text{ kg/m}^3 \]`


`\[ F_b = (1.28 - 0.178) * V * 9.81 \]`

`\[ F_b = (1.102) * 14.137 * 9.81 \]`

`\[ F_b \approx 1.102 * 14.137 * 9.81 \]`

`\[ F_b \approx 153.357 \text{ N} \]`

Calculate the weight of the balloon's skin.
The weight W of the balloon's skin can be calculated by:

`\[ W = \text{mass of the balloon's skin} * g \]`

Given:

`\[ \text{mass of the balloon's skin} = 2.00 \text{ kg} \]`


`\[ W = 2.00 * 9.81 \]`

`\[ W = 19.62 \text{ N} \]`

Calculate the tension in the string.
At equilibrium, the tension in the string T is the difference between the buoyant force and the weight of the balloon's skin, since the buoyant force must support both the weight of the helium and the skin.

`\[ T = F_b - W \]`


`\[ T = 153.357 \text{ N} - 19.62 \text{ N} \]`

`\[ T \approx 153.357 - 19.62 \]`

`\[ T \approx 133.737 \text{ N} \]`