Many machines employ cams for various purposes, such as opening and closing valv

Question

Many machines employ cams for various purposes, such as opening and closing valves. In the figure below, the cam is a circular disk of radius R with a hole of diameter R cut through it. As shown in the figure, the hole does not pass through the center of the disk. The cam with the hole cut out has mass M. The cam is mounted on a uniform, solid, cylindrical shaft of diameter R and also of mass M. What is the kinetic energy of the cam—shaft combination when it is rotating with angular speed about the shaft's axis? (Use any variable or symbol stated above as necessary.)

Explanation / Answer

Cam with hole has mass M
=> Mass of solid in hole = M/3
=> Mass of cam without the hole = 4M/3

M.I. of cam without the hole about its center
= (1/2) (4M/3) R^2
=> M.I. of cam without the hole about the axis of rotation
= (1/2) (4M/3) R^2 + (4M/3)(R/2) ^2
= (4/3) MR^2 * (1/2 + 1/4)
= MR^2

M.I. of hole with mass (M/3) about the axis of rotation
= (1/2) (M/3) (R/2)^2
= (1/24)MR^2

M.I. of cylindrical shaft
= (1/2)M(R/2)^2
= (1/8)MR^2

=> Total M.I. of the system
= MR^2 (1 - 1/24 + 1/8)
= (13/12) MR^2

=> K..E.. = (1/2) Iw^2
= (1/2) * (13/12) MR^2w^2
= (13/24)MR^2w^2.

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