1. a ball thrown vertically upwards attains a velocity v after experiencing a retarding force f = λv. determine the dimension of λ. 3
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
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) 3
4. a certain simple pendulum has a period of 1.75 seconds. determine its period when the length is quadruple? 3
5. state: (a) three uses of optical fibers in manufacturing industries; (b) explain the basic principle of operation of a fiber optic sensor. 3
6. in a tabular form, state three differences between crystalline solids and amorphous solids. 3
7. provide three evidence that support the molecular nature of matter. 3

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 Range Formula
The range $ R $ of a projectile is given by $ 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’s orbit radius $ r = $ Earth’s radius $ + $ altitude $ = 6.4\times10^{3}\ km + 2.5\times10^{3}\ km = 8.9\times10^{3}\ km $.
## Step 2: Find Orbit Circumference
Circumference $ C = 2\pi r = 2\times\pi\times8.9\times10^{3}\ km\approx5.59\times10^{4}\ km $.
## Step 3: Calculate Speed
Time $ t = 15.5\ h $, speed $ v=\frac{C}{t}=\frac{5.59\times10^{4}\ km}{15.5\ h}\approx3606\ km/h\approx1.002\ km/s $.
# Answer: The orbital speed is approximately $ \boldsymbol{3600\ km/h} $ (or $ 1.0\ km/s $, more accurately $ \approx3606\ km/h $).

### Question 4
# Explanation:
## Step 1: Recall Pendulum Period Formula
The period $ T $ of a simple pendulum is $ T = 2\pi\sqrt{\frac{l}{g}} $, so $ T\propto\sqrt{l} $. Let initial length be $ 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**: Transmit control signals/operational data between machines.
2. **Sensing/Monitoring**: In fiber - optic sensors to detect temperature/strain in manufacturing processes.
3. **Quality Control**: In non - destructive testing (e.g., inspecting internal structures of components).
# Answer:
1. Transmit operational data between manufacturing machines.
2. Used in fiber - optic sensors for process monitoring (e.g., temperature/strain).
3. Aid in non - destructive testing for quality control.

### Question 5 (b)
# Brief Explanations:
Fiber optic sensors work on the principle of **modulation of light properties** (intensity, phase, wavelength, polarization) as light travels through an optical fiber. When a physical quantity (e.g., temperature, pressure, strain) acts on the fiber or the light, a property of the light changes. For example, in an intensity - based sensor, bending/stretching the fiber (due to external force) changes light transmission, which is detected and related to the measured quantity.
# Answer: Fiber optic sensors operate by modifying a light property (intensity/phase/wavelength) as light propagates through the fiber, with the change correlating to the measured physical quantity (e.g., temperature, strain).

### Question 7
1. **Diffusion**: Mixing of gases (e.g., perfume spreading) or solutes in solvents shows particles move and interact, supporting molecular nature.
2. **Brownian Motion**: Random movement of colloidal particles (e.g., pollen in water) due to collision with invisible molecules, proving molecular existence and motion.
3. **Compressibility of Gases**: Gases can be compressed as molecules have empty space between them, indicating discrete molecular structure.

# Answer:
1. Diffusion (e.g., perfume spreading) shows molecular movement.
2. Brownian motion (e.g., pollen in water) shows molecular collision - induced particle motion.
3. Gas compressibility shows molecular spacing (empty space between molecules).

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 1

Step 1: Recall Dimensional Analysis

Step 2: Substitute Dimensions

Step 1: Identify Projectile Range 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