Why banking of road is necessary? Obtain formula for maximum permissible
speed of a vehicle on a banked road in the presence of friction with suitable
diagram.
Answers
Banking of roads is defined as the phenomenon in which the edges are raised for the curved roads above the inner edge to provide the necessary centripetal force to the vehicles so that they take a safe turn.
centripetal force : It is the force that pulls or pushes an object toward the center of a circle as it travels, causing angular or circular motion.
Angle of Banking :
Consider a vehicle of mass ‘m’ with moving speed ‘v’ on the banked road with radius ‘r’. Let ϴ be the angle of banking, with frictional force f acting between the road and the tyres of the vehicle.
Total upwards force = Total downward force
⇒ Ncosθ = mg + f sinθ
Where,
Ncosθ : one of the components of normal reaction along the verticle axis
mg : weight of the vehicle acting vertically downward
fsinθ : one of the components of frictional force along the verticle axis
∴ mg=Ncosθ − fsinθ (eq.1)
⇒ (mv²÷r) = N sinθ + fcosθ (eq.2)
Where,
N sinθ : one of the components of normal reaction along the horizontal axis.
f cosθ : one of the components of frictional force along the horizontal axis
⇒( (mv²÷r)÷ mg ) =( N sinθ + f cosθ ) ÷ ( N cosθ − f sinθ )
(after diving eq.1 and eq.2)
∴ v²÷rg =( N sinθ + f cosθ ) ÷ ( N cosθ − f sinθ )
Frictional force,
f = μs N×(v²÷rg) = ( (N sinθ + μs N cosθ) ÷ (N cosθ − μs N sinθ) )× v²÷rg
⇒( N (sinθ + μs cosθ) ÷ N(cosθ − μs sinθ) ) v²÷rg
⇒ ( (sinθ + μs cosθ) ÷ (cosθ − μs sinθ) )
⇒ v² ÷ rg = (tanθ + μs) ÷ (1 − μs tanθ)
∴ v = √ ( rg (tanθ + μs) ÷ (1−μstanΘ) )
⇒ Vmax = √(rg tanθ)
⇒ tanθ = v² ÷ rg
∴ θ = tan−1 (v² ÷ rg)
Above is the expression for the angle of banking.
HOPE THIS HELPS YOU ✌️✌️☘️☘️.