Question 458.
A particle moves from rest at point A on the surface of a smooth circular cylinder of radius R. At B the particle leaves the cylinder. Find the equation relating θ1 and θ2.
Answer 458.
Potential energy at A = mgRcosθ1
Potential energy at B = mgRsinθ2
Gain of kinetic energy = loss of potential energy
=> (1/2) mv^2 = mgR (cosθ1 - sinθ2)
=> mv^2/R = 2mg (cosθ1 - sinθ2) ... ( 1 )
At point B, the particle will lose contact if the centripetal force = radial component of weight
=> mv^2/R = mgsinθ2 ... ( 2 )
From ( 1 ) and ( 2 ),
mgsinθ2 = 2mg (cosθ1 - sinθ2)
=> 2cosθ1 - 3sinθ2 = 0.
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A particle moves from rest at point A on the surface of a smooth circular cylinder of radius R. At B the particle leaves the cylinder. Find the equation relating θ1 and θ2.
Potential energy at A = mgRcosθ1
Potential energy at B = mgRsinθ2
Gain of kinetic energy = loss of potential energy
=> (1/2) mv^2 = mgR (cosθ1 - sinθ2)
=> mv^2/R = 2mg (cosθ1 - sinθ2) ... ( 1 )
At point B, the particle will lose contact if the centripetal force = radial component of weight
=> mv^2/R = mgsinθ2 ... ( 2 )
From ( 1 ) and ( 2 ),
mgsinθ2 = 2mg (cosθ1 - sinθ2)
=> 2cosθ1 - 3sinθ2 = 0.
Link to YA!
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