What will be maximum speed of a car on a curved road of radius 30 m , If the coefficient of friction between the tyres and the road is 0.4?
`(g=9.8 m//s^(2))`

Options

A. `10.84 ms^(-1)`
B. `9.84 ms^(-1)`
C. `8.84 ms^(-1)`
D. `6.84 ms^(-1)`

Answer

To find the maximum speed of a car on a curved road, we can use the relationship between frictional force, gravitational force, and centripetal force. Here are the steps to solve the problem:

### Step 1: Understand the Forces Involved
When a car is moving on a curved road, the frictional force provides the necessary centripetal force to keep the car moving in a circle. The forces acting on the car are:
- Gravitational force (weight) acting downwards: \( F_g = mg \)
- Frictional force acting towards the center of the curve: \( F_f = \mu mg \)
- Centripetal force required to keep the car moving in a circle: \( F_c = \frac{mv^2}{r} \)

### Step 2: Set Up the Equation
At maximum speed, the frictional force equals the centripetal force:
\[ F_f = F_c \]
This gives us the equation:
\[ \mu mg = \frac{mv^2}{r} \]

### Step 3: Simplify the Equation
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \mu g = \frac{v^2}{r} \]

### Step 4: Solve for Speed \( v \)
Rearranging the equation to solve for \( v \):
\[ v^2 = \mu g r \]
\[ v = \sqrt{\mu g r} \]

### Step 5: Substitute the Values
Now we can substitute the values given in the problem:
- Coefficient of friction \( \mu = 0.4 \)
- Acceleration due to gravity \( g = 9.8 \, \text{m/s}^2 \)
- Radius of the curve \( r = 30 \, \text{m} \)

Substituting these values into the equation:
\[ v = \sqrt{0.4 \times 9.8 \times 30} \]

### Step 6: Calculate the Value
Calculating the expression inside the square root:
\[ 0.4 \times 9.8 = 3.92 \]
\[ 3.92 \times 30 = 117.6 \]
Now, take the square root:
\[ v = \sqrt{117.6} \approx 10.84 \, \text{m/s} \]

### Final Answer
The maximum speed of the car on the curved road is approximately \( 10.84 \, \text{m/s} \).

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