Question

You see there that depending on the angle of “pull", e, three different motions can occur - the spool either rolls toward or away from the person pulling the cord or, at a critical angle of pull, does not rotate. The purpose of this exercise is to analyze this behavior using fundamental principles of mechanics. The physical system is shown in the figure and consists of two concentric spools of radius R, and R,, respectively. Cord is wrapped around the inner cylinder and is pulled upward at an angle (relative to the horizontal) by a force F (the cord is assumed to be massless so that the tension force has magnitude F. IN [1] Define in words the three external forces shown in the figure. fs : Fg: N: [2] Denote the horizontal and angular accelerations of the system by a and a , respectively, and the total mass of the two spools, and their moment of inertia about their center axis, by m and Ior. Write Newton's 2nd Laws for translational and rotational motions about 0: ma = Tua [3] a > 0 corresponds to a counter-clockwise rotation, and a > 0 denotes translational motion toward the right. Write a relation between a and a when the spool rolls without slipping: a= [4] Eliminate fs from the Newton's 2nd Law relations in [2], above, and find an expression for the angular acceleration, a, of the spools in terms of m, lac, R., R., and F: a= [5] Define 0. = cos'(R/R,) and use (3,4) above to find the directions of a and a when e > 0.: e<0.: 0=0.:

Answer 1

Fsing 1 > 10 a floso fs = friction force applied by ground on spool fga gravitational force acted Earth on spol ling N = Normal contact force fs by ig 2 - ma Flose - fs = fs x R2 - FXR1 = Scand (3) a = XR2 (4) 2 Floso - fs=m& R2 fs R2 - FR I can of fs Ilm = m Rzfs- mF R, R2. => I cut coso fs I am F Cosot mF RiR2 MR2 I am + Ri Cos 0o = R2/R, d = F Cose -fs R21 m Rz Floso - (1 cmf loso & M F R1 R2 = L 2 MR2 & I am MR2

d = mFRy? long + Plantoso f I can coso MFR, R2 m R2 [m R₂ + Ich) FR, Coso - FRI m R2 2 + 1 ch. at Cosoo R2 RI L =o. Net ł about contact point=o. f will pass from contact point so, ifs x R₂ = FXR Ifs FRI R₂ O > oo f will produce torque. Acw about contact point (0) x 0=0 will pass from o OLOo I will produce terque clockwise about o. 2=0