Many machines employ cams for various purposes, such as opening and closing valv
ID: 1287197 • Letter: M
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 camshaft combination when it is rotating with angular speed about the shaft's axis? (Use any variable or symbol stated above as necessary.) K =
Explanation / Answer
By using the parallel axis theorem, moment of inertia of the circular disc is
I =(1/2)Mor2+Mor2
=(1/2)MoR2+Mo(R/2)2
=(3/4)MoR2
Is =(1/2)Ms(R/2)2
=(1/8)MsR2
Mo =R2tQ
Ms=r2tQ=(R/2)2tQ
=(1/4)R2tQ
=1/4Mo
M=M0-Ms =Mo -(1/4)Mo
=(3/4)Mo
Mo=(4/3)M
Ms=-(1/3)M
Icam=I+Is
=(3/4)MoR2 +(1/8)MsR2
=(3/4)(4M/3)R2+(1/8)(-M/3)R2
=(23/24)MR2
kinetic energy
KE =(1/2)Icam2
=(23/48)MR22
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