What is the maximum speed with which a cyclist can move without skidding, in a circular path of radius 10 m, on a road where the coefficient of friction between the tyre and the road is 0.5 ?

Explanation

To find the maximum speed with which a cyclist can move without skidding on a circular path, we can follow these steps:

### Step 1: Understand the Forces Acting on the Cyclist
When a cyclist moves in a circular path, the frictional force between the tires and the road provides the necessary centripetal force to keep the cyclist moving in a circle. The maximum frictional force can be calculated using the formula:
\[ F_{\text{friction}} = \mu \cdot F_{\text{normal}} \]
where:
- \( \mu \) is the coefficient of friction,
- \( F_{\text{normal}} \) is the normal force, which for a cyclist on a flat surface is equal to the weight of the cyclist, \( mg \).

### Step 2: Set Up the Equation for Centripetal Force
The centripetal force required to keep the cyclist moving in a circle is given by:
\[ F_{\text{centripetal}} = \frac{mv^2}{R} \]
where:
- \( m \) is the mass of the cyclist,
- \( v \) is the speed of the cyclist,
- \( R \) is the radius of the circular path.

### Step 3: Relate Frictional Force to Centripetal Force
For the cyclist to avoid skidding, the frictional force must be greater than or equal to the centripetal force:
\[ \mu mg \geq \frac{mv^2}{R} \]

### Step 4: Cancel Mass from Both Sides
Since the mass \( m \) appears on both sides of the equation, we can cancel it out:
\[ \mu g \geq \frac{v^2}{R} \]

### Step 5: Rearrange the Equation to Solve for Speed
Rearranging the equation gives us:
\[ v^2 \leq \mu g R \]
Taking the square root of both sides, we find:
\[ v \leq \sqrt{\mu g R} \]

### Step 6: Plug in the Values
Now, we can substitute the given values into the equation:
- \( \mu = 0.5 \)
- \( g = 9.8 \, \text{m/s}^2 \)
- \( R = 10 \, \text{m} \)

Calculating:
\[ v \leq \sqrt{0.5 \cdot 9.8 \cdot 10} \]

### Step 7: Calculate the Maximum Speed
Calculating the value inside the square root:
\[ v \leq \sqrt{49} \]
\[ v \leq 7 \, \text{m/s} \]

### Final Answer
The maximum speed with which the cyclist can move without skidding is:
\[ \boxed{7 \, \text{m/s}} \]

---