A car is moving on a circular track of radius R. The road is banked at an angle θ. μs is the coefficient of static friction between the car and the track. Find the maximum speed with which the car can move safely on the track.
A
[rg(sinθ+μscosθ)cosθ+μssinθ]12
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B
[rg(cosθ+μssinθ)cosθ−μssinθ]12
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C
[rg(sinθ+μscosθ)cosθ−μssinθ]12
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D
None
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Solution
The correct option is C[rg(sinθ+μscosθ)cosθ−μssinθ]12 In this case, frictional force is also used to support the requirement of centripetal force along with that provided by normal reaction.
For maximum safe speed, friction will also act at its limiting value i.e fmax
Applying the equation of dynamics along horizontal and vertical direction: Nsinθ+fmaxcosθ=mv2maxr
The maximum value of friction: fmax=μsN ∴Nsinθ+μsNcosθ=mv2maxr..(i)
In vertical direction, the system is in a state of equilibrium: Ncosθ=fmaxsinθ+mg ∴Ncosθ−μsNsinθ=mg...(ii)
Dividing Eq.(i) by Eq.(ii), we get: sinθ+μscosθcosθ−μssinθ=v2maxrg