A cam of mass M is in the shape of a circular disk of diameter 2 R with an off-c

Question

A cam of mass M is in the shape of a circular disk of diameter 2R with an off-center circular hole of diameter R is mounted on a uniform cylindrical shaft whose diameter matches that of the hole (see figure below).

(a) What is the rotational inertia of the cam and shaft around the axis of the shaft? (Use the following as necessary: Mcam, Mshaft, and R.)

I = ___

(b) What is the rotational kinetic energy of the cam and shaft if the system rotates with angular speed around this axis? (Use the following as necessary: Mcam, Mshaft, R, and .)

Krot = ___

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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