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

The correct answer is 50 m/s.

When a motorcycle travels on a curved (circular) track, the friction between the tyres and the road provides the centripetal force needed to keep the motorcycle on the circular path. If the speed exceeds a critical value, the required centripetal force exceeds the maximum available frictional force, and the motorcycle starts to skid.

Step 1: Identify the condition for no skidding

For circular motion, the centripetal force required is:

F_{\mathrm{centripetal}}=\frac{m{v}^2}{r}

The maximum static frictional force available is:

F_{\mathrm{friction}}=\mu mg

To avoid skidding, the centripetal force must not exceed the maximum frictional force:

\frac{m{v}^2}{r}\le \mu mg

Step 2: Derive the formula for maximum safe speed

Cancelling mass m from both sides and rearranging:

v^2\le \mu rg

Therefore, the maximum speed to avoid skidding is:

v_{\max }=\sqrt{\mu rg}

Step 3: Substitute the given values

Given:

- Radius, r=500\text{m}
- Coefficient of friction, \mu =0.5
- Acceleration due to gravity, g=10\text{m/s²}

v_{\max }=\sqrt{0.5\times 500\times 10}

v_{\max }=\sqrt{2500}

v_{\max }=50\text{m/s}

Conclusion: The maximum speed at which the motorcycle can travel on the curved track without skidding is 50 m/s. Any speed beyond this value means friction can no longer provide sufficient centripetal force, causing the tyres to lose grip and the motorcycle to skid outward.