1. **Problem Statement:** We want to understand why it is easier to do chin-ups when your hands are 30 cm apart compared to 90 cm apart, assuming the force exerted by your arms is the same in both cases. 2. **Concepts and Formula:** The difficulty of chin-ups relates to the torque (rotational force) around your shoulder joints. Torque $\tau$ is given by: $$\tau = F \times r$$ where $F$ is the force exerted by your arms and $r$ is the perpendicular distance from the axis of rotation (shoulder joint) to the line of action of the force. 3. **Vector Diagram Explanation:** - When your hands are closer (30 cm apart), the horizontal distance $r$ from the shoulder joint to the force application point is smaller. - When your hands are wider apart (90 cm), $r$ is larger. 4. **Effect on Torque:** - Since $\tau = F \times r$, a larger $r$ means greater torque is required to lift your body. - With the same force $F$, a larger $r$ (wider grip) increases the torque, making the chin-up harder. - A smaller $r$ (narrower grip) reduces the torque, making it easier. 5. **Conclusion:** Because the torque needed to lift your body increases with the distance between your hands, chin-ups are easier when your hands are 30 cm apart than when they are 90 cm apart, assuming the same force exerted by your arms.