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A uniform disc of mass m and radius r is rotated about an axis passing through its centre and perpendicular to its plane with an angular velocity w. It is placed on a rough horizontal plane with the axis of the disc keeping vertical. Co-efficient of friction between the disc and the surface is u. Find: (1)the time when the disc stops rotating (2) the angle rotated by the disc before stopping
A uniform disc of mass m and radius r is rotated about an axis passing through its centre and perpendicular to its plane with an angular velocity w. It is placed on a rough horizontal plane with the axis of the disc keeping vertical. Co-efficient of friction between the disc and the surface is u. Find:
(1)the time when the disc stops rotating
(2) the angle rotated by the disc before stopping
consider a elemental ring of radius r... a small retarding torque is acting on this element say dT due to friction... dT = df(r) (df is friction force acting on ring) =u(dm)gr ..................1 (dm is the mass of ring ) mass per unit area = M/piR2 (M is mass of disk) mass of elemental disk (dm)= (2pir dr)(M/piR2) (dr is the thickness of ring) dm =2Mrdr/R2 ................2 now putting dm in eq 1 we get dT = 2uMgr2dr/R2 integrating both sides T = 2uMgR/3 .....................3 T = I(alfa) alfa = 4ug/3R ..........5 (alfa = angular accleration) w = wo - (alfa)t finally disk stops so w=0 t = Wo/(alfa) = 3wR/4ug @ = Wot - (alfa)t2 /2 put t in above eq u will get the angle rotated before coming to rest.... approve my ans if u like
consider a elemental ring of radius r...
a small retarding torque is acting on this element say dT due to friction...
dT = df(r) (df is friction force acting on ring)
=u(dm)gr ..................1 (dm is the mass of ring )
mass per unit area = M/piR2 (M is mass of disk)
mass of elemental disk (dm)= (2pir dr)(M/piR2) (dr is the thickness of ring)
dm =2Mrdr/R2 ................2
now putting dm in eq 1 we get
dT = 2uMgr2dr/R2
integrating both sides
T = 2uMgR/3 .....................3
T = I(alfa)
alfa = 4ug/3R ..........5 (alfa = angular accleration)
w = wo - (alfa)t
finally disk stops so w=0
t = Wo/(alfa) = 3wR/4ug
@ = Wot - (alfa)t2 /2
put t in above eq u will get the angle rotated before coming to rest....
approve my ans if u like
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