An automobile enters a turn whose radius is R. The road is banked at angle `theta`. Fricton is negligible between wheels of the automobile and road, Mass of automobile is `m` and speed is `v`. Select the correct alternative.

Options

A. net force on the automobile is zero
B. normal reaction on the automobile is `mg cos theta`
C. normal reaction on the automobile is `mg sec theta`
D. net force on the automobile is `sqrt((mg)^(2)+(mv^(2)//R)^(2))`

Answer

To solve the problem of an automobile entering a banked turn with negligible friction, we can analyze the forces acting on the automobile and apply the concepts of circular motion. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Forces Acting on the Automobile
When the automobile is on a banked road, the following forces act on it:
- The weight of the automobile (mg) acting downward.
- The normal force (N) acting perpendicular to the surface of the road.

### Step 2: Resolve the Normal Force into Components
Since the road is banked at an angle θ, we can resolve the normal force into two components:
- The vertical component: \( N \cos \theta \)
- The horizontal component: \( N \sin \theta \)

### Step 3: Set Up the Equations for Forces
1. In the vertical direction, the forces must balance since there is no vertical acceleration:
\[ N \cos \theta = mg \]
From this, we can express the normal force:
\[ N = \frac{mg}{\cos \theta} \]

2. In the horizontal direction, the horizontal component of the normal force provides the necessary centripetal force for circular motion:
\[ N \sin \theta = \frac{mv^2}{R} \]

### Step 4: Substitute the Expression for Normal Force
Substituting \( N = \frac{mg}{\cos \theta} \) into the horizontal force equation:
\[ \frac{mg}{\cos \theta} \sin \theta = \frac{mv^2}{R} \]

### Step 5: Simplify the Equation
Canceling \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{g \sin \theta}{\cos \theta} = \frac{v^2}{R} \]
This can be rewritten using the definition of tangent:
\[ g \tan \theta = \frac{v^2}{R} \]

### Step 6: Conclusion
From the derived equation, we can analyze the relationship between the speed \( v \), the radius \( R \), and the angle \( \theta \). The correct expression derived from the forces indicates that the normal force is not simply \( mg \) or \( mg \cos \theta \), but rather \( N = \frac{mg}{\cos \theta} \).

### Final Answer
Thus, the correct alternative is:
- **Option C: The normal reaction on the automobile is \( mg \sec \theta \)**.