A boy is spinning on a chair and holding a paper bag so that it is horizontal, and perpendicular to the axis of the rotation. What should be the angular velocity of the boy in radians/second so that the bottom of the bag breaks?
Details and assumptions
Assume paper breaks at an overpressure of 1.2 atm.
The molecular mass of the air is 29 g/mol.
The boy's arm is L=70 cm long.
When horizontal, the bag has a shape of a cylinder of length 30 cm.
Outside pressure is 1 atm and temperature 25∘C.
Assume that the boy won't get sick no matter how fast he is spinning. Hint: he whizzes around.
To determine the angular velocity at which the bottom of the bag breaks, we need to consider the centrifugal force experienced by the air inside the bag.
The centrifugal force acting on the air inside the bag creates extra pressure that can potentially cause the bag to break. This extra pressure can be calculated using the formula:
ΔP = ρ * ω² * R
Where:
ΔP is the extra pressure inside the bag in Pascals (Pa)
ρ is the density of air in the bag (in kg/m³)
ω is the angular velocity of the boy spinning (in radians/second)
R is the distance from the axis of rotation to the bottom of the bag (in meters)
First, we need to calculate the density of air inside the bag. To do this, we can use the ideal gas law equation:
PV = nRT
Where:
P is the pressure outside the bag (in Pascals)
V is the volume of the bag (in cubic meters)
n is the number of moles of air inside the bag
R is the ideal gas constant (8.314 J/(mol*K))
T is the temperature outside the bag (in Kelvin)
Since the bag is shaped like a cylinder, we can find its volume using the formula:
V = π * r² * h
Where:
r is the radius of the bag (in meters)
h is the length of the bag (in meters)
Given that the bag has a length of 30 cm, we convert it to meters:
h = 0.3 m
To find the radius of the bag, we assume it has a circular cross-section and divide its diameter by 2:
d = 2r
d = 2 * 0.3 m
r = 0.15 m
Now, we can calculate the volume of the bag:
V = π * (0.15 m)² * 0.3 m
V = 0.02121 m³
Given that the pressure outside the bag is 1 atm, we convert it to Pascals:
P = 1 atm * 1.01325 * 10^5 Pa/atm
P = 1.01325 * 10^5 Pa
The temperature outside the bag is given as 25°C, so we convert it to Kelvin:
T = 25°C + 273.15
T = 298.15 K
Now we can substitute these values into the ideal gas law to find the number of moles of air inside the bag:
n = (P * V) / (R * T)
n = (1.01325 * 10^5 Pa) * (0.02121 m³) / (8.314 J/(mol*K) * 298.15 K)
n ≈ 0.0093 mol
Next, we calculate the density of air inside the bag:
ρ = (mass of air) / V
ρ = (n * molar mass of air) / V
The molar mass of air is given as 29 g/mol, so we convert it to kg/mol:
molar mass of air = 29 g/mol * (1 kg / 1000 g)
molar mass of air = 0.029 kg/mol
Substituting this value, we can find the density of air inside the bag:
ρ = (0.0093 mol) * (0.029 kg/mol) / (0.02121 m³)
ρ ≈ 0.0124 kg/m³
Now, we can calculate the extra pressure inside the bag using the given formula:
ΔP = ρ * ω² * R
Given that the overpressure that causes the bag to break is 1.2 atm, we convert it to Pascals:
ΔP = 1.2 atm * 1.01325 * 10^5 Pa/atm
ΔP = 1.2159 * 10^5 Pa
With all the values determined, we can rearrange the formula to solve for the angular velocity:
ω² = ΔP / (ρ * R)
Substituting the given values:
ω² = (1.2159 * 10^5 Pa) / (0.0124 kg/m³ * 0.7 m)
ω² ≈ 1717741935.48 rad²/s²
Taking the square root of both sides, we find:
ω ≈ √(1717741935.48 rad²/s²)
ω ≈ 41463.8696 rad/s
Therefore, the angular velocity of the boy spinning on the chair should be approximately 41463.87 radians/second for the bottom of the bag to break.