(III) On a level billiards table a cue ball, initially at rest at point O on the table, is struck so that it leaves the cue stick with a center-of-mass speed v 0 and a “reverse” spin of angular speed ω 0 (sec Fig. 11–40). A kinetic friction force acts on the ball as it initially skids across the table. ( a ) Explain why the ball’s angular momentum is conserved about point O. ( b ) Using conservation of angular momentum, find the critical angular speed ω C , such that, if ω 0 = ω C , kinetic friction will bring the ball to a complete (as opposed to momentary) stop. ( c ) If ω 0 is 10% smaller than ω C , i.e., ω 0 = 0.90 ω C , determine the ball’s CM velocity v CM when it starts to roll without slipping. ( d ) If ω 0 is 10% larger than ω C , i.e., ω 0 = 1.10 ω C , determine the ball’s CM velocity v CM when it starts to roll without slipping. [ Hint : The ball possesses two types of angular momentum, the first due to the linear speed v CM of its CM relative to point O, the second due to the spin at angular velocity ω about its own CM. The ball’s total L about O is the sum of these two angular momenta.] FIGURE 11–40 Problem 52.
(III) On a level billiards table a cue ball, initially at rest at point O on the table, is struck so that it leaves the cue stick with a center-of-mass speed v 0 and a “reverse” spin of angular speed ω 0 (sec Fig. 11–40). A kinetic friction force acts on the ball as it initially skids across the table. ( a ) Explain why the ball’s angular momentum is conserved about point O. ( b ) Using conservation of angular momentum, find the critical angular speed ω C , such that, if ω 0 = ω C , kinetic friction will bring the ball to a complete (as opposed to momentary) stop. ( c ) If ω 0 is 10% smaller than ω C , i.e., ω 0 = 0.90 ω C , determine the ball’s CM velocity v CM when it starts to roll without slipping. ( d ) If ω 0 is 10% larger than ω C , i.e., ω 0 = 1.10 ω C , determine the ball’s CM velocity v CM when it starts to roll without slipping. [ Hint : The ball possesses two types of angular momentum, the first due to the linear speed v CM of its CM relative to point O, the second due to the spin at angular velocity ω about its own CM. The ball’s total L about O is the sum of these two angular momenta.] FIGURE 11–40 Problem 52.
(III) On a level billiards table a cue ball, initially at rest at point O on the table, is struck so that it leaves the cue stick with a center-of-mass speed v0 and a “reverse” spin of angular speed ω0 (sec Fig. 11–40). A kinetic friction force acts on the ball as it initially skids across the table. (a) Explain why the ball’s angular momentum is conserved about point O. (b) Using conservation of angular momentum, find the critical angular speed ωC, such that, if ω0 = ωC, kinetic friction will bring the ball to a complete (as opposed to momentary) stop. (c) If ω0 is 10% smaller than ωC, i.e., ω0 = 0.90 ωC, determine the ball’s CM velocity vCM when it starts to roll without slipping. (d) If ω0 is 10% larger than ωC, i.e., ω0 = 1.10 ωC, determine the ball’s CM velocity vCM when it starts to roll without slipping. [Hint: The ball possesses two types of angular momentum, the first due to the linear speed vCM of its CM relative to point O, the second due to the spin at angular velocityω about its own CM. The ball’s total L about O is the sum of these two angular momenta.]
FIGURE 11–40
Problem 52.
Definition Definition Product of the moment of inertia and angular velocity of the rotating body: (L) = Iω Angular momentum is a vector quantity, and it has both magnitude and direction. The magnitude of angular momentum is represented by the length of the vector, and the direction is the same as the direction of angular velocity.
(II) Figure 11–35 shows two masses connected by a cord
passing over a pulley of radius R, and moment of inertia I.
Mass MA slides on a frictionless surface, and Mg hangs freely.
Determine a formula for (a) the angular momentum of
the system about the pulley axis, as a function of the speed v
of mass MA or Mg,
and (b) the accelera-
MA
tion of the masses.
MB
FIGURE 11-35
Problem 41.
(II) Three particles of masses m1 = 2.4 kg, m2= 2.6 kg and m3= 3.9kg are connected by rods of length d = 58 cm. The system rotates at an angular velocity ω = 2.5 rad/s around the left end of the system. What is the angular momentum of the system (a) if the rods have negligible mass; (b) if each rod has a mass M = 3.6kg?
(II) A uniform disk turns at 3.3 rev/s around a frictionless
central axis. A nonrotating rod, of the same mass as the disk
and length equal to the disk's
diameter, is dropped onto the
freely spinning disk, Fig. 8–56.
They then turn together around
the axis with their centers
superposed. What is the angular
frequency in rev/s of the
combination?
FIGURE 8-56
Problem 72.
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