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JEE Main 2023 (31 Jan, Shift 2) · Moment of Inertia & Radius of Gyration

Moment of Inertia & Radius of Gyration question from JEE Main 2023 (31 Jan, Shift 2)

Q106JEE Main 2023 · 31 JAN · SHIFT 2Medium

Moment of Inertia & Radius of Gyration

Two discs of same mass and different radii are made of different materials such that their thicknesses are 1 cm1\text{ cm} and 0.5 cm0.5\text{ cm} respectively. The densities of materials are in the ratio 3:53:5. The moment of inertia of these discs respectively about their diameters will be in the ratio of x6\frac{x}{6}. The value of xx is ________.

Numerical answer type. Work it out, then open the answer.

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Answer: 5

  1. Thin disc about a diameter: I=14mR2I = \frac{1}{4}mR^2.
  2. Same mass: m=ρπR2t⇒R2=mπρtm = \rho \pi R^2 t \Rightarrow R^2 = \frac{m}{\pi \rho t}, so I∝1ρtI \propto \frac{1}{\rho t}.
  3. I1I2=ρ2t2ρ1t1=5×0.53×1=56\frac{I_1}{I_2} = \frac{\rho_2 t_2}{\rho_1 t_1} = \frac{5 \times 0.5}{3 \times 1} = \frac{5}{6}
  4. So x=5x = 5.
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