A 10 kg cylindrical barrel with a radius of 0.5 m rotates about a vertical axis of rotation without displacing with an initial angular velocity of 4 rad/s. The rotational inertia of the cylindrical barrel is 0.3mr2 and rain begins to fill the barrel with water as it rotates. If the barrel accumulates 4 kg of water, calculate the new angular velocity of the barrel.

A bicycle tire is spinning counterclockwise at 4.00 rad/s. During a time period Δt = 2.10 s, the tire is stopped and spun in the opposite (clockwise) direction, also at 4.00 rad/s. Calculate the change in the tire's angular velocity Δ? and the tire's average angular acceleration ?av. (Indicate the direction with the signs of your answers.)   (a)  the change in the tire's angular velocity Δ? (in rad/s)  _______rad/s (b) the tire's average angular acceleration ?av (in rad/s2) _________rad/s2

Three children are riding on the edge of a merry‑go‑round that has a mass of 105 kg and a radius of 1.40 m . The merry‑go‑round is spinning at 24.0 rpm. The children have masses of 22.0, 28.0, and 33.0 kg. If the 28.0 kg child moves to the center of the merry‑go‑round, what is the new angular velocity in revolutions per minute? Ignore friction, and assume that the merry‑go‑round can be treated as a solid disk and the children as point masses.

A wind turbine is rotating counterclockwise at 0.78 rev/s and slows to a stop in 21 s. Its blades are 27 m in length. (a) What is the angular acceleration of the turbine (in rad/s)? (Indicate the direction with the sign of your answer.) |rad/s2 (b) What is the centripetal acceleration (in m/s2) of the tip of the blades at t = 0? (Enter the magnitude. Round your answer to at least two decimal places.) m/s? (c) What is the magnitude (in m/s) of the total linear acceleration of the tip of the blades at t = 0? (Round your answer to at least two decimal places.) m/s2 What is the angle that this total linear acceleration vector makes with the centripetal acceleration vector? (Enter the magnitude of the smallest such angle in degrees.)