A 3-4-5 inclined plane is fixed to a rotating turntable. A block rests on the inclined plane and the coefficient of static friction of static friction between the inclined plane and the block is µ s = 1/4 . The block is to remain at a position 40 cm from the center of rotation of the turntable (see Fig.). Find the minimum angular velocity 
to keep the block from sliding down the plane (towards the center).
Text Solution
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Sol. As shown in fig. the forces acting on the block are the gravitational force mg , the normal reaction N , the static friction f , and the centrifugal force with f = µ s N, P = m ω 2 r. Thus conditions for equilibrium are
mg sin θ = P cos θ + µ s N,
N = mg cos θ + P sin θ .
Hence, mg sin θ = P cos θ + µ s mg cos θ + µ s P sin θ ,
giving P = 
mg = m ω 2 r,
or ω 2 = 
= 
. 
= 10.3.
i.e. ω = 3.2 rad/s.
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