A bend in level road has a radius of 180 m. Find the maximum speed which a car turning this bend may without skidding if coefficient of friction between the car and the road is 1.5

Options

A. 51 m/s
B. 62 m/s
C. 48 m/s
D. 49 m/s

Explanation

To solve the problem of finding the maximum speed at which a car can turn a bend without skidding, we can follow these steps:

### Step 1: Understand the forces involved
When a car turns on a bend, two main forces are acting on it:
- The centripetal force required to keep the car moving in a circular path.
- The frictional force between the tires and the road, which provides the necessary centripetal force.

### Step 2: Write the equation for centripetal force
The centripetal force \( F_c \) required to keep the car moving in a circle is given by the formula:
\[ F_c = \frac{mv^2}{r} \]
where:
- \( m \) is the mass of the car,
- \( v \) is the speed of the car,
- \( r \) is the radius of the bend.

### Step 3: Write the equation for frictional force
The maximum frictional force \( F_f \) that can act on the car without skidding is given by:
\[ F_f = \mu mg \]
where:
- \( \mu \) is the coefficient of friction,
- \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \), but we can use \( 10 \, \text{m/s}^2 \) for simplicity).

### Step 4: Set the forces equal for maximum speed
For the car to turn safely without skidding, the maximum frictional force must be equal to the centripetal force:
\[ \mu mg = \frac{mv^2}{r} \]

### Step 5: Cancel the mass \( m \) from both sides
Since mass \( m \) appears on both sides of the equation, we can cancel it out:
\[ \mu g = \frac{v^2}{r} \]

### Step 6: Rearrange the equation to solve for speed \( v \)
Rearranging the equation gives:
\[ v^2 = \mu g r \]
Taking the square root of both sides, we find:
\[ v = \sqrt{\mu g r} \]

### Step 7: Substitute the known values
Given:
- \( \mu = 1.5 \)
- \( g = 10 \, \text{m/s}^2 \)
- \( r = 180 \, \text{m} \)

Substituting these values into the equation:
\[ v = \sqrt{1.5 \times 10 \times 180} \]

### Step 8: Calculate the value
Calculating the expression inside the square root:
\[ 1.5 \times 10 = 15 \]
\[ 15 \times 180 = 2700 \]
Now, taking the square root:
\[ v = \sqrt{2700} \]
Calculating the square root:
\[ v \approx 51.96 \, \text{m/s} \]

### Final Answer
The maximum speed at which the car can turn the bend without skidding is approximately \( 51.96 \, \text{m/s} \).

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