.................... A cylinder of uniform mass M and radius r is released from

Question

....................

A cylinder of uniform mass M and radius r is released from point A of height h on a track as shown in the figure. The end of the track is a circular path of radius R. The cylinder rolls without slipping. Determine the radial and tangential acceleration of the cylinder at points B, C (90 degree from B on the circular arc) and D as shown. Express your answers in terms of given variables, M, h, r, R and/or g. Assume h is high enough that cylinder will make it to point D without falling off.

Explanation / Answer

at point B

mgh=1/2 mv^2 +1/2 Iw^2

since it is rolling

v=wr

I= 1/2 mr^2

==> mgh=1/2 mv^2 +1/2 * 1/2 mv^2

==> mgh=3/4 mv^2

==> v=sqrt(4gh/3)

tangential acceleration=0

radial acceleration=v^2/(R-r)=4gh/3*(R-r)

at point C

mg(h-R)=1/2 mv^2 +1/2 Iw^2

==> mg(h-R)=1/2 mv^2 +1/2 * 1/2 mv^2

==> mg(h-R)=3/4 mv^2

==> v=sqrt(4g(h-R)/3)

tangential acceleration=g

radial acceleration=v^2/(R-r)=4g(h-R)/3*(R-r)

at point D

mg(h-2R)=1/2 mv^2 +1/2 Iw^2

==> mg(h-2R)=1/2 mv^2 +1/2 * 1/2 mv^2

==> mg(h-2R)=3/4 mv^2

==> v=sqrt(4g(h-2R)/3)

tangential acceleration=0

radial acceleration=v^2/(R-r)=4g(h-2R)/3*(R-r)

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