Fing the maximum speed with which an automobile can round a curve of radius 8 m without slipping of the road is unbanked and he coefficient of friction between the orad an the tyres is `0.8 (g=10 m//^(2))`

Options

A. `8ms^(-1)`
B. `10ms^(-1)`
C. `20ms^(-1)`
D. none of these

Answer

To find the maximum speed with which an automobile can round a curve of radius 8 m without slipping on an unbanked road, we can use the following steps:

### Step 1: Understand the forces involved
When a car rounds a curve, there are two main forces acting on it:
- The centripetal force required to keep the car moving in a circular path, which is given by \( F_c = \frac{mv^2}{r} \).
- The frictional force that provides this centripetal force, which is given by \( F_f = \mu N \), where \( \mu \) is the coefficient of friction and \( N \) is the normal force.

### Step 2: Set up the equations
Since the road is unbanked, the normal force \( N \) is equal to the weight of the car, \( mg \). Therefore, we can write:
\[ F_f = \mu mg \]
The car will not slip as long as the frictional force is equal to or greater than the required centripetal force:
\[ \mu mg = \frac{mv^2}{r} \]

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

### Step 4: Solve for \( v \)
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 given values into the equation:
- \( \mu = 0.8 \)
- \( g = 10 \, \text{m/s}^2 \)
- \( r = 8 \, \text{m} \)

Substituting these values:
\[ v = \sqrt{0.8 \times 10 \times 8} \]

### Step 6: Calculate the value
Calculating the expression inside the square root:
\[ v = \sqrt{0.8 \times 10 \times 8} = \sqrt{64} = 8 \, \text{m/s} \]

### Final Answer
The maximum speed with which the automobile can round the curve without slipping is **8 m/s**.

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