Question

Derive a relation for the optimum velocity of negotiating a curve by a body in a banked curve when friction is taken into consideration?

Answer 1

So, we need a derivation for optimum velocity along a banked road,

Here we go,

Suppose an object of mass 'm', Radius Of the curved road 'r'....

In this case of circular motion, Centripetal force is necessary

-->> Centripetal Force = Force of Friction

(Which means Centripetal force is provided by the force of friction between road and tyres)

Lets ¶ be the coefficient of friction(I cant find the symbol here)

'v' the velocity(optimum, What u need)

So, by definition

mv^2/r = ¶mg

v^2 = ¶rg

Hence, V=ærg

Hope it helps Dear friend!!

Answer 2

Explanation:

The attached figure shows the motion of car on the banked curve. The forces acting on the car in this position are :

• Weight of the car i.e. mg
• The reaction N of the ground to the vehicle.

The vertical component of the car is balanced by the weight of car such that,

$N\cos \theta=mg$ .........(1)

m is mass of car

g is acceleration due to gravity

and

The horizontal component is balanced by the centripetal force i.e.

$N\sin \theta=\dfrac{mv^2}{r}$  .........(2)

v is the velocity of car

r is the radius of curve

Dividing equation (2) by (1) we get :

$\tan\theta=\dfrac{v^2}{rg}$

$\theta$ is the angle of banking

$v=\sqrt{rg\tan \theta}$

Hence, the above formula is the relation for the optimum velocity of negotiating a curve by a body in a banked curve.

Learn more,

Banking of curve

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