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 of the boy in radians/second required for the bottom of the bag to break, we can use the concept of centrifugal force. The centrifugal force acting on the bottom of the bag must exceed the force holding the bag together in order for it to break.
Let's break down the problem step by step:
Step 1: Calculate the force required to break the bottom of the bag.
The force required to break the bottom of the bag is equivalent to the pressure difference between the outside and inside of the bag multiplied by the area of the bag's bottom surface. Since the bag's shape is cylindrical, the area of the bottom surface can be calculated as the product of its radius and length.
Step 2: Compute the pressure difference.
The pressure difference is calculated as the difference between the external pressure and the pressure inside the bag. Since the molecular mass of air is given, we can use the ideal gas equation to determine the pressure inside the bag.
Step 3: Calculate the rotational velocity required.
The centrifugal force acting on the bottom of the bag is given by the formula F = m * ω² * r, where F is the force, m is the mass of the air inside the bag, ω is the angular velocity, and r is the distance from the axis of rotation to the bottom of the bag.
Step 4: Set up an equation for the force required to break the bag.
Equate the force required to the centrifugal force acting on the bottom of the bag. This will give us an equation to solve for the angular velocity.
Step 5: Solve the equation for the angular velocity.
Solve the equation derived in step 4 for the angular velocity ω.
Let's apply these steps to calculate the angular velocity:
Step 1: Calculate the force required to break the bottom of the bag.
The area of the bottom surface of the bag can be calculated using the formula for the area of a circle: A = π * r², where A is the area and r is the radius. The radius can be calculated as half of the bag's length.
Given:
Length of the bag (L) = 30 cm = 0.3 m
Radius of the bag (r) = L/2 = 0.15 m
Area of the bottom surface (A) = π * 0.15²
Step 2: Compute the pressure difference.
The pressure difference is given by the external pressure minus the internal pressure. The internal pressure can be calculated using the ideal gas equation:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
Given:
External pressure (P_ext) = 1 atm (convert to Pa)
Internal pressure (P_int) = P_ext + 1.2 atm (convert to Pa)
Temperature (T) = 25°C = 298 K
Molecular mass of air (M) = 29 g/mol (convert to kg/mol)
Step 3: Calculate the rotational velocity required.
The force required to break the bottom of the bag is equal to the centrifugal force acting on it. This force can be calculated using the formula F = m * ω² * r, where m is the mass of air inside the bag, ω is the angular velocity, and r is the distance from the axis of rotation to the bottom of the bag.
The mass of air inside the bag can be calculated using the ideal gas equation:
m = (P_int * V) / (R * T)
Given the length (L) and radius (r) of the bag, the distance from the axis of rotation to the bottom of the bag (r) is equal to L/2.
Step 4: Set up an equation for the force required to break the bag.
Equating the force required to the centrifugal force gives us the equation:
F = m * ω² * r
Step 5: Solve the equation for the angular velocity.
Solve the derived equation for the angular velocity ω.
By following these steps, you should be able to calculate the required angular velocity to break the bottom of the bag.