Question

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A ball with a radius r and a mass m starts from rest and then rolls without slipping a distance L down a roof sloped at an angle θ. The ball comes off the roof a height H above the ground. Derive an expression for the ball’s speed v, just as it leaves the roof, in terms of g, L, and θ:

A. If the ball is a solid sphere.

B.If the ball is a hollow sphere.

C. For each ball, find the horizontal distance that the ball will travel from the point directly below the edge of the roof. Let L = 11.0 feet, H = 9.50 feet, and θ = 30.0◦.

D.If a solid cylinder rolled from the same spot on the roof, would it land farther than the farthest ball? Closer than the closest ball? Or between the two balls

Answer 1

Acceleration of - gsino continuous body t sphere, er Cylinder) Wher K = Rading of gegration - It KR2 (A.) - . For solid sphere MK² = 2 / MR² a = gsino » K²= 2 R2 - glin Fsing = sg sino + YRE V² Vo²+QaL - v² = 02+2 ( sg sings x2 = v= J log Sino Ans (B.) for Hollow sphere - MK2 = 2 / 3 MR2 = K² = 2 R2 = 39 sino v=J Egising 7 (C) velocity of elorety of sole solid sphene ball at pant (B) lowest V. Poston. Jiogesino - J 10x98 X0:30s xlixSinzó of roof Lill feet v=4.85 mis =118.305

too v 30 9 30 Vx 2 v cosso = 4.85x csso = 4.16 mis Vy = rsinzoo = 4858 Sinzo = 2.4 m/s Sy=H=9.5*0.305= 2.897 mes § 2.90 + Sy = Vyt + hh ayt? 2.9 = 2.4xt + x (9.8) Xt2 2.9=2.44 +4.972 4.9+²+2.4t-2-g=0 t- -2-4 I √ (2-4) 2-4X4 9X.(+29) 2x4.9 -2-4+ 7.9 .. gig , time can't be negative so, t= -2-4 +7.9 9-8 -2.75 see x = Vaxt = 4.16x2.75 NOTE = 11.44m. Repeat this process for Hollow sphere By taking va v Ggesino = 16x9 8 x 11 x 0 305 x Singo D For Solid Cylinder, I MR2 = MK² » K²=.R2 O for Sold cue = 3.75ma ac gsino. ltkor - gsino 1/2 i 29sino uglsino 3. 7 = 4.68 m/s Solid & Hollow spherical ball, .. als velocity between the Et lean distance blw turo balls