A motorcycle is travelling on a curved track of radius 500 m. If the coefficient of friction between road and tyres is 0.5, the speed avoiding skidding will be

Options

A. 50 m/s
B. 75m/s
C. 25m/s
D. 35m/s

Explanation

To solve the problem of finding the speed at which a motorcycle can travel on a curved track without skidding, we can follow these steps:

### Step 1: Identify the forces acting on the motorcycle
When a motorcycle travels in a circular path, the forces acting on it include:
- The gravitational force (weight) acting downwards, \( mg \).
- The normal force \( N \) acting upwards.
- The frictional force that provides the necessary centripetal force to keep the motorcycle on the curved path.

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

### Step 3: Write the equation for frictional force
The maximum frictional force that can act on the motorcycle without skidding is given by:
\[ F_f = \mu N \]
where:
- \( \mu \) is the coefficient of friction,
- \( N \) is the normal force. For a flat surface, \( N = mg \).

Thus, we can write:
\[ F_f = \mu mg \]

### Step 4: Set the centripetal force equal to the frictional force
To avoid skidding, the frictional force must be equal to the centripetal force:
\[ \mu mg = \frac{mv^2}{r} \]

### Step 5: Simplify the equation
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \mu g = \frac{v^2}{r} \]

### Step 6: Solve for \( v^2 \)
Rearranging the equation gives us:
\[ v^2 = \mu g r \]

### Step 7: Substitute the known values
Given:
- \( \mu = 0.5 \)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( r = 500 \, \text{m} \)

Substituting these values into the equation:
\[ v^2 = 0.5 \times 10 \times 500 \]
\[ v^2 = 0.5 \times 5000 = 2500 \]

### Step 8: Calculate \( v \)
Taking the square root of both sides gives:
\[ v = \sqrt{2500} = 50 \, \text{m/s} \]

### Conclusion
The speed at which the motorcycle can travel on the curved track without skidding is:
\[ \boxed{50 \, \text{m/s}} \]