Question

A vehicle is moving with uniform speed along 3 different shaped roads as horizontal, concave and convex. The surface of road on which, the normal reaction on vehicle is maximum is

• A. convex
• B. concave
• C. horizontal
• D. same on all the 3 surface

Hint:

For circular motion on roads:
• convex road \(\Rightarrow N < mg\)
• horizontal road \(\Rightarrow N = mg\)
• concave road \(\Rightarrow N > mg\) e} So the maximum normal reaction occurs on a concave road.

Answer 1

The Correct Option is B

Concept:
The normal reaction depends on the shape of the road because the required centripetal force may act upward or downward depending on the motion. For a vehicle moving on:
• a horizontal road, there is no vertical centripetal acceleration,
• a convex road, centripetal acceleration is downward,
• a concave road, centripetal acceleration is upward. e} ip

Step 1: Find the normal reaction on a horizontal road.
On a horizontal road, the vehicle has no vertical circular motion. So the forces balance vertically: \[ N = mg \] ip

Step 2: Find the normal reaction on a convex road.
At the top of a convex road, the centripetal force acts downward. Hence, \[ mg - N = \frac{mv^2}{r} \] So, \[ N = mg - \frac{mv^2}{r} \] This means the normal reaction is less than \(mg\). ip

Step 3: Find the normal reaction on a concave road.
At the bottom of a concave road, the centripetal force acts upward. Hence, \[ N - mg = \frac{mv^2}{r} \] So, \[ N = mg + \frac{mv^2}{r} \] This means the normal reaction is greater than \(mg\). ip

Step 4: Compare the three cases.
We have: \[ N_{\text{convex}} = mg - \frac{mv^2}{r} \] \[ N_{\text{horizontal}} = mg \] \[ N_{\text{concave}} = mg + \frac{mv^2}{r} \] Thus, \[ N_{\text{concave}} > N_{\text{horizontal}} > N_{\text{convex}} \] ip Hence, the correct answer is:
\[ \boxed{(B)\ \text{concave}} \]