More practice questions

4. Q4. Four particles of 1 kg each sit at the corners of a square of side 1 m. Find the moment of inertia of the system about (a) an axis through the centre of the square perpendicular to its plane, (b) an axis along one side and (c) an axis along a diagonal.

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   (a) Each particle is half a diagonal, 1/√2 m, from the centre, so I = 4 × 1 × ½ = 2 kg m^². (b) The two particles on the side are on the axis; the other two are 1 m away: I = 2 × 1 × 1^² = 2 kg m^². (c) Two particles are on the diagonal; the other two are 1/√2 m from it: I = 2 × 1 × ½ = 1 kg m^².

5. Q5. (Beyond NCERT.) Show by integration that a uniform rod of mass M and length L has I = ML^²/3 about a perpendicular axis through one end.

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   With the axis at x = 0, an element dx at x has mass (M/L) dx, so

   $$\begin{aligned} I &= \int_0^L x^{2}\,\frac{M}{L}\,dx \\ &= \frac{M}{L}\cdot\frac{L^{3}}{3} = \frac{ML^{2}}{3} \end{aligned}$$

   This is four times ML^²/12, the value about the middle: more of the rod is far from an axis at the end.

6. Q6. A flywheel is a uniform disc of mass 50 kg and radius 0.4 m, turning at 600 rpm about its axis. How much kinetic energy does it store?

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   I = ½ × 50 × 0.4^² = 4 kg m^², and ω = 2π × 600/60 = 20π, about 62.8 rad s^⁻¹. So K = ½ × 4 × (20π)^² = 800π^², about 7.9 × 10^³ J.