A block is placed inside a horizontal hollow cylinder. The cylinder is rotating with constant angular speed one revolution per second about its axis. The angular position of the block at which it begins to slide is 30° below the horizontal level passing through the center. Find the radius of the cylinder if the coefficient of friction is 0.6. What should be the minimum constant angular speed of the cylinder so that the block reaches the highest point of the cylinder?
Text Solution
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(0.24m, 8.9rad/sec)
Sol.
N – mg sin 30º = m ω 2 R .....(1)
mg cos 30º = μ N .....(2)
ω = 2 π rad/s
⇒ 
= μ 
⇒ 
R = 
R = 
= 0.24 m Ans.
For minimum angular velocity, normal should be zero at heighest point
m ω 2 R = mg
ω = 
= 
= 6.4 rad/second
Also, condition for which block will not slip on cylinder is
N – mg cos θ = m ω 2 R
N = mg cos θ + m ω 2 R
f r max = μ N = μ (mg cos θ + m ω 2 R)
For the block does not slip over cylinder,
mg sin θ ≤ f rmax
mg sin θ ≤ μ mg cos θ + μ m ω 2 R
block will not shift anywhere if ω is greater than maximum possible value of RHS which is
; ω ≥ 8.9 rad/sec.;
ω min = 8.9 rad/sec.
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