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. 0.1
B. 0.2
C. 0.15
D. 1

Explanation

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

### Step 1: Identify the given values
- Radius of the circular road (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:
\[ a_c = \frac{(10 \, \text{m/s})^2}{100 \, \text{m}} \]
\[ a_c = \frac{100 \, \text{m}^2/\text{s}^2}{100 \, \text{m}} \]
\[ a_c = 1 \, \text{m/s}^2 \]

### Step 4: Relate the frictional force to centripetal force
The frictional force provides the necessary centripetal force to keep the car moving in a circle. The maximum 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), so:
\[ F_f = \mu \cdot mg \]

### Step 5: Set the frictional force equal to the centripetal force
The centripetal force (F_c) required for circular motion is given by:
\[ F_c = m \cdot a_c \]
Setting the two forces equal gives:
\[ \mu \cdot mg = m \cdot a_c \]

### 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 (μ)
Now, rearranging the equation to solve for μ:
\[ \mu = \frac{a_c}{g} \]

### Step 8: Substitute the values of a_c and g
Substituting the known values:
\[ \mu = \frac{1 \, \text{m/s}^2}{10 \, \text{m/s}^2} \]
\[ \mu = 0.1 \]

### Conclusion
The minimum coefficient of friction required for the car to complete the circular path without slipping is **0.1**.

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