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Homework Problems

 

HW#33   Ch 11: Conservation of Angular Momentum

Instructions: Be sure to write formulas and show substitutions to earn full credit.

1.    Armed with an in-depth understanding of rolling objects, Tayden heads down to the bowling lane to try his luck. He bowls a ball with no rotational speed and initial linear speed of 8 m/s down the center of the lane. If the bowling ball has a radius of 10.85 cm and a mass of 7 kg, and the lane has a length of 18 m and a coefficient of kinetic friction of μk = 0.12, a) find how far his ball travels before it starts to roll. b) Find the fraction of kinetic energy that it has lost due to heat.

2.    Ethan wants to play a backspin on a cue ball (\(I = \dfrac{2}{5}m{R^2}\)) by hitting the ball a distance R/2 below the center of the cue ball, as shown in the figure below. Assume that the cue stick has mass M and initial speed vcue and that the ball has mass m and is initially at rest. After the collision, the cue stick has speed v′cue and the ball has speed v0. The collision is entirely elastic. a) Using linear momentum and angular momentum equations show that the angular speed of the ball after the collision is equal to \(\omega = \dfrac{{5{v_0}}}{{4R}}\). b) Find the time that it takes for the ball to start rolling without slipping on the table if the coefficient of kinetic friction is μk. c) Find the speed of the ball when it is rolling without slipping.

Cue ball and stick

3.    The position vector of a chicken with a mass of 2 kg is given by \(\vec r = \left( {1 + 2t} \right)\hat i + \left( {2 + t - 5{t^2}} \right)\hat j\) , where \(\vec r \) is in meters. Calculate the angular momentum and torque about the point (1m, 1m) in unit vector notation.

4.    Noah has mass M and runs with speed v towards a merry-go-round of mass 5M. The merry-go round has a radius R, moment of inertia \(\dfrac{5}{2}M{R^2}\), and is initially stationary. Noah approaches the merry-go-round in a direction tangent to the edge and jumps on it, causing it to start spinning on its frictionless bearings. a) What is the angular speed of the merry-go-round after Noah jumps on? b) He then walks radially inward from the edge of the merry-go round to a point R/4 from the center. What is his new rotational speed? c) What is the source of energy for the increase in kinetic energy of the merry-go-round? (Saying that Noah is the source of energy will earn no credit, despite the fact that he is full of energy.)