How do the Banked Curve & Banking Angle help a car? | banking angle formula with derivation

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Banking Angle – what is the banking angle and why is it important?

When a car travels without skidding around an unbanked curve, the static frictional force between the tires and the road provides the centripetal force. But the wear and tear of tires caused by this friction increases the maintenance cost of the vehicles and increases the risk of sudden accidents at the curved points of the roads.

However, if the curve of a road is banked at an angle relative to the horizontal, much in the same way that a plane is banked while making a turn, the reliance on friction to provide the required centripetal force can be eliminated completely for a given speed. This angle the curve of the road makes here with respect to the horizontal is called banking angle or banked angle.

Banking angle at the curved turns of the roads reduces friction between the tires and the road and this, in turn, reduces maintenance cost and accidents of the vehicles.

banking angle
[(a) A car travels on a circle of radius r on a frictionless banked road. The
banking angle is θ, and the center of the circle is at C.]

Banking angle formula with derivation

Figure (a) shows a car going around a friction-free banked curve. The radius of the curve is r, where r is measured parallel to the horizontal and not to the slanted surface.

Banking angle formula with derivation
Banking angle formula with derivation

Figure (b) shows the normal force FN that the road applies to the car, the normal force being perpendicular to the road.

Because the roadbed makes an angle θ with respect to the horizontal, the normal force has a component FN sin θ that points toward the center C of the circle and provides the centripetal force:
Fc = FN sin θ = (mv2)/r ……… (1)

The vertical component of the normal force is FN cos θ and, since the car does not accelerate in the vertical direction, this component must balance the weight mg of the car.
Therefore, FN cos θ = mg …… (2)

Dividing equation 2 with equation 1 shows that:
(FN sin θ)/(FN cos θ) = [(mv2)/r]/mg
=> tan θ = v2/(rg) ...……………………… (3). This gives the equation or formula of the Banking angle.
And thus we can derive the banking angle formula.

Different observations related to the Banking angle of the road

1 ) Equation 3 indicates that, for a given speed v, the centripetal force needed for a turn of radius r can be obtained from the normal force FN by banking the turn at an angle θ.

2 ) This banking angle is independent of the mass of the vehicle. The formula doesn’t contain any mention of the mass m in it.

3 ) Greater speeds and smaller radii require more steeply banked curves—that is, larger values of θ.

4 ) At a speed that is too small for a given θ, a car would slide down a frictionless banked curve.

5 ) At a speed that is too large, a car would slide off the top.

Suggested Reading:

Centripetal force formula

Centripetal force and washing machine

How do the Banked Curve & Banking Angle help a car? | banking angle formula with derivation
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