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
To determine the angular velocity at which the bottom of the bag will break, we need to consider the forces acting on the bag.
When the boy spins on the chair, his body creates a centrifugal force that pushes outward. This force acts on the bag, exerting tension in the bottom part, trying to tear it apart. The gravitational force also acts on the bag, exerting a downward force.
To find the angular velocity at which the bottom of the bag breaks, we need to equate the centrifugal force and the gravitational force:
Centrifugal Force = Gravitational Force
The centrifugal force can be calculated using the equation: Centrifugal Force = Mass * Radius * Angular Velocity^2
The gravitational force can be calculated using the equation: Gravitational Force = Mass * Gravity
Since we are looking for the angular velocity (ω) at which the bottom of the bag breaks, we rearrange the equation:
(Mass * Radius * Angular Velocity^2) = (Mass * Gravity)
The mass of the bag cancels out from both sides of the equation:
Radius * Angular Velocity^2 = Gravity
Now, solve for the angular velocity (ω):
Angular Velocity^2 = Gravity / Radius
Taking the square root of both sides:
Angular Velocity = √(Gravity / Radius)
Therefore, the angular velocity required for the bottom of the bag to break can be calculated by taking the square root of the ratio of the gravitational acceleration (9.8 m/s^2) to the radius of the bag from the axis of rotation.