A car goes on a circular horizontal road of radius 100 m. What should be the minimum coefficient of friction between the tyres and the road so that it completes the circle with velocity 10 m/s without slipping ? (Take `g=10m//s^(2)`)

Options

A. (a)0.1
B. (b)0.2
C. (c)0.15
D. (d)1

Explanation

To solve the problem, we need to find the minimum coefficient of friction (\( \mu \)) required for a car to complete a circular path without slipping. Here are the steps to arrive at the solution:

### Step 1: Identify the given values
- Radius of the circular path (\( r \)) = 100 m
- Velocity of the car (\( v \)) = 10 m/s
- Acceleration due to gravity (\( g \)) = 10 m/s²

### Step 2: Write the formula for centripetal acceleration
The centripetal acceleration (\( a_c \)) required to keep the car moving in a circle is given by the formula:
\[ a_c = \frac{v^2}{r} \]

### Step 3: Substitute the known values into the centripetal acceleration formula
Substituting the values of \( v \) and \( r \):
\[ a_c = \frac{(10 \, \text{m/s})^2}{100 \, \text{m}} = \frac{100}{100} = 1 \, \text{m/s}^2 \]

### Step 4: Relate centripetal force to frictional force
The frictional force provides the necessary centripetal force to keep the car moving in a circle. The frictional force (\( F_f \)) can be expressed as:
\[ F_f = \mu \cdot N \]
where \( N \) is the normal force. For a car on a horizontal surface, the normal force is equal to the weight of the car:
\[ N = mg \]
Thus,
\[ F_f = \mu \cdot mg \]

### Step 5: Set the centripetal force equal to the frictional force
For the car to not slip, the frictional force must equal the required centripetal force:
\[ \mu \cdot mg = m \cdot a_c \]

### Step 6: Cancel mass (\( m \)) from both sides
Since mass (\( m \)) appears on both sides of the equation, we can cancel it out:
\[ \mu \cdot g = a_c \]

### Step 7: Solve for the coefficient of friction (\( \mu \))
Rearranging the equation gives:
\[ \mu = \frac{a_c}{g} \]
Substituting the values of \( a_c \) and \( g \):
\[ \mu = \frac{1 \, \text{m/s}^2}{10 \, \text{m/s}^2} = 0.1 \]

### Conclusion
The minimum coefficient of friction required is \( \mu = 0.1 \).