A smooth sphere rests on a horizontal plane. A point particle slides frictionless down the sphere, starting at the top. Let R be the radius of the sphere. Describe the particle’s path up to the time it strikes the plane.
Text Solution
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Sol. A shown in fig. conservation of energy gives
mv 2 = mg R (1 – cos θ ).
The radial force the sphere exerts on the particle is
F = mg cos θ – 
.
When F = 0, the constraint vanishes and the particle leaves the sphere. At this instant, we have
= g cos θ ,
v 2 = 2gR (1 – cos θ )
giving cos θ = 
, or q = 48.2º,
v = 
.
The particle leaves the sphere with a speed v = 
at an angle θ = 48.2º. After leaving the sphere the particle follows a parabolic trajectory until it hits the plane.
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