(a)(i) State Hooke’s law. (ii) A spring has a length of 0.20 m when a mass of 0.30 kg hangs on it, and a length of 0.75 nm when a mass of 1.95 kg hangs on it. Calculate the: (i) force constant of the spring; (ii) length of the spring when it is unloaded. [g = 10m/s\(^2\)]
(b)(i) What is diffusion? (ii) State two factors that affect the rate of diffusion of a substance. (iii) State the exact relationship between the rate of diffusion of a gas and its density.
(c) A satellite of mass, m orbits the earth of mass. M with a velocity, v at a distance R from the centre of the earth. Derive the relationship between the period T, of orbit and R.
Explanation
(a)(i) Hooke's Law: This states that provided the elastic limit is not exceeded, the extension or compression of an elastic material is directly proportional to the applied force. (ii) Calculation of force constant (K): From Hooke's law:
Mg = Ke = k(l - lo)
0.3 x 10 = k(0.20- lo).. ..(i)
1.95x 10 = k(0.75 - lo) 1) ...(ii)
From (i) lo = \(\frac{3+0.20 k}{lo}\)
From (ii) 19.5 k[0.75 - 0.2 - 3/k]
: K = 30N/m
lo = 0.2 - 3/20
: lo = 0.2 - 0.1 = 0.1m
(b)(i) Definition of diffusion: This is the intimate mixing of substance due to random motion of their molecules/particles.
(b)(ii) Factors that affect the rate of diffusion: (i) Temperature of the environment. (ii) Density of the substance. (iii) Medium of diffusion. (iv) Concentration gradient.
(b)(iii) Relationship between rate of diffusion of a gas and its density:
Rate of diffusion = \(\frac{1}{βdensity of gas}\)
(c) Derivation of the relationship between the period T, of orbit and R:
\(\frac{MV^2}{R}\) = \(\frac{GMM}{R^2}\)
\(\frac{2ΟR}{T}\) = \(\frac{GM}{R}\)
\(T^2\) = \(\frac{4Ο^2R^3}{GM}\)
\(T^2\) Ξ± \(R^3\)