derive expression for minimum velocity on banked road
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Answer:
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. Now, let us recall, what is 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. In the next few sections, let us discuss the angle of banking and the terminologies used in the banking of roads.
Another terminology used is banked turn which is defined as the turn or change of direction in which the vehicle inclines towards inside. The angle at which the vehicle is inclined is defined as the bank angle. The inclination happens at the longitudinal and horizontal axis.
Let's consider a vehicle of mass "m" moving along a banked road with a circular track of radius "r". The road is banked at an angle "θ" with respect to the horizontal.
At the minimum velocity, the centripetal force acting on the vehicle is equal to the force of friction between the tires and the road.
Centripetal force = mv^2/r, where v is the velocity of the vehicle.
The force of friction is given by Ff = μN, where μ is the coefficient of friction between the tires and the road, and N is the normal force acting on the vehicle.
The normal force N is resolved into two components: N cosθ, which acts perpendicular to the banked road, and N sinθ, which acts parallel to the banked road.
The weight of the vehicle is resolved into two components: mg cosθ, which acts perpendicular to the banked road, and mg sinθ, which acts parallel to the banked road.
Since the vehicle is moving along a circular path, the net force acting on the vehicle must be towards the center of the circle.
Therefore, we can write:
mv^2/r = Ff sinθ + mg cosθ
Substituting Ff = μN and N = mg cosθ + mv^2/r sinθ, we get:
mv^2/r = μ(mg cosθ + mv^2/r sinθ)sinθ + mg cosθ
Simplifying this equation, we get:
v^2 = gr tanθ / (1 - μ tanθ)
where g is the acceleration due to gravity.
This is the expression for the minimum velocity on a banked road.
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