Using the equation T=la,andl=MR2=×1.92×118×10-2=0.113kgm2 Thus the magnitude of the angular acceleration of the cylinder is 102 90.3rad/s2 0.113 The direction of he angular acceleration is counterclockwise, point out of the page 7. A disk of mass m and radius r is free to turn about a fixed. horizontal axle. The disk has an ideal string wrapped around its periphery from which another mass m (equal to the mass of the disk) is suspended, as indicated in Figure 3. The magnitude of the acceleration of the falling mass is 2g/3, the magnitude of the angular acceleration of the disk is 2g/3R Solution: Assume the acceleration of the falling mass is a, and the angular acceleration of the disk is a We have 3 a= Ra TP≈mR2a R 3R 8. A uniform beam of length / is in a vertical position with its lower end on a rough surface that prevents this end from slipping. The beam topples. At the instant before impact with the floor, the angular speed of the beam about its fixed end is Solution: Use the Cwe theorem. Assume the zero pe is at the point of the end of the beam. So E l,2+0 2 2 23m12D=mgl→0=y1 E==mgl+0 III. Give the Solutions of the following problems 1. A pulley having a rotational inertia of 1. 14x10 kg m" and a radius of 9.88cm is acted on by force, applied tangentially at its rim that varies in time as F=At+ BI where A=0.496N/s and B=0.305N/s". If the pulley was initially at rest, find its angular speed after 3. 60s SolutionUsing the equation τ α v v I total = , and 2 2 2 1.92 11.8 10 0.113 kg m 2 1 2 1 = = × × × = ⋅ − I MR Thus the magnitude of the angular acceleration of the cylinder is 2 90.3 rad/s 0.113 10.2 = = = I τ α The direction of he angular acceleration is counterclockwise, point out of the page. 7. A disk of mass m and radius R is free to turn about a fixed, horizontal axle. The disk has an ideal string wrapped around its periphery from which another mass m (equal to the mass of the disk) is suspended, as indicated in Figure 3. The magnitude of the acceleration of the falling mass is 2g/3 , the magnitude of the angular acceleration of the disk is 2g/3R . Solution: Assume the acceleration of the falling mass is a, and the angular acceleration of the disk isα . We have ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = = = ⇒ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = − = = R g R a g a mR TR mg T ma a R 3 2 3 2 2 2 α α α 8. A uniform beam of length l is in a vertical position with its lower end on a rough surface that prevents this end from slipping. The beam topples. At the instant before impact with the floor, the angular speed of the beam about its fixed end is l 3g . Solution: Use the CWE theorem. Assume the zero PE is at the point of the end of the beam. So l g I mgl ml mgl E mgl E I CM i f CM 3 2 1 3 1 2 1 2 1 2 1 0 2 1 0 2 1 2 2 2 2 2 ⇒ = ⇒ ⋅ ⋅ = ⇒ = ⎪ ⎪ ⎭ ⎪ ⎪ ⎬ ⎫ = + = + ω ω ω ω III. Give the Solutions of the Following Problems 1. A pulley having a rotational inertia of 1.14×10-3kg⋅m 2 and a radius of 9.88cm is acted on by a force, applied tangentially at its rim that varies in time as F=At+Bt 2 , where A=0.496N/s and B=0.305N/s2 . If the pulley was initially at rest, find its angular speed after 3.60s. Solution: m m R Fig.3