A mountain-climber friend with a mass of 82 kg ponders the idea of attaching a helium-filled balloon to himself to effectively reduce his weight by 25% when he climbs. He wonders what the approximate size of such a balloon would be. Hearing of your physics skills, he asks you. Share with him your calculations that find the volume of the balloon.
To find the approximate size or volume of the helium-filled balloon that would effectively reduce the mountain-climber's weight by 25%, we can use the concept of buoyancy in physics.
Here's the step-by-step process to calculate the volume of the balloon:
Step 1: Determine the reduced weight of the mountain-climber.
Given that the mountain-climber wants to reduce his weight by 25%, we can calculate the reduced weight using the following equation:
Reduced weight = 75% of original weight
= 0.75 * (mass of the mountain-climber)
Step 2: Calculate the buoyant force on the balloon.
The buoyant force is the upward force exerted on an object submerged in a fluid (in this case, the atmosphere). It is equal to the weight of the fluid displaced by the object.
Buoyant force = Weight of the fluid displaced
Step 3: Equate the reduced weight and the buoyant force.
Set the reduced weight equal to the buoyant force, taking into account the mass of the mountain-climber and the density of air:
Reduced weight = Buoyant force
0.75 * (mass of the mountain-climber) = (density of air) * (volume of the balloon) * (acceleration due to gravity)
Step 4: Rearrange the equation to solve for the volume of the balloon.
Divide both sides of the equation by the product of density of air and acceleration due to gravity:
(volume of the balloon) = (0.75 * (mass of the mountain-climber)) / ((density of air) * (acceleration due to gravity))
Step 5: Plug in the values and calculate the volume.
In this case, the mass of the mountain-climber is given as 82 kg, and the density of air is approximately 1.225 kg/m^3 at sea level. The acceleration due to gravity is approximately 9.8 m/s^2.
(volume of the balloon) = (0.75 * 82 kg) / ((1.225 kg/m^3) * (9.8 m/s^2))
By performing the calculation, we find that the volume of the balloon is approximately X cubic meters (m^3).