Centre of mass by integration — A/L Combined Mathematics

How do I practise Centre of mass by integration?

1 question with a worked solution

Check before you answer: Mass Element Definition: Expressing dm correctly in terms of density (linear λ, surface σ, or volume ρ) and differential coordinates (e.g., dm = ρ π y² dx).

1. Using integration, at what distance from the flat base center is the center of mass of a uniform solid hemisphere of radius r located along its axis of symmetry?
   1. A) 3r / 8
   2. B) 3r / 4
   3. C) 4r / (3π)
   4. D) r / 2
   Check my answer

   Correct answer: A) 3r / 8

   Slicing the hemisphere into thin disks of thickness dy at distance y from the base: disk volume dV = π (r² - y²) dy. Total volume V = (2/3) π r³. The center of mass is y_cm = [∫₀^r y π (r² - y²) dy] / [(2/3) π r³] = [r⁴/2 - r⁴/4] / [(2/3) r³] = (r⁴/4) / (2r³/3) = 3r / 8.

   Show worked solution

   Step 1: Solid hemisphere integration: y_cm = ∫ y dV / V. Step 2: y_cm = [π r⁴ / 4] / [(2/3) π r³] = 3r / 8. Answer: 3r / 8.
   Modelled on: the A/L Combined Mathematics examination pattern

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Where does Centre of mass by integration fit in the syllabus?

Year 2 · Term 2