A L shaped rod whose one rod is horizontal and other vertical is rotating about a vertical axis as shown with angular speed o. The sleeve shown in figure has mass 2m and friction coefficient between rod and sleeve is u. The minimum angular speed o for which sleeve cannot slip on rod is ?.......
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Explanation:
At first draw the free body diagram of the device as shown. The forces, acting on the sleeve are its weight, acting vertically downward, spring force, along the length of the spring and normal reaction by the rod, perpendicular to its length.
Let F be the spring force, and Δl be the elongation.
From F
n
=mw
n
Nsinθ+Fcosθ=mω
2
r (1)
where rcosθ=(l
0
+Δl).
Similary from F
t
=mw
t
Ncosθ−Fsinθ=0 or, N=
cosθ
Fsinθ
(2)
From (1) and (2)
F(
cosθ
sinθ
)⋅sinθ+Fcosθ=mω
2
r
=mω
2
cosθ
(l
0
+Δl)
On putting F=κΔl,
κΔlsin
2
θ+κΔlcos
2
θ=mω
2
(l
0
+Δl)
on solving, we get,
Δl=mω
2
κ−mω
2
l
0
=
(
m
κ
ω
2
−1)
l
0
and it is independent of the direction of rotation.
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