The centre of mass of solid hemisphere of radius 8 cm is x from the centre of the flat surface. Then value of x is __________.

Solution

Physics sir · 3 steps

Slice the hemisphere into thin discs stacked on the flat face, and the integral gives 3R/8.

1. Slice into discs. Put the flat face on the xy-plane with its centre at the origin. At height y, a thin disc of thickness dy has radius √(R² - y²), so its mass is dm = ρ π (R² - y²) dy.

   y_(cm) = (∫₀^R y ρπ (R² - y²) dy)/(2/3π R³ ρ) = (ρπ R⁴/4)/(2/3π R³ ρ) = 3R/8

2. Put in numbers. With R = 8 cm, x = (3 × 8)/8 = 3 cm.
3. So. The centre of mass lies on the symmetry axis, 3 cm from the centre of the flat surface.

The common slip: 4. Using the hollow hemisphere result R/2 gives 4 cm, not 3. This hemisphere is solid, so use 3R/8.

The idea to keep. Remember the pair: a solid hemisphere has its centre of mass at 3R/8 from the flat face, a hollow one at R/2. Solid puts more mass near the base, so its value is smaller.

Answer from the NTA final answer key. Solution by PrepWiser.