Figure shows a hemisphere and a supported rod. Hemisphere is moving right with a uniform velocity v2 and the end of rod which is in contact with ground is moving left with a velocity v1. The rate at which the angle θ is decreasing will be
A
(v1+v2)sin2θRcosθ
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B
(v1+v2)tanθRcosθ
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C
(v1+v2)cos2θRsinθ
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D
(v1+v2)cotθRsinθ
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Solution
The correct option is A(v1+v2)sin2θRcosθ From right angled triangle formed by the rod, sinθ=Rx x=Rcosecθ
Differentiating above equation, we get dxdt=−Rcosecθcotθdθdt
dxdt can be witten as v1+v2
Hence, dθdt=−(v1+v2)sin2θRcosθ
(−ve sign shows that θ decreasing with time).