Question

a ball of mass M and radius R start from rest at a height of 5.0 m and rolls down a 20 degree slope as he fig below. what is the linear speed of the ball when it leaves the incline? assuming that the ball rolls without slipping B) a disk of M and radius R start from rest at a height of 5.0 m and rolls down a 20 degree slope as he fig below. what is the linear speed of the ball when it leaves the incline? assuming that the disk rolls without slipping

Answer 1

kinetic energy of rolling object K = (1/2)*m*v^2 + (1/2)*I*w^2

angular speed w = v/r

kinetic energy of rolling cylinder K = (1/2)m*v^2 + (1/2)*I*v^2/r^2

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

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

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

from energy conservation

total energy at the top = total energy at the bottom

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

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

part(A)

for ball I = (2/5)*m*r^2

speed v = sqrt(2*m*g*h/(m + (2/5)*m))

speed v = sqrt(10*g*h/7)

speed v = sqrt(10*9.8*5/7) = 8.37 m/s

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part B

for disk

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

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

speed v = sqrt(4*g*h/3)

speed v = sqrt(4*9.8*5/3) = 8.1 m/s