Question

10. A balloon is filled with helium (density =0.178 kg/m3) 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 4

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Answer 1

To find the tension in the string of a helium balloon at equilibrium, calculate the volume of the balloon, the buoyancy force using the volume and density of air, the weight of the balloon, and then find the difference between the buoyancy force and the balloon's weight.

Step-by-step explanation:

Calculating the Tension in the String of a Helium Balloon

The problem involves finding the tension in the string to which a helium-filled balloon is attached when the balloon is in equilibrium. To solve this, we need to apply principles of buoyancy and the forces acting on the balloon in air. The buoyancy force can be calculated using Archimedes' principle, which states that the buoyant force is equal to the weight of the displaced fluid. In this case, the fluid is air.

First, calculate the volume of the helium inside the balloon using the volume formula for a sphere (V = ¾πr³). Then, calculate the weight of the air displaced by this volume, which is the buoyancy force (F_b = volume * density of air * g), and the weight of the balloon itself (W_balloon = mass of skin * g). At equilibrium, the tension in the string (T) must balance the difference between the buoyancy force and the weight of the balloon.

Steps to calculate the tension:

1. Calculate the volume of the sphere: V = ¾π(1.50 m)³.
2. Calculate the buoyancy force: F_b = V * density of air * g.
3. Calculate the weight of the balloon: W_balloon = mass of skin * g.
4. Calculate the tension in the string: T = F_b - W_balloon.

Remember that the tension must provide the necessary force to keep the balloon floating steadily, offsetting the gravitational pull on the balloon's mass.

Answer 2

To find the tension in the string when the balloon is at equilibrium, we need to consider the forces acting on the balloon and use the equations for buoyant force and weight. Substituting the given values into these equations will allow us to calculate the tension.

Step-by-step explanation:

To determine the tension in the string when the balloon is at equilibrium, we need to consider the forces acting on the balloon. The buoyant force exerted by the helium is equal to the weight of the displaced air. The weight of the balloon itself and the tension in the string will balance this buoyant force.

The buoyant force can be calculated using the formula:

Buoyant force = density of air * volume of displaced air * gravitational acceleration

The weight of the balloon and the tension in the string can be calculated using the formula:

Weight of balloon + Tension in string = weight of displaced air

Substituting the given values into these equations will allow us to solve for the tension in the string.