Elasticity

(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.

Answer Details

(a)

1. Hooke's law: states that the force exerted by a spring is directly proportional to the amount it is stretched or compressed, provided that the elastic limit is not exceeded. Mathematically, it can be represented as F = kx, where F is the force applied, k is the force constant, and x is the displacement of the spring from its equilibrium position.
2. Calculation of force constant and length of the unloaded spring: First, we can calculate the force constant of the spring using the equation F = kx. Using the first length and mass, the force can be calculated as F = mg = 0.30 kg * 10 m/s² = 3 N. Substituting these values in the equation, we get 3 = k * 0.20 m. Solving for k, we get k = 15 N/m. Next, we can calculate the length of the spring when it is unloaded using the equation F = kx, where F = 0 as there is no mass hanging from the spring. Thus, kx = 0, and x = 0, which means the spring is at its equilibrium length when there is no force applied to it.

(b)

1. Diffusion: is the movement of particles or substances from an area of high concentration to an area of low concentration. This occurs until the concentration of the particles is equal throughout the system.
2. Factors affecting the rate of diffusion: Two factors that affect the rate of diffusion are the concentration gradient and the temperature. The concentration gradient refers to the difference in concentration of the particles, with a larger difference leading to a faster rate of diffusion. Higher temperature means that the particles have more energy, leading to a faster rate of diffusion.
3. Relationship between rate of diffusion and gas density: The exact relationship between the rate of diffusion of a gas and its density is not well defined. However, it can be said that the rate of diffusion of a gas is inversely proportional to its density.

(c) The relationship between the period T of orbit and R:

The equation for the period, T, of an object in orbit can be represented as T² = 4π² * R³ / (G * (M + m)), where G is the gravitational constant and M and m are the masses of the earth and satellite, respectively.

A piano wire 50 cm long has a total mass of 10 g and its stretched with a tension of 800 N. Find the frequency of the wire when it sounds its third overtone note.

Answer Details

T=800N; I=50cm=0.5m,

m=10g=0.01kg

fundamental freq: fo ${\mathrm{<br>}}_{}$ =?

fo ${\mathrm{<br>}}_{}$ = 121√Tμ $\frac{1}{21}<br>$

μ =m1 $\frac{\mathrm{<br>}}{}$=0.010.5 $\frac{0.01}{0.5}$

⇒ fo ${\mathrm{<br>}}_{}$ =12×0.5 $\frac{1}{2\times 0.5}$√8000.02 $\frac{800}{0.02}$

                  fo ${\mathrm{<br>}}_{}$  ⇒√ 40,000

⇒1st overtone = 2fo ${\mathrm{<br>}}_{}$ =2×200 = 400Hz

⇒2nd overtone =3fo ${\mathrm{<br>}}_{}$ =3×200=600Hz

∴3rd over tone= 4fo ${\mathrm{<br>}}_{}$  =4×200=800Hz