The center of mass of a thin rectangular plate (fig - x ) with sides of length a and b, whose mass per unit area (σ) varies as σ=(σ₀ x)/(a b) (where σ₀ is a constant), would be __________.

Solution

Physics sir · 3 steps

Density grows only with x, so the plate is heavier on the right but still balanced top to bottom.

1. Take vertical strips. Cut the plate into thin vertical strips of width dx at distance x from O. Every point in a strip has the same density, so its mass is dm = (σ₀ x)/ab × b dx = σ₀/a x dx.

   x_(cm) = (∫₀ᵃ x dm)/(∫₀ᵃ dm) = (∫₀ᵃ x² dx)/(∫₀ᵃ x dx) = (a³/3)/(a²/2) = 2a/3

2. The y-coordinate. σ does not depend on y. Each strip is uniform along its length, so its own centre is at y = b/2. That makes y_(cm) = b/2 for the whole plate.
3. So. The centre of mass is at (2a/3, b/2), which is option (b).

The idea to keep. Check which coordinate the density depends on. Only that coordinate shifts away from the middle. Along any direction where the density stays uniform, the centre of mass sits at the midpoint.

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