1. a ball thrown vertically upwards attains a velocity v after experiencing a retarding force f = λv. determine the dimension of λ.
2. a particle is projected from the ground with a speed of 40 m s⁻¹ at an angle of 60° with the horizontal. calculate the distance midway between the point of projection and the point at which it landed which is the projection plane.
3. a satellite orbits at 2.5 × 10⁴ km above the earth taking 15.5 hours to complete its orbit. at what speed will the orbit take place? (radius of the earth = 6.4 × 10³ km)
4. a certain simple pendulum has a period of 1.75 seconds. determine its period when the length is quadruple?
5. state: (a) three uses of optical fibers in manufacturing industries; (b) explain the basic principle of operation of a fiber optic sensor.
6. in a tabular form, state three differences between crystalline solids and amorphous solids.
7. provide three evidence that support the molecular nature of matter.

Question

Accepted Answer

### Question 1
# Explanation:
## Step 1: Recall Dimensional Analysis
Force $ F $ has dimensions $ [MLT^{-2}] $, velocity $ v $ has dimensions $ [LT^{-1}] $. Given $ F = \lambda v $, so $ \lambda=\frac{F}{v} $.
## Step 2: Substitute Dimensions
Substitute dimensions of $ F $ and $ v $: $ \lambda=\frac{[MLT^{-2}]}{[LT^{-1}]} = [ML^{0}T^{-1}] $.
# Answer: The dimension of $ \lambda $ is $ \boldsymbol{[ML^{0}T^{-1}]} $ (or $ [MLT^{-1}] $ with $ L^0 $ implicit).

### Question 2
# Explanation:
## Step 1: Identify Projectile Motion Formula
Range $ R $ of projectile: $ R=\frac{u^{2}\sin2\theta}{g} $, where $ u = 40\ m/s $, $ \theta = 60^\circ $, $ g = 9.8\ m/s^2 $.
## Step 2: Calculate $ \sin2\theta $
$ 2\theta = 120^\circ $, $ \sin120^\circ=\frac{\sqrt{3}}{2}\approx0.866 $.
## Step 3: Substitute Values
$ R=\frac{(40)^{2}\times0.866}{9.8}=\frac{1600\times0.866}{9.8}=\frac{1385.6}{9.8}\approx141.4\ m $.
# Answer: The range is approximately $ \boldsymbol{141\ m} $ (or $ 141.4\ m $).

### Question 3
# Explanation:
## Step 1: Find Orbit Radius
Satellite altitude $ h = 2.5\times10^{4}\ km $, Earth radius $ R = 6.4\times10^{3}\ km $. Orbit radius $ r=R + h=6.4\times10^{3}+2.5\times10^{4}=3.14\times10^{4}\ km $.
## Step 2: Find Orbit Circumference
Circumference $ C = 2\pi r=2\times\pi\times3.14\times10^{4}\approx1.97\times10^{5}\ km $.
## Step 3: Calculate Speed
Time $ t = 15.5\ h $, speed $ v=\frac{C}{t}=\frac{1.97\times10^{5}}{15.5}\approx12709.68\ km/h\approx1.27\times10^{4}\ km/h $ (or convert to $ m/s $: $ \approx3530\ m/s $).
# Answer: The speed is approximately $ \boldsymbol{1.27\times10^{4}\ km/h} $ (or $ 3530\ m/s $).

### Question 4
# Explanation:
## Step 1: Recall Pendulum Period Formula
Period $ T = 2\pi\sqrt{\frac{l}{g}} $, so $ T\propto\sqrt{l} $. Let initial length $ l_1 $, period $ T_1 = 1.75\ s $; final length $ l_2 = 4l_1 $, period $ T_2 $.
## Step 2: Relate Periods
$ \frac{T_2}{T_1}=\sqrt{\frac{l_2}{l_1}}=\sqrt{4}=2 $, so $ T_2 = 2\times T_1 $.
## Step 3: Calculate Final Period
$ T_2 = 2\times1.75 = 3.5\ s $.
# Answer: The new period is $ \boldsymbol{3.5\ s} $.

### Question 5 (a)
# Brief Explanations:
1. **Data Transmission**: In manufacturing, optical fibers transmit control signals, sensor data (e.g., from temperature/position sensors) between machines, enabling precise automation.
2. **Quality Control (Imaging)**: Used in endoscopes for internal inspection of machinery (e.g., pipes, sealed components) to detect defects without disassembly.
3. **Laser Cutting/Processing**: Guide high - power laser beams for cutting, welding, or marking in manufacturing (fiber - optic - delivered lasers offer precision and flexibility).

# Answer:
1. Transmit control/sensor data in automation.
2. Internal machinery inspection (e.g., endoscopes).
3. Guide laser beams for cutting/welding.

### Question 5 (b)
# Brief Explanations:
A fiber - optic sensor works on the **principle of modulation of light** (intensity, phase, wavelength, or polarization) as it travels through an optical fiber. When a physical quantity (e.g., temperature, pressure, strain, or presence of a substance) acts on the fiber or the light within it, the light’s properties change. For example, in an intensity - based sensor: if a fiber is bent/stretched (due to external force) or a substance absorbs/scatters light (in a sensing region), the light intensity exiting the fiber changes. This change is detected and correlated to the magnitude of the physical quantity. In phase - based sensors (e.g., interferometric), external factors (like temperature - induced refractive index change) alter the optical path length, causing a phase shift between two light waves, which is measured.

# Answer: A fiber - optic sensor operates by modulating light (intensity/phase/wavelength/polarization) as it travels through the fiber, with external physical quantities (e.g., temp, pressure, strain) altering light properties, which are detected and related to the quantity’s magnitude.

### Question 6
| Property | Crystalline Solids | Amorphous Solids |
|--|--|--|
| **Order of Arrangement** | Atoms/molecules are arranged in a long - range, repeating (periodic) lattice structure. | Atoms/molecules have short - range order (no long - range periodic arrangement). |
| **Melting Point** | Have a sharp, definite melting point (e.g., ice melts at $ 0^\circ C $). | Do not have a sharp melting point; they soften over a range of temperatures (e.g., glass softens gradually). |
| **Heat of Fusion** | Have a definite heat of fusion (energy required to melt a unit mass at melting point). | Do not have a definite heat of fusion (since no sharp melting point). |
| **Anisotropy/Isotropy** | Often anisotropic (physical properties like electrical conductivity, refractive index vary with direction). | Usually isotropic (physical properties are same in all directions). |

(Note: Any three differences can be chosen. Here, order, melting point, and anisotropy/isotropy are used as three.)

### Question 7
1. **Brownian Motion**: Observation of pollen grains (or smoke particles) in water under a microscope shows random, erratic movement. This is due to the bombardment of the visible particles by invisible water molecules, providing evidence for the existence and motion of molecules.
2. **Diffusion**: When a drop of ink is added to water, the ink spreads throughout the water over time. This is because ink molecules and water molecules move and intermix, indicating that molecules are in constant motion and have space between them.
3. **Compressibility of Gases**: Gases can be compressed (e.g., air in a syringe). This is because there are large intermolecular spaces between gas molecules, which can be reduced by applying pressure, supporting the idea that matter is made of molecules with spaces between them.
4. **Evaporation and Cooling**: When a liquid evaporates (e.g., alcohol on the skin), it cools the surface. This is because the more energetic (faster - moving) liquid molecules escape into the air, leaving behind slower - moving (cooler) molecules, showing that molecules have different kinetic energies (a molecular property).

(Any three of these can be used. Here, Brownian motion, diffusion, and compressibility of gases are used.)

# Answer:
1. Brownian motion (pollen in water: random movement from molecular bombardment).
2. Diffusion (ink in water: intermixing of molecules).
3. Compressibility of gases (large intermolecular spaces).

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 1

Step 1: Recall Dimensional Analysis

Step 2: Substitute Dimensions

Step 1: Identify Projectile Motion Formula

Step 2: Calculate \( \sin2\theta \)

Step 3: Substitute Values

Step 1: Find Orbit Radius

Step 2: Find Orbit Circumference

Step 3: Calculate Speed

Answer:

Question 2