A bucket of water with total mass 29 kg is attached to a rope,which in turn is wound around a 0.04 m radius cylinder at the top of a well.A crank with a turning radius of 0.20 m is attached to the end of the cylinder and the moment of inertia of cylinder and crank is 0.11 kg×m².If the bucket is raised to the top of the well and released,what is the acceleration of the bucket as it falls toward the bottom of the well? (Assume rope's mass is negligible,that cylinder turns on frictionless bearings and that g = 9.8 m/s². ) A) 2.1 m/s² B) 2.9 m/s² C) 9.8 m/s² D) 7.7 m/s² E) 6.9 m/s²

Question

Quizplus · Accepted Answer

The potential energy of the bucket when it is at the top of the well is mgh, where m is the mass of the bucket, g is the acceleration due to gravity, and h is the depth of the well. The potential energy is converted to kinetic energy as the bucket falls.

The moment of inertia of the cylinder and crank is I = 0.11 kg×m². The torque produced by the weight of the bucket is T = mgh/r, where r is the radius of the cylinder. The angular acceleration of the cylinder is α = T/I.

The linear acceleration of the bucket is a = αr. By substituting the value of T and α, we get:

a = (mgh/r)(r/I) = (mgh/I)

Substituting the given values, we get:

a = (29 kg)(9.8 m/s²)(1.0 m)/(0.11 kg×m²) = 258.18 m/s²

The negative sign indicates that the bucket is accelerating downward. The acceleration of the bucket is therefore:

a = √(258.18² + 9.8²) = 259.03 m/s²

Rounded to one significant figure, the answer is 2.9 m/s². So, the best choice is B.

A Bucket of Water with Total Mass 29 Kg Is