## PHY 001: MECHANICS AND PROPERTIES OF MATTER
**1. (a)** Mention the **THREE** types of modulus.
**(b)** Briefly explain the following: **[4 Marks]**
- i. Elasticity
- ii. Plasticity
- iii. Ductility
- iv. Malleability
**(c)** A particle is moving along the x-axis such that its position is given by:
**x = 4t³ − 16t + 12**
- i. Find the instantaneous velocity when t = 5s
- ii. Find the instantaneous acceleration when t = 5s
- iii. At what time is the particle stationary?
**[4½ Marks]**
**[Total = 10 Marks]**
---
**2. (a)**
- i. State the Archimedes' Principle.
- ii. List **FOUR** characteristics of pressure in a fluid. **[3 Marks]**
**(b)** Distinguish between a compressible fluid and an incompressible fluid. **[2 Marks]**
**(c)** Two divers M and N are at a depth of 60 m and 80 m respectively below the water surface of a sea. The pressure on M is P₁ and the pressure on N is P₂. If the atmospheric pressure is equivalent to 20 m of water, find the value of P₂/P₁. **[2 Marks]**
**(d)** Differentiate between conservative force and non-conservative force, giving one example each. **[3 Marks]**
**[Total = 10 Marks]**
---
## PHY 002: HEAT, WAVES AND OPTICS
**3. (a)** State Huygens' principle. **[1 Mark]**
**(b)**
- i. Explain the term "electromagnetic spectrum". List any **THREE** of its components.
- ii. In a Young's slit experiment, the separation between the first and the fifth bright fringes is 2.5 mm when the wavelength used is 4.5 × 10⁻⁷ m. The distance from the slits to the screen is 0.9 m. Calculate the separation of the two slits. **[4 Marks]**
**(c)**
- i. Define internal energy.
- ii. State the second law of thermodynamics. **[2 Marks]**
**(d)**
- i. Using the first law of thermodynamics, write expressions for adiabatic and isochoric processes.
- ii. State **FOUR** factors which affect heat loss by convection. **[3 Marks]**
**[Total = 10 Marks]**
---
**4. (a)** State **TWO** similarities and **TWO** differences between the image formed by a converging mirror and a converging lens. **[2 Marks]**
**(b)**
- i. Mention any **TWO** uses each of plane mirror, concave mirror and convex mirror.
- ii. Copy and complete the table below for the image formed by a concave mirror for different positions of the object:
| Position of Object | Position of Image | Size of Image | Nature of Image |
|---|---|---|---|
| At infinity | | | |
| At C | | | |
| Between C and F | | | |
| At F | | | |
| Between F and P | | | |
**[8 Marks]**
**[Total = 10 Marks]**
---
## PHY 003: ELECTRICITY AND MAGNETISM
**5. (a)**
- i. Explain what is meant by relative permittivity.
- ii. State **TWO** physical desirable properties in a material considered for dielectric in a capacitor. **[3 Marks]**
**(b)** A long magnet is removed from the centre of a coil of 30 turns. The speed of the magnet is controlled to maintain an induced e.m.f. of 80 μV across the coil. Removing the magnet in this way takes 3 minutes. Calculate the change of flux through the coil. **[3 Marks]**
**(c)** An electron enters the region of a uniform electric field as shown in Figure 1 below, with v₀ = 3.00 × 10⁶ m/s and E = 200 N/C. The horizontal length of each plate is l = 0.100 m.
- i. Find the acceleration of the electron while it is in the electric field.
- ii. Assuming the electron enters the field at time t = 0, find the time at which it leaves the field. (mass of an electron = 1.67 × 10⁻³¹ kg, charge of an electron = 1.60 × 10⁻¹⁹ C)
**[4 Marks]**
**[Total = 10 Marks]**
---
**6. (a)** How can the motion of a moving charged particle be used to distinguish between a magnetic field and an electric field? **[2 Marks]**
**(b)** Consider the circuit diagram in Figure 2. Calculate the current in each resistor. **[5 Marks]**
*(Circuit diagram with two voltage sources: 24.0 V and 12.0 V, with resistors 80.0 Ω, 120.0 Ω, and 180.0 Ω)*
**(c)** A proton is moving in a circular orbit of radius 14 cm in a uniform 0.35 T magnetic field perpendicular to the velocity of the proton. Determine the speed of the proton. **[3 Marks]**
---
## PHY 004: MODERN PHYSICS
**7. (a)** Calculate the de Broglie wavelength for a particle moving with a speed of 2.20 × 10⁶ m/s if the particle is:
- i. an electron
- ii. a proton
- iii. a 200g bullet **[5 Marks]**
**(b)**
- i. Give the properties of α-, β- and γ-radiation in terms of charge, mass, ionizing effect and field effect.
- ii. Explain the effect of temperature and pressure on the rate of disintegration of a radioactive nucleus. **[7 Marks]**
**[Total = 10 Marks]**
**(c)**
- i. State Heisenberg's Uncertainty principle.
- ii. List **FOUR** practical applications of X-rays. **[3 Marks]**
**(d)** A photon has a wavelength of 1Å. Calculate the:
- i. energy of the photon in electron volts
- ii. momentum of the photon
*(1Å = 10⁻¹⁰ m)* **[3 Marks]**
**(e)**
- i. Define binding energy.
- ii. Calculate the atomic binding energy per nucleon of ⁵⁶Fe which has a mass number of 55.934927 u. (mass of proton = 1.007825 u, mass of neutron = 1.008665 u, 1u = 931.5 MeV) **[4 Marks]**
**[Total = 10 Marks]**
# SOLUTIONS — PHY 001–004
---
## PHY 001: MECHANICS AND PROPERTIES OF MATTER
### Question 1
**(a) THREE Types of Modulus:**
1. **Young's Modulus (E)** – ratio of tensile stress to tensile strain
2. **Bulk Modulus (K)** – ratio of volumetric stress to volumetric strain
3. **Shear/Rigidity Modulus (G)** – ratio of shear stress to shear strain
---
**(b) Brief Explanations:**
**i. Elasticity:** The ability of a material to return to its original shape and size after the deforming force is removed.
**ii. Plasticity:** The property of a material whereby it permanently deforms after the elastic limit is exceeded and does not return to its original shape.
**iii. Ductility:** The ability of a material to be drawn into wires without breaking (e.g. copper).
**iv. Malleability:** The ability of a material to be hammered or rolled into thin sheets without fracturing (e.g. gold, aluminium).
---
**(c)** Given: x = 4t³ − 16t + 12
**i. Instantaneous velocity at t = 5s:**
$$v = \frac{dx}{dt} = 12t^2 - 16$$
$$v(5) = 12(5)^2 - 16 = 12(25) - 16 = 300 - 16 = \boxed{284 \ \text{m/s}}$$
**ii. Instantaneous acceleration at t = 5s:**
$$a = \frac{dv}{dt} = 24t$$
$$a(5) = 24(5) = \boxed{120 \ \text{m/s}^2}$$
**iii. Time when particle is stationary (v = 0):**
$$12t^2 - 16 = 0$$
$$t^2 = \frac{16}{12} = \frac{4}{3}$$
$$t = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}} \approx \boxed{1.15 \ \text{s}}$$
---
### Question 2
**(a)**
**i. Archimedes' Principle:** When a body is wholly or partially immersed in a fluid, it experiences an upthrust (buoyant force) equal to the weight of fluid displaced.
**ii. FOUR Characteristics of Pressure in a Fluid:**
1. Pressure acts equally in all directions at a point
2. Pressure increases with depth
3. Pressure depends on the density of the fluid
4. Pressure at the same horizontal level in a connected fluid is equal
---
**(b) Compressible vs Incompressible Fluid:**
| | Compressible | Incompressible |
|---|---|---|
| Density | Changes with pressure | Remains constant |
| Example | Gases (air) | Liquids (water) |
---
**(c)** Depth of M = 60 m, Depth of N = 80 m, Atmospheric pressure = 20 m of water
$$P_1 = \rho g(20 + 60) = 80\rho g$$
$$P_2 = \rho g(20 + 80) = 100\rho g$$
$$\frac{P_2}{P_1} = \frac{100\rho g}{80\rho g} = \boxed{\frac{5}{4} = 1.25}$$
---
**(d) Conservative vs Non-Conservative Force:**
| | Conservative Force | Non-Conservative Force |
|---|---|---|
| Definition | Work done is independent of path; energy is conserved | Work done depends on path; energy is lost |
| Example | Gravitational force, spring force | Friction, air resistance |
---
## PHY 002: HEAT, WAVES AND OPTICS
### Question 3
**(a) Huygens' Principle:**
Every point on a wavefront acts as a source of secondary wavelets that spread out in the forward direction. The new wavefront is the tangent (envelope) to all these secondary wavelets.
---
**(b)**
**i. Electromagnetic Spectrum:**
The electromagnetic spectrum is the range of all types of electromagnetic radiation arranged in order of frequency or wavelength.
THREE components:
1. Visible light
2. X-rays
3. Radio waves
**ii. Young's Slit — separation of slits:**
Given:
- Fringe separation between 1st and 5th bright fringe: y = 2.5 mm = 2.5 × 10⁻³ m
- Number of fringe spacings = 5 − 1 = 4
- So fringe width: β = 2.5×10⁻³ / 4 = 6.25 × 10⁻⁴ m
- λ = 4.5 × 10⁻⁷ m
- D = 0.9 m
$$\beta = \frac{\lambda D}{d} \Rightarrow d = \frac{\lambda D}{\beta}$$
$$d = \frac{4.5 \times 10^{-7} \times 0.9}{6.25 \times 10^{-4}}$$
$$d = \frac{4.05 \times 10^{-7}}{6.25 \times 10^{-4}} = \boxed{6.48 \times 10^{-4} \ \text{m} \approx 0.648 \ \text{mm}}$$
---
**(c)**
**i. Internal Energy:** The total energy (kinetic + potential) of all the molecules within a system due to their random motion and intermolecular forces.
**ii. Second Law of Thermodynamics:** Heat cannot spontaneously flow from a colder body to a hotter body. Alternatively: the entropy of an isolated system always increases or remains constant.
---
**(d)**
**i. First Law expressions:**
- **Adiabatic process** (no heat exchange, Q = 0):
$$\Delta U = -W \quad \Rightarrow \quad W = -\Delta U$$
- **Isochoric process** (constant volume, W = 0):
$$\Delta U = Q$$
**ii. FOUR Factors Affecting Heat Loss by Convection:**
1. Temperature difference between the surface and the fluid
2. Surface area of the object
3. Density and viscosity of the fluid
4. Velocity/flow of the fluid (natural vs forced convection)
---
### Question 4
**(a) Converging Mirror vs Converging Lens:**
**TWO Similarities:**
1. Both can produce real, inverted images
2. Both can produce magnified or diminished images depending on object position
**TWO Differences:**
1. A converging mirror uses reflection; a converging lens uses refraction
2. A converging mirror has a single focal point in front; a converging lens has focal points on both sides
---
**(b i.) Uses of Mirrors:**
| Mirror | Uses |
|---|---|
| Plane mirror | Dressing/grooming; periscopes |
| Concave mirror | Shaving/makeup mirror; car headlights/torches (as reflector) |
| Convex mirror | Rear-view mirror in vehicles; security/shop mirrors |
**ii. Concave Mirror Table:**
| Position of Object | Position of Image | Size of Image | Nature of Image |
|---|---|---|---|
| At infinity | At F | Highly diminished (point) | Real, Inverted |
| At C | At C | Same size | Real, Inverted |
| Between C and F | Beyond C | Magnified | Real, Inverted |
| At F | At infinity | Highly magnified | Real, Inverted |
| Between F and P | Behind mirror | Magnified | Virtual, Erect |
---
## PHY 003: ELECTRICITY AND MAGNETISM
### Question 5
**(a)**
**i. Relative Permittivity (εᵣ):**
The ratio of the permittivity of a material to the permittivity of free space (vacuum). It indicates how much the material reduces the electric field compared to a vacuum:
$$\varepsilon_r = \frac{\varepsilon}{\varepsilon_0}$$
**ii. TWO Desirable Physical Properties of a Dielectric:**
1. High electrical insulation (low electrical conductivity)
2. High dielectric strength (ability to withstand strong electric fields without breakdown)
---
**(b)** Given: N = 30 turns, EMF = 80 μV = 80 × 10⁻⁶ V, t = 3 min = 180 s
$$\text{EMF} = N\frac{\Delta\Phi}{\Delta t}$$
$$\Delta\Phi = \frac{\text{EMF} \times \Delta t}{N} = \frac{80 \times 10^{-6} \times 180}{30}$$
$$\Delta\Phi = \frac{0.0144}{30} = \boxed{4.8 \times 10^{-4} \ \text{Wb}}$$
---
**(c)** Given: v₀ = 3.00 × 10⁶ m/s, E = 200 N/C, l = 0.100 m
m = 9.11 × 10⁻³¹ kg (standard), q = 1.60 × 10⁻¹⁹ C
**i. Acceleration of electron:**
$$a = \frac{qE}{m} = \frac{1.60 \times 10^{-19} \times 200}{9.11 \times 10^{-31}}$$
$$a = \frac{3.20 \times 10^{-17}}{9.11 \times 10^{-31}} = \boxed{3.51 \times 10^{13} \ \text{m/s}^2}$$
**ii. Time to cross the field:**
$$t = \frac{l}{v_0} = \frac{0.100}{3.00 \times 10^6} = \boxed{3.33 \times 10^{-8} \ \text{s}}$$
---
### Question 6
**(a)** A moving charged particle in an **electric field** experiences a force parallel (or antiparallel) to the field regardless of its direction of motion — it follows a parabolic path if moving perpendicular to E.
In a **magnetic field**, the force is always perpendicular to both the velocity and the field (F = qv × B), causing circular/helical motion, and a particle moving parallel to B experiences no force. These distinct behaviors distinguish the two fields.
---
**(b)** Circuit: 24.0 V and 12.0 V sources with resistors 80.0 Ω, 120.0 Ω, 180.0 Ω.
Using Kirchhoff's Voltage Law (KVL) with two loops:
**Assume:**
- Loop 1 (top): 24.0 V, 80.0 Ω (I₁), 180.0 Ω (shared, I₃)
- Loop 2 (bottom): 12.0 V, 120.0 Ω (I₂), 180.0 Ω (shared, I₃)
- KCL: I₁ + I₂ = I₃ (currents through 180 Ω)
**Loop 1:**
$$24 = 80I_1 + 180I_3 \quad ...(1)$$
**Loop 2:**
$$12 = 120I_2 + 180I_3 \quad ...(2)$$
**KCL:** I₃ = I₁ + I₂, so I₂ = I₃ − I₁
Substitute into (2):
$$12 = 120(I_3 - I_1) + 180I_3 = 300I_3 - 120I_1 \quad ...(3)$$
From (1): 24 = 80I₁ + 180I₃ → I₁ = (24 − 180I₃)/80
Substitute into (3):
$$12 = 300I_3 - 120\left(\frac{24 - 180I_3}{80}\right)$$
$$12 = 300I_3 - \frac{2880 - 21600I_3}{80}$$
$$12 = 300I_3 - 36 + 270I_3$$
$$48 = 570I_3$$
$$I_3 = \frac{48}{570} = \boxed{0.0842 \ \text{A} \approx 84.2 \ \text{mA}}$$
$$I_1 = \frac{24 - 180(0.0842)}{80} = \frac{24 - 15.16}{80} = \frac{8.84}{80} = \boxed{0.1105 \ \text{A} \approx 110.5 \ \text{mA}}$$
$$I_2 = I_3 - I_1 = 0.0842 - 0.1105 = -0.0263 \ \text{A}$$
The negative sign means I₂ flows opposite to our assumed direction: **|I₂| ≈ 26.3 mA**
---
**(c)** Given: r = 14 cm = 0.14 m, B = 0.35 T
Mass of proton = 1.67 × 10⁻²⁷ kg, q = 1.60 × 10⁻¹⁹ C
For circular motion: qvB = mv²/r → v = qBr/m
$$v = \frac{qBr}{m} = \frac{1.60 \times 10^{-19} \times 0.35 \times 0.14}{1.67 \times 10^{-27}}$$
$$v = \frac{7.84 \times 10^{-21}}{1.67 \times 10^{-27}} = \boxed{4.69 \times 10^{6} \ \text{m/s}}$$
---
## PHY 004: MODERN PHYSICS
### Question 7
**(a)** de Broglie wavelength: λ = h/mv, h = 6.626 × 10⁻³⁴ J·s, v = 2.20 × 10⁶ m/s
**i. Electron** (m = 9.11 × 10⁻³¹ kg):
$$\lambda = \frac{6.626 \times 10^{-34}}{9.11 \times 10^{-31} \times 2.20 \times 10^6} = \frac{6.626 \times 10^{-34}}{2.004 \times 10^{-24}} = \boxed{3.31 \times 10^{-10} \ \text{m}}$$
**ii. Proton** (m = 1.67 × 10⁻²⁷ kg):
$$\lambda = \frac{6.626 \times 10^{-34}}{1.67 \times 10^{-27} \times 2.20 \times 10^6} = \frac{6.626 \times 10^{-34}}{3.674 \times 10^{-21}} = \boxed{1.80 \times 10^{-13} \ \text{m}}$$
**iii. 200 g bullet** (m = 0.200 kg):
$$\lambda = \frac{6.626 \times 10^{-34}}{0.200 \times 2.20 \times 10^6} = \frac{6.626 \times 10^{-34}}{4.40 \times 10^5} = \boxed{1.51 \times 10^{-39} \ \text{m}}$$
*(Negligibly small — no quantum effects)*
---
**(b i.) Properties of α, β, γ radiation:**
| Property | α | β | γ |
|---|---|---|---|
| Charge | +2 | −1 | 0 |
| Mass | 4 u (heavy) | ~1/1836 u | 0 (massless photon) |
| Ionising effect | Strongly ionising | Moderately ionising | Weakly ionising |
| Field effect | Deflected (towards −ve plate) | Deflected (towards +ve plate) | Not deflected |
**ii. Effect of Temperature and Pressure on Radioactive Decay:**
Radioactive decay is a **nuclear process** and is completely unaffected by temperature and pressure. These are external physical conditions that only influence electron interactions (chemical bonds), not the nucleus. The decay constant λ remains unchanged regardless of temperature or pressure.
---
**(c)**
**i. Heisenberg's Uncertainty Principle:**
It is impossible to simultaneously determine, with perfect precision, both the position (x) and momentum (p) of a particle. The product of their uncertainties satisfies:
$$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$$
**ii. FOUR Practical Applications of X-rays:**
1. Medical diagnosis (imaging bones and internal organs)
2. Cancer treatment (radiotherapy)
3. Security screening at airports (luggage scanning)
4. Crystallography (determining crystal structures)
---
**(d)** Photon wavelength λ = 1 Å = 1 × 10⁻¹⁰ m
**i. Energy in electron volts:**
$$E = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{1 \times 10^{-10}}$$
$$E = 1.988 \times 10^{-15} \ \text{J}$$
$$E = \frac{1.988 \times 10^{-15}}{1.6 \times 10^{-19}} = \boxed{12,425 \ \text{eV} \approx 12.4 \ \text{keV}}$$
**ii. Momentum of the photon:**
$$p = \frac{h}{\lambda} = \frac{6.626 \times 10^{-34}}{1 \times 10^{-10}} = \boxed{6.63 \times 10^{-24} \ \text{kg·m/s}}$$
---
**(e)** ⁵⁶Fe: Mass number A = 56, Atomic number Z = 26 → Neutrons N = 30
**i. Binding Energy:** The energy required to completely separate all nucleons (protons and neutrons) in a nucleus, or equivalently, the energy released when nucleons come together to form the nucleus.
**ii. Binding energy per nucleon of ⁵⁶Fe:**
Mass of nucleus = 55.934927 u
Mass of 26 protons = 26 × 1.007825 = 26.20345 u
Mass of 30 neutrons = 30 × 1.008665 = 30.25995 u
Total nucleon mass = 56.46340 u
**Mass defect:**
$$\Delta m = 56.46340 - 55.934927 = 0.528473 \ \text{u}$$
**Total Binding Energy:**
$$BE = 0.528473 \times 931.5 = 492.27 \ \text{MeV}$$
**Binding Energy per nucleon:**
$$\frac{BE}{A} = \frac{492.27}{56} = \boxed{8.79 \ \text{MeV/nucleon}}$$
*(This is one of the highest binding energies per nucleon, making ⁵⁶Fe one of the most stable nuclei)*