Centre of Mass

CM of discrete and continuous bodies.

Concept

Centre of Mass is a foundational idea in this chapter. The defining relationship is:

$\vec R_{\rm cm} = (1/M)\int \vec r\,dm$

CM of standard bodies: rod at midpoint; triangle at centroid; hemisphere shell at 3R/8 from base.

Worked Application

Apply the formula to a typical problem from your textbook. Identify the knowns, plug into the equation, and check that the units work out. Always sanity-check the magnitude before accepting an answer.

Common Confusions

• Confusing the variables in the formula — re-read the symbol definitions in the chapter.
• Forgetting unit conversions (cm vs m, g vs kg).
• Ignoring sign conventions where direction matters.

Key Takeaways

• Master the formula $\vec R_{\rm cm} = (1/M)\int \vec r\,dm$ and one canonical example.
• See the chapter index for full derivations, traps and exam-grade problems.