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.
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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 required for the bottom of the bag to break, we need to consider the centrifugal force acting on the air inside the bag.
Here are the steps to find the answer:
1. Find the mass of the air inside the bag:
- Given that the molecular mass of air is 29 g/mol, we can calculate the mass of air using the ideal gas law.
- The ideal gas law is represented as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature.
- In this case, the pressure is 1.2 atm, and the temperature is 25°C, which we need to convert to Kelvin (25 + 273 = 298 K).
- The volume can be calculated as the product of the bag's length (30 cm or 0.3 m) and the length of the boy's arm (70 cm or 0.7 m).
- Rearranging the ideal gas law equation to solve for the number of moles (n), we get n = PV / RT.
- Plugging in the values, we get n = (1.2 atm) * (0.3 m * 0.7 m) / ((0.082 L * atm / mol * K) * 298 K).
- Calculate the mass by multiplying the number of moles with the molecular mass of air: mass = n * molecular mass of air.
2. Calculate the centrifugal force acting on the air inside the bag:
- Centrifugal force is given by F = m * ω^2 * r, where m is the mass, ω is angular velocity, and r is the radial distance from the center of rotation.
- In this case, the radial distance (r) is half of the length of the bag (0.15 m), and the mass (m) is calculated in the previous step.
- Rearranging the equation to solve for ω, we get ω = sqrt(F / (m * r)).
3. Convert the angular velocity to radians per second:
- Angular velocity is measured in radians per second. If you are given the angular velocity in revolutions per minute (rpm), you need to convert it.
- Since 1 revolution is equal to 2π radians, we can convert ω from rpm to radians per second by multiplying it by 2π / 60.
4. Calculate the required angular velocity:
- Plug in the calculated values into the above equations to find the angular velocity (ω) in radians per second.
Following these steps will provide the answer to the question. Let me know if you have any further questions!