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Concepts Of Physics MCQ Edition [Volume 1]PhysicsCentre of Mass, Linear momentum, Collision

Determine the position of the centre of mass for a uniform plate that has semicircular inner and outer boundaries with radii R₁ and R₂ , as shown in the figure.

Options

  1. A4(R₁^2 + R₂^2) 3 (R₁ + R₂) above the centre
  2. B2(R₁^2 + R₁ R₂ + R₂^2) 3 (R₁ + R₂) above the centre
  3. C4(R₁^2 + R₁ R₂ + R₂^2) 3 (R₁ + R₂) above the centre
  4. D4(R₂^3 + R₁^3) 3 (R₂^2 + R₁^2) above the centre

Correct answer

C. 4(R₁^2 + R₁ R₂ + R₂^2) 3 (R₁ + R₂) above the centre

Step-by-step solution

Let the surface mass density of the uniform plate be . By symmetry, the centre of mass lies on the y-axis, perpendicular to the straight edge and passing through the centre. Consider an elemental semicircular ring of radius r and thickness dr . The mass of this elemental ring is dm = ( r dr) . The y-coordinate of the centre of mass of a semicircular ring of radius r is y_c = 2r . The y-coordinate of the centre of mass of the entire plate is given by: y_ cm = y_c dm dm Substituting the expressions for y_c and dm : y

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