On a horizontal curved road , the skidding or overturning of a vehicle will occur when

Options

A. `v gt sqrt (murg)`
B. the radius r is small
C. coefficient of friction `mu` is small
D. all of these

Answer

To determine when a vehicle will skid or overturn on a horizontal curved road, we need to analyze the forces acting on the vehicle. Here’s a step-by-step solution:

### Step 1: Identify the Forces Acting on the Vehicle
When a vehicle is on a curved road, two primary forces act on it:
1. **Centripetal Force (Fc)**: This force is required to keep the vehicle moving in a circular path and is directed towards the center of the curve.
2. **Frictional Force (Ff)**: This force opposes the tendency of the vehicle to skid outward due to inertia (which we can refer to as centrifugal force in the vehicle's frame of reference).

### Step 2: Establish the Condition for Skidding
The vehicle will skid when the required centripetal force exceeds the maximum frictional force available. The maximum frictional force can be expressed as:
\[ F_f = \mu \cdot N \]
where \( \mu \) is the coefficient of friction and \( N \) is the normal force. On a flat surface, the normal force equals the weight of the vehicle:
\[ N = mg \]
Thus, the maximum frictional force becomes:
\[ F_f = \mu \cdot mg \]

### Step 3: Relate Centripetal Force to Vehicle Motion
The centripetal force required to keep the vehicle moving in a circular path is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( v \) is the velocity of the vehicle and \( r \) is the radius of the curve.

### Step 4: Set Up the Inequality for Skidding
For the vehicle to not skid, the centripetal force must be less than or equal to the maximum frictional force:
\[ \frac{mv^2}{r} \leq \mu mg \]

### Step 5: Simplify the Inequality
We can cancel the mass \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{v^2}{r} \leq \mu g \]
Rearranging gives us:
\[ v^2 \leq \mu g r \]
Taking the square root of both sides, we find:
\[ v \leq \sqrt{\mu g r} \]

### Step 6: Determine the Condition for Skidding
The vehicle will skid when:
\[ v > \sqrt{\mu g r} \]

### Conclusion
From our analysis, we conclude that the vehicle will skid or overturn on a horizontal curved road when:
- The speed of the vehicle exceeds \( \sqrt{\mu g r} \).
- A smaller radius \( r \) increases the required speed for skidding.
- A smaller coefficient of friction \( \mu \) also increases the likelihood of skidding.

### Final Answer
The vehicle will skid or overturn when:
1. The speed \( v \) is greater than \( \sqrt{\mu g r} \).
2. The radius \( r \) is small.
3. The coefficient of friction \( \mu \) is small.

Thus, the correct option is that all these conditions contribute to skidding.

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