Question

A sphere of radius R can rotate about a vertical axis on frictionless bearings (see figure below). Let the rotational inertia of the sphere be Isphere. A massless cord passes around the equator of the sphere, over a pulley with rotational inertia I pulley and radius r, and is attached to a small object of mass m. There is no friction on the pulley's axle and the cord does not slip on the pulley. At t = 0, the mass m has speed Vy. The system is subject to the regular force of gravity. (a) Using conservation of energy, derive an expression for the speed of mass m after it has fallen for a distance h. Your expression should depend on Isphere and pulley (b) For the 3 cases below, evaluate the speed of the falling object in m/s with the following parameters: Vo = 0, m = 3.0 kg, h = 1.0 m. Mass of "sphere": Msphere = 5.5 kg, radius of "sphere" R = 0.20 m Mass of pulley: Mulley = 1.5 kg. Case 1: a. Solid sphere of uniform density. b. Pulley is a solid disk of uniform density with radius r = 0.10 m. Case 2: ucharo" ir thin chorical chall

Answer 1

AnA.) radius of sphere R radius of Pulley = e at t=0 speed of mass mev. = Ei= - (a) initial energy of the system 14 mv ² t / Isphere west 2 I pulley Wro soit ngh um Who angular speed of sphere Whe angular speed of pulley now since the cold does not slip hence WR Vo/R and Who = voſal ² Ei mitt. Isphere + 1 pulley 2 vo + migh Ei taha 1

final energy Ef Es ( m+ & Isphere it I pulley & J V Vf final speed of mass m . Contenvation of energy gives Ei Et m+ t Isphere Beth 12 B + Ipulley 2 2 + righ = [2 m + / Ispaene to the pulley & Vo & mgh /24m+. Isphere t + Inglesey (67 Bkg Msphere 5-5kg R = 0.2m Iskg 2 ta VA R² + ma him -pulley =

case 1 2 72 Sphere is solid = Ispires = = ? MB 5 Pulley is a solid disk of radius i = Int 2 hence Uf = amgh m + 2 misphen R² + + pulley 2² 2 mgh m+ 2 msphere + / mpulky. -2x389.8x1 3+ 2/575.5+ 2 x 1.5 R² 22 2. V2. Vf 3:143621 mln

2 case Sphere is thin spherical shell = Espinne 2/3 Mspree R² Pulley is a solid disk of radius i - I pulley = 1/2 Mpulley i V¢ /2 2 mgh m 2 Msphere t Mrulley 72 2X2X 9.8 X 3 + 3x5.5+ 1/2X15) Vf 2.81569 mla cases! sphere is solid = Isphere somente Pulley is an annular cylinder with R, = Onlm ,. R2=0.09m

V = mt 2 li Ipulley - Ź Mouttert Rit) this time we becomes. [ 2mgh 1/2 23 Msphere + / Moulley (Rita) Ri 2x3 x4.8 X1 V2 3 + 2/5x5.5+ bax!:51 (0:12+ (00902) Vf - 2.99 45 m/s c) in both in both the cales the moment of inestia of the system (is there & pulley) is greater than that in case 1, hence cases 2 and I have final speed dower than case 1. (6 9² 2 and 3