More practice questions

4. Q4. An L-shaped plate is made of three squares of side 1 m: one with its corners at (0, 0) and (1, 1), one to its right and one above it. NCERT Example 6.3 (page 98) takes all three of mass 1 kg and finds the centre of mass at (5/6 m, 5/6 m). Find it instead when the squares have masses of 1 kg, 2 kg and 3 kg, in that order.

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   Each square acts as a point mass at its own centre: (0.5, 0.5), (1.5, 0.5) and (0.5, 1.5). The total mass is 6 kg.

   X = (1 × 0.5 + 2 × 1.5 + 3 × 0.5)/6 = 5/6 m, about 0.833 m.

   Y = (1 × 0.5 + 2 × 0.5 + 3 × 1.5)/6 = 6/6 = 1 m.

   The centre of mass is at (5/6 m, 1 m): pulled upwards, towards the heaviest square.

5. Q5. (Beyond NCERT.) A rod of length L lies along the x-axis from x = 0 to x = L. It is not uniform: its mass per unit length is λ = kx, where k is a constant. Find its centre of mass.

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   An element of length dx at x has mass dm = kx dx. Then

   $$\begin{aligned} X &= \frac{\int_0^L x\,(kx)\,dx}{\int_0^L kx\,dx} \\ &= \frac{kL^{3}/3}{kL^{2}/2} = \frac{2L}{3} \end{aligned}$$

   The centre of mass is at 2L/3 from the light end, nearer the heavier end, as expected.

6. Q6. From a uniform square plate of side 2a, a square of side a is cut from one corner. Taking the opposite corner as the origin and the sides as axes, locate the centre of mass of the remaining plate.

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   The full plate (mass M) has its centre at (a, a). The removed square has a quarter of the area, so mass M/4, at (3a/2, 3a/2). Treating it as negative mass:

   $$\begin{aligned} X &= \frac{Ma - \tfrac{M}{4}\cdot\tfrac{3a}{2}}{\tfrac{3M}{4}} \\ &= \frac{5Ma/8}{3M/4} = \frac{5a}{6} \end{aligned}$$

   By symmetry Y is the same, so the centre of mass is at (5a/6, 5a/6). With a = 1 m this is NCERT’s L-shape, and the answer matches its (5/6 m, 5/6 m).