A race track banked at an angle of 37∘ has a coefficient of static friction 0.4. What is the range of the safe velocities for the cars travelling on the track, for a turn of radius 40m? Take g=10m/s2.
A
9m/sto12m/s
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B
10m/sto20m/s
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C
√188.46m/sto√243.32m/s
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D
√107.69m/s to √657.14m/s
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Solution
The correct option is D√107.69m/s to √657.14m/s For case of vmin, car will have a tendency to slip down. Hence, friction fmax will act up the plane.
Applying the equations of circular dynamics: Nsinθ−fmaxcosθ=mvmin2r...(i)
where fmax=μsN....(ii)
Ncosθ+fmaxsinθ=mg.....(iii)
From Eq. (i), (ii), (iii) :- vmin=√rg(tanθ−μs)1+μstanθm/s =√(tan37∘−0.41+0.4×tan37∘)40×10 =√107.69m/s
For the case of vmax, car will have a tendency to skid up the incline. Hence, friction will act down the plane:
Simillarly, solving the equations of circular dynamics, we get: vmax=√(μs+tanθ1−μstanθ)rg vmax=√(0.4+tan37∘1−0.4×tan37∘)40×10 vmax=√657.14m/s
Hence the range of safe velocities on the track is √188.46m/s to √657.14m/s ∴ option (D) is correct.