Obtain the expression of maximum velocity of safe turn for a vehicle moving on a circular level road
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JEE-MAINPhysicsGrade 11Circular Motion
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Derive an expression for maximum speed of a car on a banked road in circular motion.
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Hint:In this question, we need to find the maximum speed of the car at which car doesn’t skid while moving in a circular banked road. We basically balance all the factors of the forces exerted on a car in x-axis direction and y-axis direction.
To solve any balancing problems, whether motion is translational or rotational, we always balance forces acting along x-axis and y-axis.
In case of rotational motion, we balance torques acting on the body.
Complete step-by-step answer:
Several formulas used in our calculations are-
Gravitational attraction force – mg
Frictional force (f) – acting between car’s tire and road, and parallel to the slope of road
Normal force (N)- It is a reaction force of the gravitational force exerted by the road on car
Centripetal force (mv2r)" role="presentation" style="box-sizing: border-box; font-family: "Open Sans", sans-serif; -webkit-tap-highlight-color: rgba(255, 255, 255, 0); display: inline; font-style: normal; font-weight: normal; line-height: 49px; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">(mv2r)(mv2r) - acting towards the center of the circular path followed by the car.
Let’s take inclination angle A of road with respect to the x-axis as shown in the above diagram. Now we can balance all the x-axis and y-axis factors of forces. Refer to the figure.
(1) Balancing all the forces acting along the x-axis-
fcos⁡(A)+Nsin⁡(A)=mv2r" role="presentation" style="box-sizing: border-box; font-family: "Open Sans", sans-serif; -webkit-tap-highlight-color: rgba(255, 255, 255, 0); display: inline; font-style: normal; font-weight: normal; line-height: 49px; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">fcos(A)+Nsin(A)=mv2
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Explanation:
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