Question 8.14: THE FALLING BUCKET GOAL Combine Newton's second law with its...
THE FALLING BUCKET
GOAL Combine Newton’s second law with its rotational analog.
PROBLEM A solid, uniform, frictionless cylindrical reel of mass M=3.00 \mathrm{~kg} and radius R=0.400 \mathrm{~m} is used to draw water from a well (Fig. 8.28a). A bucket of mass m=2.00 \mathrm{~kg} is attached to a cord that is wrapped around the cylinder. (a) Find the tension T in the cord and acceleration a of the bucket. (b) If the bucket starts from rest at the top of the well and falls for 3.00 \mathrm{~s} before hitting the water, how far does it fall?
STRATEGY This problem involves three equations and three unknowns. The three equations are Newton’s second law applied to the bucket, m a=\Sigma F_{i}; the rotational version of the second law applied to the cylinder, I \alpha=\sum \tau_{i}; and the relationship between linear and angular acceleration, a=r \alpha, which connects the dynamics of the bucket and cylinder. The three unknowns are the acceleration a of the bucket, the angular acceleration a of the cylinder, and the tension T in the rope. Assemble the terms of the three equations and solve for the three unknowns by substitution. Part (b) is a review of kinematics.
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