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4. Q4. The speed v of sound in a medium depends on its bulk modulus B, [M L^⁻¹ T^⁻²], and its density ρ. Use dimensions to find the form of v.

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   Assume v = k B^x ρ^y: [L T^⁻¹] = [M L^⁻¹ T^⁻²]^x [M L^⁻³]^y. Mass: 0 = x + y. Time: −1 = −2x, so x = ½ and y = −½. Length: −x − 3y = −½ + 3/2 = 1, as it should be. So v = k√(B/ρ), the form of the speed of sound.

5. Q5. Convert G = 6.67 × 10^⁻¹¹ N m^² kg^⁻² into CGS units, cm^³ s^⁻² g^⁻¹ (NCERT Exercise 1.2(d), page 10).

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   [G] = [M^⁻¹ L^³ T^⁻²], so n₂ = 6.67 × 10^⁻¹¹ × (1 kg/1 g)^⁻¹ × (1 m/1 cm)^³ × (1 s/1 s)^⁻² = 6.67 × 10^⁻¹¹ × 10^⁻³ × 10^⁶ = 6.67 × 10^⁻⁸. So G = 6.67 × 10^⁻⁸ cm^³ s^⁻² g^⁻¹.

6. Q6. The frequency ν of a vibrating string depends on its length l, its tension F and its mass per unit length μ. Find the form of ν.

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   Assume ν = k l^x F^y μ^z: [T^⁻¹] = [L]^x [M L T^⁻²]^y [M L^⁻¹]^z. Time: −1 = −2y, so y = ½. Mass: 0 = y + z, so z = −½. Length: 0 = x + y − z, so x = −1. Hence ν = (k/l)√(F/μ); for the lowest note of a string fixed at both ends, theory gives k = ½.