1. The angular position of a point on a rotating wheel is given by θ = 3.37 + 8.91t2 + 2.64t3, where θ is in radians and t is in seconds. At t = 0, what are (a) the point’s angular position and (b) its angular velocity? (c) What is its angular velocity at t = 8.48 s? (d) Calculate its angular acceleration at t = 1.17 s. (e) Is its angular acceleration constant?
2. The angular speed of an automobile engine is increased at a constant rate from 1480 rev/min to 3940 rev/min in 13.9 s. (a) What is its angular acceleration in revolutions per minute-squared? (b) How many revolutions does the engine make during this 13.9 s interval?
3. At t = 0, a flywheel has an angular velocity of 5.4 rad/s, an angular acceleration of -0.33 rad/s2, and a reference line at θ0 = 0. (a) Through what maximum angle θmax will the reference line turn in the positive direction? What are the (b) first and (c) second times the reference line will be at θ = θmax/8? At what (d) negative time and (e) positive time will the reference line be at θ = -6.8 rad?
4. The flywheel of a steam engine runs with a constant angular velocity of 230 rev/min. When steam is shut off, the friction of the bearings and of the air stops the wheel in 1.8 h. (a)What is the constant angular acceleration, in revolutions per minute-squared, of the wheel during the slowdown? (b) How many revolutions does the wheel make before stopping? (c)At the instant the flywheel is turning at 115 rev/min, what is the tangential component of the linear acceleration of a flywheel particle that is 30 cm from the axis of rotation? (d)What is the magnitude of the net linear acceleration of the particle in (c)?
5. In the figure, two particles, each with mass m = 0.83 kg, are fastened to each other, and to a rotation axis at O, by two thin rods, each with length d = 5.9 cm and mass M = 1.1 kg. The combination rotates around the rotation axis with angular speed ω = 0.30 rad/s. Measured about O, what is the combination’s (a) rotational inertia and (b) kinetic energy?
6. In the figure, block 1 has mass m1 = 450 g, block 2 has mass m2 = 560 g, and the pulley is on a frictionless horizontal axle and has radius R = 5.2 cm. When released from rest, block 2 falls 73 cm in 4.6 s without the cord slipping on the pulley. (a) What is the magnitude of the acceleration of the blocks? What are (b) tension T2 (the tension force on the block 2) and (c) tension T1 (the tension force on the block 1)? (d) What is the magnitude of the pulley’s angular acceleration? (e) What is its rotational inertia? Caution: Try to avoid rounding off answers along the way to the solution. Use g = 9.81 m/s2.
7. A 25.0 kg wheel, essentially a thin hoop with radius 1.30 m, is rotating at 166 rev/min. It must be brought to a stop in 16.0 s. (a) How much work must be done to stop it? (b) What is the required average power? Give absolute values for both parts.
8. A uniform spherical shell of mass M = 19.0 kg and radius R = 0.560 m can rotate about a vertical axis on frictionless bearings (see the figure). A massless cord passes around the equator of the shell, over a pulley of rotational inertia I = 0.220 kg•m2 and radius r = 0.0680 m, and is attached to a small object of mass m = 2.50 kg. There is no friction on the pulley’s axle; the cord does not slip on the pulley. What is the speed of the object when it has fallen a distance 1.06 m after being released from rest? Use energy considerations.
9. The thin uniform rod in the figure has length 5.0 m and can pivot about a horizontal, frictionless pin through one end. It is released from rest at angle θ = 30° above the horizontal. Use the principle of conservation of energy to determine the angular speed of the rod as it passes through the horizontal position. Assume free-fall acceleration to be equal to 9.83 m/s2.
10. At 7:14 A.M. on June 30, 1908, a huge explosion occurred above remote central Siberia, at latitude 61° N and longitude 102° E; the fireball thus created was the brightest flash seen by anyone before nuclear weapons. The Tunguska Event, which according to one chance witness “covered an enormous part of the sky,” was probably the explosion of a stony asteroidabout 140 m wide. (a) Considering only Earth’s rotation, determine how much later the asteroid would have had to arrive to put the explosion above Helsinki at longitude 25° E. This would have obliterated the city. (b) If the asteroid had, instead, been a metallic asteroid, it could have reached Earth’s surface. How much later would such an asteroid have had to arrive to put the impact in the Atlantic Ocean at longitude 20° W? (The resulting tsunamis would have wiped out coastal civilization on both sides of the Atlantic.)
