A spherical uniform shell rotates in a vertical axis without friction.One string without mass passing through sphere equator and a sheave has a small object at its end.what is the velocity of object,when he down a distance H.

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A spherical uniform shell rotates in a vertical axis without friction.One string without mass passing through sphere equator and a sheave has a small object at its end.what is the velocity of object,when he down a distance H.

Physics
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Use work energy theorem
rotational kinetic energy of spherical shell + rotational K.E. of pulley + K.E. of object = change in P.E. of object

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since string is not stretchable: velocity of object = angular speed of shell* Radius of shell = angular speed of pulley * radius of pulley. now put the known values and get the answer.
\[1/2mv^2+1/2I_{sphere}\omega_{sphere}^2+1/2I_{pulley}\omega_{pulley}^2=mgH\]
@CarlosGP ,but we dont have w,and v yet
w= v/r
you will get a equation in one variable and will be able to find solution also
@carlos i get it i will try
\[I_{sphere}=2/5MR^2\] then, for the sphere \[1/2(2/5MR^2)w_{sphere}^2=1/5Mv^2\] and for the pulley: \[1/2Iw_{pulley}^2=1/2I(v/r)^2\] with the data given in the drawing: \[mgH=v^2[m/2+M/5+I/(2r^2)]\]
Isphere=2/3MR²
@CarlosGP thanks
@RaphaelFilgueiras you are welcome. Isphere is 2/5MR^2
@CarlosGP it's just a shell
You are right: I=2/3MR^2 for the sphere

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