(7%) Problem 3: Consider two cylindrical objects of the same mass and radius. Ob

Question

(7%) Problem 3: Consider two cylindrical objects of the same mass and radius. Object A s a solid cylinder, whereas object B s a hollow cylinder 33 % Part (a) If these objects roll without slipping down a ramp, which one will reach the bottom of the ramp first? > X 33% Part (b) I low fast, in meters per second, is object A moving at the end of the ramp if it's mass is 95 g, it's radius 25 cm, and the height of the end of the ramp is 54 cm? Grade Summary Deductions 20 Potential 80% tan0 sin coS cotanasin acos( atan acotan sinh0 cosh) tanhcotanhO Submissions Attempts remaining: S (10% per attempt) detailed view 0 END Degrees O Radians 10% 10% BACKSPACE Submit lint I give up! Hints: 5% deduction per hint. Hints remaining: 2 Feedback: 0% deduction per feedback. 33% Part (c) How fast, in meters per second, is object B moving at the end of the ramp if it rolls down the same ramp?

Explanation / Answer

initial energy at the top of ramp Ei = m*g*h

final energy at the bottom of the ramp Ef = (1/2)*m*v^2 + (1/2)*I*w^2

I = moment of inertia

w = angular speed = v/r

v = linear speed

Ef = (1/2)*m*v^2 + (1/2)*I*v^2/r^2

Ef = (1/2)*v^2*(m + I/r^2)

from energy conservation

Ef = Ei

(1/2)*v^2*(m + I/r^2) = m*g*h

v = sqrt( 2*m*g*h/(m + I/r^2) )

for object A

IA = (1/2)*m*r^2

VA = sqrt( 2*m*g*h/(m + (1/2)*m ) )

VA = sqrt((4/3)*g*h)

for object B

IB = m*r^2

VB = sqrt( 2*m*g*h/(m + m ) )

VB = sqrt(g*h)

VA > VB

object A will reach the bottom of ramp first

==========================

part(b)

VA = sqrt((4/3)*9.8*0.54)

VA = 2.66 m/s

========================

part(c)

VB = sqrt(9.8*0.54)

VB = 2.3 m/s

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