Determine the translational speed of the cylinder

[k-rod AP 2010]

Homework Statement

A solid cylinder w/ mass m, radius r, and rotational inertia 1/25mr² rolls without slipping down an inclined plane. THe cylinder starts form rest at height h. The inclined plane makes an angle θ w/ the horizontal.Determine the translational speed of the cylinder when it reaches the bottom of the inclined plane.

Homework Equations

v=velocity
ω=angular speed
I=moment of inertia
U=gravitational potential energy
K=Kinetic Energy

The Attempt at a Solution

U=K_{translational} + K_rotational

mgh=(1/2mv²) + (1/2Iω²)

mgh=(1/2mv²) + (1/2 (1/2mr²)(v²/r²))

gh=(v²/2) + (v²/4)

gh=(2v²/4) + (v²/4)

gh= (4/3)v²

v²=(4/3)gh

v= √[(4/3)gh] would this be the correct procedure to solve this?

 

[tiny-tim]

hi k-rod AP 2010!

A solid cylinder w/ mass m, radius r, and rotational inertia 1/25mr² rolls without slipping down an inclined plane. THe cylinder starts form rest at height h. The inclined plane makes an angle θ w/ the horizontal.Determine the translational speed of the cylinder when it reaches the bottom of the inclined plane.
…
v²=(4/3)gh

v= √[(4/3)gh] would this be the correct procedure to solve this?

(i assume you meant rotational inertia 1/2 mr² ? )

yes, that's the right method and result!

(btw, another way is to say the "rolling mass" is I/r², = m/2, so the total "effective mass" is 3m/2, so the "effective gravity" is 2g/3, so 1/2 mv² = 2mgh/3 … but i don't think the examiners would like that! )