A car moves along a horizontal circular road of radius r with velocity u -The coefficient of friction between the wheels and the road is `mu`. Which of the following statement is not true?

Options

A. The car will slip if `v gt sqrt(mu rg)`
B. The car will slip if `mu lt v^(2)/(rg)`
C. The car will slip if `mu gt v^(2)/(rg)`
D. the car will slip at lower speed, if it moves with some tangential acceleration, than if it moves at constant speed.

Answer

To solve the problem, we need to analyze the motion of a car moving along a horizontal circular road and determine which statement about the conditions of motion is not true.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Car**:
- The car experiences a gravitational force acting downwards (mg).
- There is a normal force (N) acting upwards, which balances the gravitational force.
- The car also experiences a centripetal force required for circular motion, which is provided by the frictional force (f) between the tires and the road.

2. **Centripetal Force Requirement**:
- For an object moving in a circle of radius \( r \) with velocity \( u \), the required centripetal force is given by:
\[ f = \frac{mu^2}{r} \]
- This frictional force must be equal to or less than the maximum static friction force to prevent slipping.

3. **Maximum Static Friction**:
- The maximum static friction force can be expressed as:
\[ f_{max} = \mu N = \mu mg \]
- For the car to not slip, the frictional force must satisfy:
\[ \frac{mu^2}{r} \leq \mu mg \]

4. **Setting Up the Inequality**:
- From the above inequality, we can simplify:
\[ \frac{u^2}{r} \leq g \]
- Rearranging gives:
\[ u^2 \leq \mu rg \]
- This means that the maximum speed \( u \) for the car to avoid slipping is:
\[ u \leq \sqrt{\mu rg} \]

5. **Analyzing the Statements**:
- We need to evaluate the statements provided in the question to find which one is not true based on our derived conditions.
- The statements typically relate to the relationship between speed, friction, and the possibility of slipping.

6. **Conclusion**:
- After evaluating the statements, we find that one of them contradicts the derived relationship \( u \leq \sqrt{\mu rg} \). This statement is the one that is not true.