Question

A curved road of 50m radius is banked at
correct angle for a given speed. If the
speed is to be doubled keeping the same
banking angle, the radius of curvature of
the road should be changed to
25 m
100 m
150 m
O
200 m​

Answer 1

Answer:

Option 4 : 200 m

Explanation:

Given:

• The radius of the curved road = 50 m
• The speed is doubled without changing the banking angle

To Find:

• The new radius of the curved road

Solution:

Let the banking angle be θ.

Let the initial radius of curvature of the road be r₁

Let the new radius of curvature of the road be r₂

Let the speed/velocity of the vehicle be v₁

By given,

New speed = Double the initial speed

Hence,

New speed (v₂) = 2 v₁

We know that banking angle of a road is given by,

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

where v is the velocity of the vehicle,

r = radius of curvature of the road

g is the acceleration due to gravity

θ is the banking angle.

By given there is no change in the banking angle of the road in both cases.

Hence,

$\sf \dfrac{(v_1)^{2} }{r_1g} =\dfrac{(v_2)^{2} }{r_2g}$

Substitute the data,

$\sf \dfrac{(v_1)^{2} }{50g} =\dfrac{(2v_1)^{2} }{r_2g}$

Cancelling g on both sides,

$\sf \dfrac{(v_1)^{2} }{50} =\dfrac{4(v_1)^{2} }{r_2}$

$\sf \dfrac{1}{50} =\dfrac{4}{r_2}$

r₂ =  50 × 4

r₂ = 200 m

Hence the new radius of curvature of the road is 200 m.

Therefore option 4 is correct.