Physics 2023 Jamb Examination

This problem involves three distinct stages of heat absorption: heating the ice, melting the ice, and heating the resulting water.

1. Heat required to raise the temperature of ice from \(-10\,^{\circ}\text{C}\) to \(0\,^{\circ}\text{C}\):\[Q_1 = m \cdot c_{ice} \cdot \Delta\theta_1\]\[Q_1 = 0.02 \cdot 2100 \cdot (0 - (-10)) = 0.02 \cdot 2100 \cdot 10 = 420\,\text{J}\]2. Heat required to melt the ice at \(0\,^{\circ}\text{C}\) (latent heat of fusion):\[Q_2 = m \cdot L_f\]\[Q_2 = 0.02 \cdot 3.34 \times 10^5 = 6680\,\text{J}\]3. Heat required to raise the temperature of the resulting water from \(0\,^{\circ}\text{C}\) to \(10\,^{\circ}\text{C}\):\[Q_3 = m \cdot c_{water} \cdot \Delta\theta_2\]\[Q_3 = 0.02 \cdot 4200 \cdot (10 - 0) = 0.02 \cdot 4200 \cdot 10 = 840\,\text{J}\]The total heat required is the sum of these three values:\[Q_{total} = Q_1 + Q_2 + Q_3\]\[Q_{total} = 420 + 6680 + 840 = 7940\,\text{J}\]

A pyrometer is a type of remote-sensing thermometer used to measure the temperature of distant surfaces. It works by detecting the intensity of thermal radiation (infrared or visible light) emitted by the object. Unlike other thermometers listed, such as the platinum resistance thermometer (which uses electrical resistance) or the thermocouple (which uses the Seebeck effect at a junction of two metals), the pyrometer does not need to be in thermal contact with the object, making it ideal for measuring extremely high temperatures or moving parts.

To find the coefficient of kinetic friction (\(\mu_k\)), we analyze the forces acting on the box. Since the box moves at a constant speed, it is in dynamic equilibrium, meaning the net force in both the horizontal and vertical directions is zero.

First, resolve the applied force \(P\) into components:
Horizontal component: \(P_x = P \cos 30^{\circ} = 100 \cdot 0.866 = 86.6\,\text{N}\)
Vertical component (acting downward): \(P_y = P \sin 30^{\circ} = 100 \cdot 0.5 = 50\,\text{N}\)

Next, find the Normal force (\(N\)):\[N = \text{Weight} + P_y\]\[N = 400 + 50 = 450\,\text{N}\]The frictional force (\(f_k\)) must equal the horizontal pushing force for constant speed:\[f_k = P_x = 86.6\,\text{N}\]Finally, use the friction formula:\[\mu_k = \frac{f_k}{N}\]\[\mu_k = \frac{86.6}{450} \approx 0.192\]

A fundamental property of light is that its frequency remains constant when it moves from one medium to another. While the speed and wavelength change depending on the refractive index of the medium, the frequency depends only on the source. [Image showing light refraction and constant frequency across media]

We can calculate the frequency using the properties of light in a vacuum:\[c = f \cdot \lambda\]\[f = \frac{c}{\lambda}\]\[f = \frac{3.00 \times 10^8}{589 \times 10^{-9}}\]\[f \approx 5.09 \times 10^{14}\,\text{Hz}\]This frequency will be the same inside the fused quartz.

The period \(T\) of a simple pendulum is given by \(T = 2\pi \sqrt{\frac{L}{g}}\), which implies that \(T^2 \propto L\).

Let the initial length be \(L_1\) and the final length be \(L_2\). We are told \(L_2 = L_1 - 3\).\[\frac{T_1^2}{L_1} = \frac{T_2^2}{L_2}\]\[\frac{5.77^2}{L_1} = \frac{4.60^2}{L_1 - 3}\]\[\frac{33.2929}{L_1} = \frac{21.16}{L_1 - 3}\]\[33.2929(L_1 - 3) = 21.16L_1\]\[33.2929L_1 - 99.8787 = 21.16L_1\]\[12.1329L_1 = 99.8787\]\[L_1 = \frac{99.8787}{12.1329} \approx 8.23\,\text{m}\]Since the new length \(L_2 = L_1 - 3\):\[L_2 = 8.23 - 3 = 5.23\,\text{m}\]

To find the resultant of the three forces, we resolve each force into its horizontal ($x$) and vertical ($y$) components.

Sum of x-components ($\Sigma F_x$)\[ \Sigma F_x = 12\cos(32^{\circ}) - 16\cos(26^{\circ}) \]\[ \Sigma F_x = 10.18 - 14.38 = -4.20\,\text{N} \]
\textbf{Sum of y-components ($\Sigma F_y$):}\[ \Sigma F_y = 12\sin(32^{\circ}) + 16\sin(26^{\circ}) - 21 \]\[ \Sigma F_y = 6.36 + 7.01 - 21 = -7.63\,\text{N} \]
Resultant magnitude $R$:\[ R = \sqrt{(\Sigma F_x)^2 + (\Sigma F_y)^2} \]\[ R = \sqrt{(-4.20)^2 + (-7.63)^2} = \sqrt{17.64 + 58.22} = \sqrt{75.86} \approx 8.71\,\text{N} \]
The angle $\theta$ with the x-axis:\[ \tan \theta = \frac{|\Sigma F_y|}{|\Sigma F_x|} = \frac{7.63}{4.20} \approx 1.817 \]\[ \theta = \tan^{-1}(1.817) \approx 61^{\circ} \]

The wave number (\(k\)) is defined as the number of radians per unit distance and is mathematically expressed as:\[k = \frac{2\pi}{\lambda}\]First, we need to find the wavelength (\(\lambda\)) using the wave speed (\(v\)) and frequency (\(f\)):\[v = f \cdot \lambda\]\[\lambda = \frac{v}{f} = \frac{20}{16} = 1.25\,\text{m}\]Now, substitute the wavelength into the formula for the wave number:\[k = \frac{2\pi}{1.25} = \frac{2 \cdot 3.14159}{1.25} \approx 5.02\]Rounding to the nearest whole number, we get 5.

The patient suffers from hypermetropia (long-sightedness) because their near point is further away than the standard 25 cm. To correct this, we need a lens that takes an object at 25 cm (the desired near point) and forms a virtual image at 50 cm (the patient's actual near point).

Using the lens formula:\[\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\]Where:
\(u = 25\,\text{cm}\) (object distance)
\(v = -50\,\text{cm}\) (image distance is negative because it is virtual)
\[\frac{1}{f} = \frac{1}{25} + \frac{1}{-50}\]\[\frac{1}{f} = \frac{2 - 1}{50} = \frac{1}{50}\]\[f = 50\,\text{cm} = 0.5\,\text{m}\]The power \(P\) in diopters is the reciprocal of the focal length in metres:\[P = \frac{1}{f} = \frac{1}{0.5} = 2\,\text{diopters}\]

Heat capacity (\(C\)) is the amount of heat required to raise the temperature of the entire mass of a substance by one Kelvin. Because it depends on the amount of matter present (mass), it is an extensive property. Therefore, statement (i) is false. The S.I. unit for heat capacity is \(\text{JK}^{-1}\). Statement (iv) is false because \(\text{JKg}^{-1}\) (or more correctly \(\text{JKg}^{-1}\text{K}^{-1}\)) is the unit for *specific* heat capacity, which is an intensive property. Thus, the incorrect statements are (i) and (iv).

A pinhole camera is a simple camera without a lens and with a single small aperture. It forms an inverted image of an object on the opposite side of the box. This occurs because light travels in straight lines, a phenomenon known as the rectilinear propagation of light. Since light from the top of an object can only travel in a straight line through the pinhole to reach the bottom of the screen, and light from the bottom travels to the top, the resulting image is inverted.