**Correct Option: B** ### Explanation of the Correct Answer In uniform circular motion, an object moves in a circular path at a constant speed. To maintain this circular motion, a net force must act on the object, directed towards the center of the circle. This force is known as the **centripetal force**. The relationship between centripetal force (F_c), mass (m), and velocity (v) can be described by the formula: \[ F_c = \frac{mv^2}{r} \] Where: - \( F_c \) is the centripetal force, - \( m \) is the mass of the object, - \( v \) is the tangential velocity of the object, - \( r \) is the radius of the circular path. From this formula, we can derive the following relationships: 1. **Directly Proportional to Mass**: The centripetal force increases linearly with an increase in mass. If you double the mass of the object while keeping the velocity and radius constant, the centripetal force will also double. 2. **Directly Proportional to the Square of Velocity**: The centripetal force increases with the square of the velocity. If you double the velocity, the centripetal force increases by a factor of four (since \( (2v)^2 = 4v^2 \)). 3. **Inversely Proportional to Radius**: The centripetal force is inversely proportional to the radius of the circular path. If the radius increases while keeping mass and velocity constant, the centripetal force decreases. ### Why the Other Options are Incorrect - **Option A**: "Centripetal force increases with mass and decreases with velocity." - **Incorrect**: While it is true that centripetal force increases with mass, it does not decrease with velocity. Instead, it increases with the square of the velocity. This option misrepresents the relationship with velocity. - **Option C**: "Centripetal force is directly proportional to both mass and the square of velocity." - **Incorrect**: This option is misleading because it does not mention the radius, which is a crucial factor in the centripetal force equation. While it correctly states that centripetal force is directly proportional to mass and the square of velocity, it fails to provide a complete picture of the relationship. - **Option D**: "Centripetal force is directly proportional to mass and inversely proportional to velocity squared." - **Incorrect**: This option incorrectly states the relationship with velocity. Centripetal force is not inversely proportional to velocity squared; rather, it is directly proportional to the square of the velocity. This is a fundamental misunderstanding of the relationship. ### Summary of Key Points - **Centripetal Force Formula**: \( F_c = \frac{mv^2}{r} \) - **Directly Proportional to Mass**: More mass means more centripetal force needed. - **Directly Proportional to the Square of Velocity**: Higher speed means much more centripetal force is required. - **Inversely Proportional to Radius**: A larger radius means less centripetal force is needed for the same mass and speed. This understanding of centripetal force is crucial for solving problems related to circular motion in physics.