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Question

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 40 m? Take g=10 m/s2.

A
9 m/s to 12 m/s
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B
10 m/s to 20 m/s
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C
188.46 m/s to 243.32 m/s
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D
107.69 m/s to 657.14 m/s
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Solution

The correct option is D 107.69 m/s to 657.14 m/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:
Nsinθfmaxcosθ=mvmin2r...(i)
where fmax=μsN....(ii)

Ncosθ+fmaxsinθ=mg.....(iii)

From Eq. (i), (ii), (iii) :-
vmin=rg(tanθμs)1+μstanθ m/s
=(tan370.41+0.4×tan37)40×10
=107.69 m/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+tanθ1μstanθ)rg
vmax=(0.4+tan3710.4×tan37)40×10
vmax=657.14 m/s

Hence the range of safe velocities on the track is 188.46 m/s to 657.14 m/s
option (D) is correct.

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