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 fig 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 w about the shaft's axis? (Use any variable or symbol stated above an necessary.) K = _____

Explanation / Answer

Mass of cam with hole = M

Mass of solid in hole = M/3

Mass of cam without the hole = 4M/3

Moment of inertia of cam without the hole about its center = (1/2) * (4M/3) * R^2

MOI of cam without the hole about the axis of rotation = (1/2) * (4M/3) * R^2 + (4M/3) * (R/2)^2

lo = MR^2

MOI of hole with mass about the axis of rotation

l' = (1/2) * (M/3) * (R/2)^2

l' = (1/24)*MR^2

MOI of cylindrical shaft

l'' = (1/2)*M(R/2)^2

l'' = (1/8)*MR^2

Total MOI of system,

l = lo - l' - l''

l = MR^2 (1 - 1/24 - 1/8)

l = (13/12)*MR^2

Kinetic energy of cam-shaft combination,

KE = (1/2)*l*w^2

KE = (1/2) *  (13/12)*MR^2 * w^2

KE = (13/24)*MR^2*w^2

Answer

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