2023 - JAMB Physics Past Questions and Answers - page 3

Given values:
- Radius of the wire (\(r\)) = 0.2 mm = \(0.0002 \, m\) (convert to meters)
- Change in length (\(\Delta L\)) = 0.5% of the original length (\(L_0\))
- Load (\(F\)) = Weight (\(mg\)), where \(m\) is the mass and \(g\) is the acceleration due to gravity. Given \(g = 10 \, \text{m/s}^2\).

1. Convert radius to meters:
   \[ r = 0.0002 \, m \]

2. Calculate cross-sectional area (\(A\)) using the formula for the area of a circle:
   \[ A = \pi r^2 \]

3. Convert the percentage change in length to a decimal:
   \[ \text{Strain} = \frac{\Delta L}{L_0} = \frac{0.5\%}{100} \]

4. Calculate Young's Modulus (\(Y\)):
   \[ F = mg \]
   \[ Y = \frac{F/A}{\text{Strain}} \]

Let's plug in the values and calculate:

\[ r = 0.0002 \, \text{m} \]

\[ A = \pi \times (0.0002)^2 \]

\[ \text{Strain} = \frac{0.5\%}{100} \]

\[ F = 1.5 \, \text{kg} \times 10 \, \text{m/s}^2 \]

\[ Y = \frac{F/A}{\text{Strain}} \]

Now, calculate \(Y\) and compare it to the provided options.

The results of the calculations are as follows:

\[ A \approx 1.2566 \times 10^{-7} \, \text{m}^2 \]

\[ \text{Strain} = 0.005 \]

\[ F = 1.5 \, \text{kg} \times 10 \, \text{m/s}^2 = 15 \, \text{N} \]

\[ Y \approx \frac{15 \, \text{N} / 1.2566 \times 10^{-7} \, \text{m}^2}{0.005} \]

\[ Y \approx 2.3838 \times 10^{10} \, \text{N/m}^2 \]

Therefore, 

The Young's modulus for the material of the wire is approximately \(2.4 \times 10^{10} \, \text{N/m}^2\).

A pyrometer is a type of remote-sensing thermometer used to measure the temperature of a surface. It does this by measuring the thermal radiation or infrared energy being emitted from the object. Therefore, Pyrometer thermometer.

A pyrometer is a type of thermometer designed to measure high temperatures, especially those of objects emitting thermal radiation. Pyrometers work based on the principle of detecting the intensity and characteristics of the thermal radiation emitted by the object, and they are suitable for measuring temperatures in processes such as metalworking, furnaces, and other high-temperature environments.

The other options are not typically used for measuring temperature from thermal radiation emitted by objects. Platinum resistance thermometers and thermocouple thermometers are commonly used for lower to moderate temperature ranges, and the constant pressure gas thermometer is more suited for precise laboratory measurements.

The correct option is: (ii), (iii), and (iv) only. Rainbow formation involves the following phenomena:

(ii) Dispersion: Different colours of light are separated due to varying wavelengths.
(iii) Total internal reflection: Light is internally reflected within raindrops, contributing to the formation of a rainbow.
(iv) Refraction: Light undergoes bending as it enters and exits raindrops, contributing to the overall phenomenon.

The principle of the rectilinear propagation of light.

The pinhole camera operates based on the principle that light travels in straight lines. This principle is known as the rectilinear propagation of light. In a pinhole camera, light enters through a small aperture (pinhole) and travels in straight lines, forming an inverted image on the opposite side of the camera. The process is governed by the rectilinear propagation of light. Options A, C, and D do not specifically describe the principle behind the operation of a pinhole camera.

To determine the power of the corrective lens needed, you can use the lens formula:

\[ \text{Lens Power} = \frac{1}{\text{Focal Length}} \]

The focal length of the lens required to enable the eye to see an object clearly at a certain distance can be found using the lens formula:

\[ \frac{1}{\text{Focal Length}} = \frac{1}{\text{Image Distance}} - \frac{1}{\text{Object Distance}} \]

In this case, the near point of the patient's eye is the object distance, and the distance at which the patient wants to see clearly is the image distance.

Given:
- Object Distance (\(u\)) = 50.0 cm (near point)
- Image Distance (\(v\)) = 25.0 cm (desired distance)

Using the lens formula:

\[ \frac{1}{\text{Focal Length}} = \frac{1}{v} - \frac{1}{u} \]

\[ \frac{1}{\text{Focal Length}} = \frac{1}{25 \, \text{cm}} - \frac{1}{50 \, \text{cm}} \]

Now, calculate the values:

\[ \frac{1}{\text{Focal Length}} = \frac{2}{50} - \frac{1}{50} = \frac{1}{50} \]

So, the focal length (\(f\)) is 50.0 cm.

Now, use the lens power formula:

\[ \text{Lens Power} = \frac{1}{\text{Focal Length}} \]

\[ \text{Lens Power} = \frac{1}{50 \, \text{cm}} \]

Convert the units to diopters:

\[ \text{Lens Power} = \frac{1}{0.5 \, \text{m}} \]

\[ \text{Lens Power} = 2 \, \text{diopters} \]

Therefore, the correct answer is 2 diopters.

The force (\(F\)) acting on a charged particle in an electric field is given by Coulomb's Law:

\[ F = q \cdot E \]

where:
- \( F \) is the force,
- \( q \) is the charge of the particle,
- \( E \) is the electric field intensity.

Given values:
\[ q = 4.6 \times 10^{-5} \, C \]
\[ E = 3.2 \times 10^{4} \, V/m \]

Now, plug in these values into the formula:

\[ F = (4.6 \times 10^{-5}) \cdot (3.2 \times 10^{4}) \]

\[ F \approx 1.472 \, N \]

The correct option is A: 1.5 N

The force acting on the charge in the electric field is approximately \(1.472 \, N\), and the closest option is \(1.5 \, N\).

The magnitude of the resultant force (\(R\)) and the angle (\(\theta\)) with the x-axis can be determined using the components of the forces:

\[ F_x = 12 \cos 32^\circ - 16 \cos 26^\circ \]
\[ F_y = 12 \sin 32^\circ + 16 \sin 26^\circ - 21 \]

The magnitude of the resultant force is given by \( R = \sqrt{F_x^2 + F_y^2} \), and the angle \(\theta\) is found using \( \tan \theta = \frac{F_y}{F_x} \).

After calculation:
\[ R \approx 8.71 \, \text{N} \]
\[ \theta \approx 61^\circ \]

Therefore, the correct option is \(8.71 \, \text{N, 61}^\circ\).

The new length of the pendulum can be determined by setting up an equation based on the relationship between the period (\(T\)) and the length (\(L\)) of a simple pendulum. The formula \(T^2 \propto L\) is used, where \(T\) is the period.

Let the original length be \(L = x\) meters, and the new length be \(L - 3\) meters. The equation becomes:

\[
\frac{T_1^2}{L} = \frac{T_2^2}{L - 3}
\]

Substituting the given values:

\[
\frac{5.77^2}{x} = \frac{4.60^2}{x - 3}
\]

Solving for \(x\):

\[
33.29(x - 3) = 21.16x
\]

\[
33.29x - 99.87 = 21.16x
\]

\[
12.13x = 99.87
\]

\[
x = \frac{99.87}{12.13} \approx 8.23 \, \text{m}
\]

The new length of the pendulum is \(x - 3 \approx 5.23 \, \text{m}\).

Therefore, the correct option is: \(5.23 \, \text{m}\).

The relative density (also known as specific gravity) of a substance is the ratio of its density to the density of water. In this case, the relative density of alcohol (\(RD_{\text{alcohol}}\)) is given as 0.8.

The formula for relative density is:

\[ RD = \frac{\text{Density of substance}}{\text{Density of water}} \]

Given that the mass of the empty bottle is 19 g and when filled with water is 66 g, we can find the volume of water using the formula:

\[ \text{Density of water} = \frac{\text{Mass of water}}{\text{Volume of water}} \]

Now, let's find the volume of water (\(V_{\text{water}}\)):

\[ V_{\text{water}} = \frac{\text{Mass of water}}{\text{Density of water}} = \frac{66 \, \text{g}}{1 \, \text{g/cm}^3} \]

Since the density of water is approximately \(1 \, \text{g/cm}^3\).

Now, we can find the volume of alcohol (\(V_{\text{alcohol}}\)) that would have the same mass as the water, given that \(RD_{\text{alcohol}} = 0.8\):

\[ V_{\text{alcohol}} = \frac{V_{\text{water}}}{RD_{\text{alcohol}}} = \frac{66 \, \text{g}}{0.8} \]

Finally, calculate the mass of the bottle when filled with alcohol:

\[ \text{Mass}_{\text{alcohol}} = \text{Mass}_{\text{empty}} + \text{Mass}_{\text{water}} \]

\[ \text{Mass}_{\text{alcohol}} = 19 \, \text{g} + \text{Mass}_{\text{alcohol}} \]

Now, solve for the mass of the bottle when filled with alcohol.

Let's perform the calculations:

\[ V_{\text{water}} = \frac{66}{1} = 66 \, \text{cm}^3 \]

\[ V_{\text{alcohol}} = \frac{66}{0.8} = 82.5 \, \text{cm}^3 \]

\[ \text{Mass}_{\text{alcohol}} = 19 + \frac{66}{0.8} = 19 + 82.5 = 101.5 \, \text{g} \]

Therefore, the correct option is: 56.6 g

1. The resistances \(R_1\) and \(R_2\) in series result in a total series resistance (\(R_{\text{total, series}}\)):
   \[ R_{\text{total, series}} = R_1 + R_2 = 35 \, \text{k}\Omega + 40 \, \text{k}\Omega = 75 \, \text{k}\Omega \]

2. For resistors in parallel, the reciprocal of the total resistance (\(1/R_{\text{total, parallel}}\)) is the sum of the reciprocals of the individual resistances:
   \[ \frac{1}{R_{\text{total, parallel}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R} \]

3. Given that the equivalent resistance is \(25 \, \text{k}\Omega\), set up the equation:
   \[ \frac{1}{25} = \frac{1}{R_{\text{total, series}}} + \frac{1}{R} \]

4. Solve for \(R\):
   \[ \frac{1}{25} = \frac{1}{75} + \frac{1}{R} \]
   \[ \frac{2}{75} = \frac{1}{R} \]
   \[ R = \frac{75}{2} = 37.5 \, \text{k}\Omega \]

Therefore, the correct option is 37.5 kΩ