Work Done by a Force

Work along a path: \(W=\int \vec F \cdot d\vec r\). The dot product captures “along the motion.”

• Path + displacement element
• Dot product meaning
• Units: joules (N·m)

For constant \(\vec F\): \(W=\vec F\cdot \Delta \vec r = F\,\Delta r \cos\theta\).

• Angle dependence
• Positive vs negative work
• Simple examples

Only the component of force parallel to motion does work: \(F_\parallel = F\cos\theta\).

• Decompose force
• Interpret signs
• Common misconceptions

If \(\theta=90^\circ\), \(W=0\). Example: centripetal force in uniform circular motion.

• Perpendicular component only
• Normal force examples
• Why speed can stay constant

Practice computing work with angles, signs, and simple paths.

• Dot-product work drills
• Sign (positive/negative) sets
• Perpendicular/zero-work checks
• Mixed multi-force examples
• Mini conceptual questions

Work is a mechanism for changing energy. This leads directly to kinetic energy and the work–energy theorem.

• Energy language
• Connection to ΔK
• Setup for energy methods