Q.348. Rotational kinetic energy of cam-shaft combination.

Question 348.
Many machines employ cams for various purposes, such as opening and closing valves. In the figure, 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 a 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 w about the axis of the shaft?

Answer 348.
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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