b

To negotiate a curve safely, the frictional force must provide the necessary centripetal force. The formula for centripetal force \( F_c \) is given by:
\[ F_c = \frac{mv^2}{r} \]
Where:
- \( m \) is the mass of the motorcyclist
- \( v \) is the speed
- \( r \) is the radius of the curve
The frictional force \( F_f \) is given by:
\[ F_f = \mu mg \]
Where:
- \( \mu \) is the coefficient of friction
- \( g \) is the acceleration due to gravity
For the motorcyclist to negotiate the curve safely, the frictional force must be equal to or greater than the centripetal force:
\[ \mu mg \geq \frac{mv^2}{r} \]
Dividing both sides by \( m \) (mass of the motorcyclist) gives:
\[ \mu g \geq \frac{v^2}{r} \]
Rearranging for \( \mu \):
\[ \mu \geq \frac{v^2}{gr} \]
Therefore, the minimum value of the coefficient of friction \( \mu \) necessary for safe negotiation of the curve is:
\[ \mu = \frac{v^2}{gr} \]
Therefore, the correct answer is option (b) \( \frac{v^2}{gr} \).