Roads are generally made with an angle on the curvatures to counter the centrifugal force on vehicles moving with speed. This is called 'Banking of Road'. Figure shows two one-way roads P and Q from top view. If the maximum safe speed limit is same for both the roads, which option shows the schematic cross section of P and Q at the position SS?
Show Hint
Remember the simple rule for banked curves: Tighter turn = Steeper bank. A smaller radius of curvature means a tighter turn, which requires a larger banking angle to safely handle the same speed.
Step 1: Understanding the Concept:
The question is about the physics concept of 'Banking of Roads'. When a car turns, it experiences a centrifugal force pushing it outwards. To counteract this, the road on a curve is banked (tilted inwards). The angle of banking depends on the speed of the vehicle and the radius of the curve.
Step 2: Key Formula or Approach:
The formula for the ideal banking angle (\(\theta\)) is given by:
\[ \tan(\theta) = \frac{v^2}{rg} \]
where:
\(v\) is the speed of the vehicle.
\(r\) is the radius of the curvature of the road.
\(g\) is the acceleration due to gravity.
The problem states that the maximum safe speed limit (\(v\)) is the same for both roads P and Q. Since \(g\) is also constant, the formula simplifies to:
\[ \tan(\theta) \propto \frac{1}{r} \]
This means that the banking angle (\(\theta\)) is inversely proportional to the radius of the curve. A smaller radius (a tighter turn) requires a larger banking angle.
Step 3: Detailed Explanation:
1. Compare the radii of roads P and Q:
From the top view diagram, road P is on the inside of the curve and road Q is on the outside. Therefore, road P has a smaller radius of curvature (\(r_P\)) compared to road Q (\(r_Q\)).
\[ r_P<r_Q \]
2. Compare the required banking angles:
Since the banking angle is inversely proportional to the radius (\(\theta \propto 1/r\)), the road with the smaller radius will need a larger banking angle.
Because \(r_P<r_Q\), it follows that \(\theta_P>\theta_Q\).
Road P must have a steeper bank than road Q.
3. Analyze the options:
We need to find the option where the cross-section for P is more steeply angled than the cross-section for Q.
(A) Shows P and Q with the same banking angle. Incorrect.
(B) Shows Q banked more steeply than P. Incorrect.
(C) Shows P banked more steeply than Q. Correct.
(D) Shows P banked in the wrong direction (outwards). Incorrect.
Step 4: Final Answer:
Road P has a smaller radius and thus requires a greater banking angle than road Q for the same safe speed. Option (C) correctly depicts this relationship.
