A helium balloon (mass including fill gas 2.1 g) is tied to a long string (mass per unit length is 3.1*10^2 g/cm, i.e., one cm of the string has a mass of 3.1*10-2 g). It rises to a height x of 210 cm, with the remainder of the string laying on the floor. If the density of air is 1.19*10-3 g/cm^3, what is the volume of the balloon?
To find the volume of the balloon, we need to use the given information and apply some basic physics concepts.
The total mass of the helium balloon, including the fill gas, is 2.1 g. The mass per unit length of the string is given as 3.1 * 10^2 g/cm (or 3.1 * 10^-2 g per 1 cm).
First, let's calculate the mass of the string. Since the string extends from the height of the balloon to the floor, its length can be determined by subtracting the height of the balloon (x = 210 cm) from the total length of the string.
The mass of the string can be calculated by multiplying the length by the mass per unit length:
mass of string = length × mass per unit length
mass of string = (total length of string - x) × (3.1 * 10^-2 g/cm)
Next, we need to determine the volume of the string. The volume of an object with a uniform density can be calculated by dividing its mass by the density. In this case, the density of the string is not explicitly given, but we can calculate it using the mass of the string and the length of the string.
density of string = mass of string / length of string
Using the calculated density of the string, we can then determine the volume of the string:
volume of string = mass of string / density of string
Now let's move on to finding the volume of the balloon. The buoyant force acting on the balloon causes it to rise, and this force is equal to the weight of the air displaced by the balloon.
The weight of the air displaced by the balloon can be calculated by multiplying the density of air by the volume of the balloon:
weight of air = density of air * volume of balloon
Since the balloon is in equilibrium and not accelerating vertically, the weight of the air displaced by the balloon is equal to the combined weight of the helium balloon and the string (mass of balloon + mass of string), which is 2.1 g + mass of string.
Now we can calculate the volume of the balloon by rearranging the equation:
volume of balloon = (weight of air) / (density of air)
Let's substitute the calculated values into the equation and compute the result.