Physics 2024 Jamb Examination

Translational motion occurs when every point of a body moves through the same distance in the same time interval along a path. When a body moves from one position P to another position Q in a straight line or along a specific path without any part of it rotating about a fixed axis, it is undergoing translation. Rotational motion involves turning around a fixed point, circular motion involves following a circular path, and simple harmonic motion is a repetitive back-and-forth movement through a central equilibrium point. Since the description specifies a change from one point to another in linear motion, it is strictly translational.

Work done in a thermodynamic system is mathematically defined as the product of pressure and the change in volume: \[ W = P \Delta V \] Boyle's Law states that for a fixed mass of gas at constant temperature, the pressure is inversely proportional to the volume, expressed as \(PV = k\). The product of pressure (P) and volume (V) has the same dimensions as work. In SI units, pressure is measured in Newtons per square metre (\(N/m^2\)) and volume in cubic metres (\(m^3\)). Multiplying them yields: \[ \frac{N}{m^2} \times m^3 = N \cdot m = \text{Joules (J)} \] Since Boyle's Law explicitly deals with the relationship between pressure and volume, it is the law most closely associated with the mechanical concept of work done by or on a gas.

Pure semiconductors, known as intrinsic semiconductors, have relatively low electrical conductivity at room temperature. Doping is the intentional introduction of tiny amounts of specific 'impurity' atoms into the crystal lattice of a semiconductor to modify its electrical properties. For example, adding pentavalent atoms (like phosphorus) creates an n-type semiconductor with extra free electrons, while adding trivalent atoms (like boron) creates a p-type semiconductor with 'holes'. These extra charge carriers significantly increase the conductivity of the material, making it useful for creating electronic components like diodes and transistors. Rectification is the conversion of AC to DC, and emission refers to the release of electrons from a surface.

Glass is a poor conductor of heat. When boiling water is poured into a thick glass tumbler, the inner surface in direct contact with the water absorbs heat and expands immediately. Because glass conducts heat slowly, the heat does not reach the outer surface quickly enough for it to expand at the same rate. This creates a significant temperature gradient across the thickness of the glass, leading to differential or uneven expansion. The resulting internal stress between the rapidly expanding inner layer and the stationary outer layer becomes great enough to exceed the tensile strength of the glass, causing it to fracture or crack. Thin-walled glass tumblers are less likely to crack because the heat is transferred to the outer surface more rapidly, resulting in more uniform expansion.

The quantity of heat energy released by a substance when it cools is determined by its mass, its specific heat capacity, and the magnitude of the temperature change. \[ Q = mc\Delta\theta \]Where: \(m\) = mass of iron = 60 g = 0.06 kg (converted to kilograms for unit consistency with \(c\)) \(c\) = specific heat capacity of iron = 460 J kg\(^{-1}\) K\(^{-1}\) \(\Delta\theta\) = change in temperature = \(75^\circ\text{C} - 35^\circ\text{C} = 40 \, \text{K}\) Substituting these values into the formula: \[ Q = 0.06 \times 460 \times 40 \] \[ Q = 27.6 \times 40 \] \[ Q = 1104 \, \text{J} \]The iron releases 1104 Joules of thermal energy into its surroundings as its temperature drops by 40 degrees Kelvin.

According to Planck's quantum theory, the energy of a photon is directly proportional to the frequency of the electromagnetic radiation. This relationship is expressed by the equation: \[ E = hf \]Where: \(E\) = energy of the light (photon) in Joules \(h\) = Planck's constant = \(6.63 \times 10^{-34} \, \text{Js}\) \(f\) = frequency of the light = \(2.0 \times 10^{15} \, \text{Hz}\) Calculating the energy: \[ E = (6.63 \times 10^{-34}) \times (2.0 \times 10^{15}) \] \[ E = 13.26 \times 10^{-34+15} \] \[ E = 13.26 \times 10^{-19} \] \[ E = 1.326 \times 10^{-18} \, \text{J} \]Rounding to three significant figures as provided in the options gives \(1.33 \times 10^{-18} \, \text{J}\).

For a fixed mass of gas at constant volume, the pressure exerted by the gas is directly proportional to its absolute temperature. This is known as the Pressure Law or Gay-Lussac's Law. \[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]Where: \(P_1\) = initial pressure = 40 mmHg \(T_1\) = initial temperature = 298 K \(T_2\) = final temperature = 348 K \(P_2\) = final pressure Rearranging the formula to find the final pressure: \[ P_2 = \frac{P_1 \times T_2}{T_1} \] \[ P_2 = \frac{40 \times 348}{298} \] \[ P_2 = \frac{13920}{298} \] \[ P_2 \approx 46.7114 \, \text{mmHg} \]The final pressure is approximately 46.71 mmHg, showing an increase in pressure as the kinetic energy of the gas molecules increases with temperature.

The speed of a wave is the product of its frequency and its wavelength. Firstly, we identify the wavelength (\(\lambda\)). The distance between two successive crests is the definition of one wavelength: \[ \lambda = 0.25 \, \text{m} \]Next, we determine the frequency (\(f\)). Frequency is the number of complete oscillations per second. Since the particle makes four complete oscillations in one second: \[ f = 4 \, \text{Hz} \]Now, we calculate the wave speed (\(v\)) using the wave equation: \[ v = f\lambda \] \[ v = 4 \times 0.25 \] \[ v = 1.0 \, \text{ms}^{-1} \]Thus, the wave travels through the water at a speed of 1 metre per second.