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Question

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)cosθμssinθ]12
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C
[rg(sinθ+μscosθ)cosθμssinθ]12
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D
None
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Solution

The correct option is C [rg(sinθ+μscosθ)cosθμssinθ]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+fmax cosθ=mv2maxr
The maximum value of friction:
fmax=μsN
Nsinθ+μsN cosθ=mv2maxr..(i)

In vertical direction, the system is in a state of equilibrium:
Ncosθ=fmaxsinθ+mg
NcosθμsNsinθ=mg...(ii)

Dividing Eq.(i) by Eq.(ii), we get:
sinθ+μscosθcosθμssinθ=v2maxrg

vmax=rg(sinθ+μscosθ)(cosθμssinθ)

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