mechanics of A uniform solid disk of radius r and mass m rolls without slipping

Question

mechanics of A uniform solid disk of radius r and mass m rolls without slipping along a flat, level, frictionless horizontal surface. It then rolls up a frictionless inclined plane where the angle of incline is theta . After the disk has rolled a distance d up (along) the ramp, it comes (momentarily) to a stop. Find the linear speed of the disk at the base of the inclined plane (before heading up) in terms of theta , d and/or g. (Interestingly, both mass and radius will have dropped out in the algebra.)

Explanation / Answer

here

GPE is converted to translational KE + rotational KE
so
m*g*d = (0.5*m*v^2) + (0.5 * I * ^2)

where v = linear speed, = angular speed = v / R

but I = 0.4*m*R^2 and ^2 = v^2 / R^2

so rot.KE = 0.5 * I * ^2 = 0.2*m*v^2

so m*g*d = (0.5+0.2)*m*v^2 = 0.7*m*v^2
m's cancel, so
g*d = 0.7*v^2

so the linear (translational) speed at the bottom is

v = sqrt(g * d / 0.7)

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