A car round on unbanked curve of radius 92 m without skidding at a speed of `26 ms ^(-1)` . The smallest possible coefficient of static friction between the tyres and the road is

Options

A. (a)0.75
B. (b)`0.60`
C. (c)0.45
D. (d)`0.30`

Answer

To find the smallest possible coefficient of static friction between the tyres and the road for a car rounding an unbanked curve, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Forces Acting on the Car:**
- When a car rounds a curve, it requires a centripetal force to keep it moving in a circular path. This force is provided by the frictional force between the tyres and the road.

2. **Identify the Relevant Equations:**
- The centripetal force \( F_c \) required to keep the car moving in a circle is given by:
\[ F_c = \frac{mv^2}{r} \]
where:
- \( m \) = mass of the car
- \( v \) = speed of the car
- \( r \) = radius of the curve

- The frictional force \( F_f \) that prevents the car from skidding is given by:
\[ F_f = \mu mg \]
where:
- \( \mu \) = coefficient of static friction
- \( g \) = acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \))

3. **Set the Forces Equal:**
- Since the frictional force provides the necessary centripetal force, we can set these two equations equal to each other:
\[ \frac{mv^2}{r} = \mu mg \]

4. **Cancel the Mass \( m \):**
- The mass \( m \) appears on both sides of the equation, so we can cancel it out:
\[ \frac{v^2}{r} = \mu g \]

5. **Rearrange to Solve for \( \mu \):**
- Rearranging the equation gives us:
\[ \mu = \frac{v^2}{rg} \]

6. **Substitute the Known Values:**
- Given:
- \( v = 26 \, \text{m/s} \)
- \( r = 92 \, \text{m} \)
- \( g = 9.8 \, \text{m/s}^2 \)

- Substitute these values into the equation:
\[ \mu = \frac{(26)^2}{92 \times 9.8} \]

7. **Calculate \( \mu \):**
- First, calculate \( (26)^2 = 676 \).
- Next, calculate \( 92 \times 9.8 = 901.6 \).
- Now, substitute these values:
\[ \mu = \frac{676}{901.6} \approx 0.75 \]

8. **Conclusion:**
- The smallest possible coefficient of static friction \( \mu \) is approximately \( 0.75 \).

### Final Answer:
The smallest possible coefficient of static friction between the tyres and the road is \( \mu \approx 0.75 \).