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Review Problems From Chapter 10&11 1) At t=0, a disk has an angular velocity of 360 rev/min, and constant angular acceleration of -0.50 rad/s**2. How many rotations does the disk make before coming to rest? A1 A2 A3 A4 A5 226 180 360 90 113 360 2 0 120 rad / s, 0.50 rad / s 2 , 0 60 0 0 t t find t t 12 t 2 12 t 2 find in rad / s divide by 2 to get in revolutions. B B 2A 2 A a1 A R 2 a2 B R 2 2 a2 B R B 2 4 a1 A R A 2 Let's call the 2kg body 1, the 4 kg body 2 and the pulley p; v p v 1 v 2 2 m / s , v p is the velocity of any point at the rim of the pulley. vp 2 66.7 rad / s R 0.03 1 1 K1 m1v 12 2 22 4 J 2 2 1 1 K 2 m 2v 2 2 4 22 8 J 2 2 1 1 K p I o p 2 0.0045 66.72 10 J 2 2 K K1 K 2 K p 4 8 10 22 J p 4) A uniform rod (M = 2.0 kg, L = 2.0 m) is held vertical about a pivot at point P, a distance L/4 from one end (as in the Figure). The rotational inertia of the rod about P is 1.17 kg*m**2. If it starts rotating from rest, what is the linear speed of the lowest point of the rod as it passes again through the vertical position (v)? A1 A2 A3 A4 A5 8.7 m/s 4.8 m/s 17 m/s 2.4 m/s zero L K i 0.0, U i Mg ( ) (taking our U 0.0 at the point P) 4 L Ei K i U i Mg ( ) 4 L L K f 12 I 2 , U f Mg ( ) E f K f U f 12 I 2 Mg ( ) 4 4 L L but Ei E f Mg ( ) 12 I 2 Mg ( ) 4 4 1 1 MgL v Mg L I 2 find v 3L / 4 2 2 I 5) A uniform solid sphere of radius 0.10 m rolls smoothly across a horizontal table at a speed 0.50 m/s with total kinetic energy 0.70 J. Find the mass of the sphere. A1 A2 A3 A4 A5 4.0 8.0 2.0 1.0 5.0 kg kg kg kg kg R 0.10 m, K 0.70 J 1 1 2 Mv com I 2 (1) 2 2 vcom 2 2 sub. in (1) for I MR ( sphere ) and for (rolling ) 5 R to get M . K acom R I 1 2 acom f s R MR 2 R 1 1 f s Macom 3.0 2.0 3.0 N 2 2 F fs 7) A 2.0 kg particle is moving such that its position vector (r) relative to the origin is r =(-2.0*t**2 i + 3.0 j) m. What is the torque (about the origin) acting on the particle at t=2.0 s? A1 24 k N.m A2 -36 k N.m A3 -24 k N.m A4 -48 k N.m A5 0 dr v 4t i dt dv a 4.0 i dt F ma 2.0(4.0)i 8.0 i r (2.0) 8.0 i 3.0 j ( 2. 0) r F (8.0 i 3.0 j ) 8.0 i (3.0 j ) 8.0 i 24 k 8) In the figure, m1 = 0.50 kg, m2 = 0.40 kg and the pulley has a disk shape of radius 0.05 m and mass M = 1.5 kg. What is the linear acceleration of the block of mass m2? T1 T2 A1 0.59 m/s**2 A2 0.42 m/s**2 m1 m2 a a A3 1.46 m/s**2 A4 0.21 m/s**2 A5 0.0 m2 g m1g m1 g - T1 m1a T1 m1 g - m1a (1) T1 T2 a R net I I I 1 MR 2 2 a T1 R T2 R I (3) R from (1) and (2) we get : T1 - T2 (m1 - m2 )g - (m1 m2 )a put this in (3) to get the value of a T2 - m2 g m2 a T2 m2 a m2 g (2) 9) Consider two thin rods each of length (L = 1.5 m) and mass 30 g, arranged on a frictionless table as shown in the figure. The system rotates about a vertical axis through point O with constant angular speed of 4.0 rad/s. What is the angular momentum of the system about O? A1 A2 A3 A4 A5 0.18 kg*m**2/s 0.54 kg*m**2/s 1.5 kg*m**2/s 0.27 kg*m**2/s 0.0 I 2 I rod around O 1 L 2 1 2 2 2( ML M ( ) ) 2( ML ) 12 2 3 L( around O ) I 1 I 11 I 2 I 22 4I Li 1 2 5I I tot I 1 I 2 3I Lf I tot f 3I f Li Lf 5I 3I f 5 f 3