(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.
The thin, homogeneous, circular disk B rotates about a fixed, frictionless axis O, located at its centre, with an angular velocity of ω = 2.5 rad/s (clockwise). Object Alands on the disk at the indicated position where θ = 53o, with an absolute velocity of 1.8 m/s at an angle of ψ = 25o as shown. Determine the angular speed of disk Bimmediately after object A has landed and rotates with B. Take the mass of the object as mA = 5 kg and the radius of gyration about an axis passing perpendicularly through its centre of gravity as kG = 0.18 m. The mass of disk B is mB = 14 kg and its radius is R = 1.7 m. Initially object A is a distance of r = 1.1 m from the axis of rotation.
Choose the correct answer:
a) 2.500 rad/s
b) 1.737 rad/s
c) 2.088 rad/s
d) 2.252 rad/s
e) 1.642 rad/s
Does an object moving in a straight line have nonzero angular momentum always, sometimes, or never?
A sphere of radius 0.2 m and a weight of 100 N is released with no initial velocity on the incline making an angle of 30° with the horizontal ang rolls without skidding. Which of the following most nearly gives the velocity of the center of the sphere after the sphere has rolled 4 m?
Chapter 11 Solutions
Physics for Scientists and Engineers, Vol 1 (Chapters 1-20)
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