Physics Grade 10: Ethiopia National Curriculum Comprehensive Study Guide

Fundamental Physical Quantities: Scalars and Vectors

Scalar quantities are physical quantities that are completely specified by a single numerical value together with an appropriate unit of measurement. Examples include mass (e.g., $12.21\,kg$ ), length (e.g., $1.42\,m$ ), density (e.g., $1000\,kg/m^3$ ), time, distance, speed, volume, temperature, energy, and power. These descriptions require no additional information beyond magnitude.

Vector quantities are physical quantities that require both magnitude and direction for a complete description. For example, stating a train's velocity is $100\,km/h$ is incomplete without specifying the direction of travel. Other examples include force, displacement, acceleration, momentum, impulse, weight, and electric field strength.

Representation and Properties of Vectors

Vectors are represented algebraically using bold letters such as A or a letter with an arrow over it, such as $\vec{A}$ . The magnitude of a vector is designated as $A$ or by using absolute value notation, $|\vec{A}|$ . Geometrically, a vector is represented by an arrow-tipped line segment. The initial point is the "tail," and the final point is the "head."

In graphical representation, the length of the arrow is drawn to scale to represent the vector's magnitude, while the arrowhead indicates the direction. Proper scale must be chosen (e.g., $1\,cm$ on paper representing $2\,N$ ), and a reference direction—such as a horizontal line or compass points—must be established.

Specific types of vectors include:

Zero (Null) Vector: A vector with zero magnitude and no specific direction.
Unit Vector: A vector with a magnitude of exactly one.
Equal Vectors: Vectors sharing the same magnitude and direction.
Negative of a Vector: A vector with the same magnitude but an opposite direction compared to the original vector.

Vector Addition and Subtraction Methods

Unlike scalars, vectors cannot be added using simple arithmetic unless they are in the same direction. Only vectors of the same nature (e.g., two forces or two velocities) can be summed. The result is called the resultant vector ( $\vec{R}$ , where $\vec{R} = \vec{A} + \vec{B}$ ).

Vector subtraction is the addition of a negative vector: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$ . This involves "flipping" the vector being subtracted to its opposite direction and then adding it to the first vector.

Graphical techniques for addition include:

Triangle Method: The head of the first vector is joined to the tail of the second. The resultant is drawn from the tail of the first to the head of the second.
Parallelogram Method: The tails of two vectors are joined. A parallelogram is constructed using these vectors as adjacent sides. The diagonal starting from the common tail represents the resultant.
Polygon Method: Used for more than two vectors. Vectors are placed head-to-tail in succession. The resultant is the vector drawn from the tail of the first to the head of the last.

Mathematical special cases for two vectors $\vec{A}$ and $\vec{B}$ :

Same direction: Magnitude $|R| = |A| + |B|$ .
Opposite directions: Magnitude $|R| = |A| - |B|$ , direction follows the larger vector.
Perpendicular: Magnitude $|R| = \sqrt{A^2 + B^2}$ . The direction is given by $\theta = \tan^{-1}(\frac{B}{A})$ .

Vector Resolution and Component Method

Vector resolution is the process of breaking a single vector into multiple component vectors that, when added, equal the original vector. In a rectangular coordinate system, a vector $\vec{A}$ is resolved into horizontal ( $A_x$ ) and vertical ( $A_y$ ) components.

Using trigonometry, where $\theta$ is measured counterclockwise from the positive x-axis: $A_x = A\cos(\theta)$ $A_y = A\sin(\theta)$

The magnitude is derived using the Pythagorean Theorem: $|A| = \sqrt{A_x^2 + A_y^2}$ . The direction is derived from the tangent function: $\theta = \tan^{-1}(\frac{A_y}{A_x})$ .

Position, Displacement, and Distance in Motion

Position is the specific location of an object relative to a chosen frame of reference, which is an arbitrary set of axes. Position can be positive or negative depending on the direction from the origin.

Distance is a scalar quantity representing the total length of the path traveled. Displacement is a vector quantity representing the change in position (the shortest straight-line distance between initial and final points). Mathematically, displacement $\Delta\vec{s}$ is defined as: $\Delta\vec{s} = \mathbf{s} - \mathbf{s}_0$

It is possible for displacement to be zero (e.g., a round trip) even when the distance traveled is large.

Velocity: Average and Instantaneous

Average velocity is the total displacement divided by the time interval over which the motion occurred: $\vec{v}_{av} = \frac{\Delta\vec{s}}{\Delta t} = \frac{\mathbf{s} - \mathbf{s}_0}{t - t_0}$

Average speed is the total distance divided by total time. While speed is a scalar, velocity is a vector. Standard SI units are meters per second ( $m/s$ ).

Instantaneous velocity is the velocity at a specific point in time or over an infinitesimally small time interval. It is the reading shown on a vehicle's speedometer together with the direction of travel. Public safety, such as speed limit adherence, relies on monitoring instantaneous speed to prevent road accidents.

Acceleration and Equations of Motion

Acceleration is the rate of change of velocity. An object accelerates if it changes its speed, direction, or both. Average acceleration is defined as: $\vec{a}_{av} = \frac{\Delta\vec{v}}{\Delta t} = \frac{v - v_0}{t - t_0}$

Acceleration is positive if the velocity increases in the defined positive direction and negative (deceleration) if the velocity decreases. Standard units are meters per second squared ( $m/s^2$ ).

For Uniformly Accelerated Motion (UAM) along a straight line, five variables are related: displacement ( $s$ ), initial velocity ( $v_0$ ), final velocity ( $v$ ), acceleration ( $a$ ), and time ( $t$ ):

$v = v_0 + at$
$s = v_0t + \frac{1}{2}at^2$
$v^2 = v_0^2 + 2as$
$v_{av} = \frac{v_0 + v}{2}$

Free Fall Motion

Free fall occurs when an object moves solely under the influence of gravity, ignoring air resistance. Near the Earth's surface, all objects fall with a constant acceleration $g \approx 9.80\,m/s^2$ . In the UAM equations, $a$ is replaced by $g$ and distance $s$ by height $h$ :

$v = v_0 + gt$
$h = v_0t + \frac{1}{2}gt^2$
$v^2 = v_0^2 + 2gh$

Graphical Representation of Accelerated Motion

Kinematics can be analyzed using three types of graphs:

Position-Time ( $s-t$ ) Graph: For UAM, this graph is a parabola. The slope of the tangent at any point on this curve represents the instantaneous velocity.
Velocity-Time ( $v-t$ ) Graph: For UAM, this is a straight line. The slope of the line equals the acceleration ( $a = \frac{\Delta v}{\Delta t}$ ). The area under the graph represents the total displacement.
Acceleration-Time ( $a-t$ ) Graph: For UAM, this is a horizontal line (constant acceleration). The area under the line represents the change in velocity ( $\Delta v$ ).

Relative Velocity in One Dimension

Relative velocity is the velocity of an object as observed from a specific frame of reference. If two objects, A and B, move with velocities $v_A$ and $v_B$ relative to the ground:

Relative velocity of A with respect to B: $v_{AB} = v_A - v_B$ .
Moving in same direction: Magnitude is the difference ( $v_A - v_B$ ).
Moving in opposite directions: Magnitude is the sum ( $v_A + v_B$ ).

Elasticity, Plasticity, and Density

A rigid body is an idealized object that maintains its shape; however, real materials deform under a deforming force.

Elasticity is the property of a material that allows it to regain its original shape and size after the deforming force is removed. Plasticity is the property whereby a material remains permanently deformed (e.g., clay or putty). The maximum force a material can withstand and still return to its original shape is the elastic limit.

Density ( $\rho$ ) is the mass per unit volume of a substance: $\rho = \frac{m}{V}$ SI unit: $kg/m^3$ . Conversion: $1\,kg/m^3 = 10^{-3}\,g/cm^3$ . Specific Gravity (SG) is the unitless ratio of a substance's density to the density of water at $4\,^oC$ ( $\rho_{water} = 1000\,kg/m^3$ ): $SG = \frac{\rho_{substance}}{\rho_{water}}$

Stress, Strain, and the Young Modulus

Stress is the deforming force per unit cross-sectional area: $\text{Stress} = \frac{F}{A}$ Unit: $N/m^2$ or Pascal ( $Pa$ ).

Strain is the measure of deformation, defined as the fractional change in dimension:

Tensile/Linear Strain: $\frac{\Delta L}{L_0}$
Volumetric Strain: $\frac{\Delta V}{V_0}$
Shearing Strain: $\frac{\Delta x}{L_0}$

Hooke's Law states that within the elastic limit, stress is directly proportional to strain. The constant of proportionality is the Modulus of Elasticity. The Young Modulus ( $Y$ ) specifically relates tensile stress and strain: $Y = \frac{\text{Tensile Stress}}{\text{Tensile Strain}} = \frac{F/A}{\Delta L / L_0} = \frac{F L_0}{A \Delta L}$ Steel ( $20 \times 10^{10}\,N/m^2$ ) has a higher Young Modulus than copper ( $11 \times 10^{10}\,N/m^2$ ), making steel more elastic and resistant to stretching.

Static Equilibrium of Rigid Bodies

An object is in static equilibrium if it is at rest and remains so. Two conditions must be met:

First Condition (Translational Equilibrium): The vector sum of all external forces must be zero. $\sum \vec{F} = 0 \implies \sum F_x = 0, \sum F_y = 0$
Second Condition (Rotational Equilibrium): The sum of all torques about any axis must be zero. $\sum \vec{\tau} = 0$

Torque ( $\tau$ ) is the twisting effect of a force, measured in Newton-meters ( $Nm$ ): $\tau = Fr\sin(\theta)$ where $r$ is the distance from the pivot (lever arm) and $\theta$ is the angle between the force and the lever arm. In equilibrium, clockwise torques must equal counter-clockwise torques.

Electrostatics: Charges, Fields, and Forces

There are two types of electric charges: positive (protons) and negative (electrons). The SI unit is the Coulomb ( $C$ ). The charge of one electron is $-1.6 \times 10^{-19}\,C$ . Charge is quantized ( $q = ne$ ) and conserved (it cannot be created or destroyed, only transferred).

Methods of charging include:

Rubbing (Friction): Transfer of electrons between different materials.
Conduction (Contact): Charging a neutral object by touching it with a charged one; the same sign of charge is shared.
Induction: Charging without contact by grounding a conductor near a charged presence; the opposite sign of charge is acquired.

Coulomb's Law states the force between two point charges is: $F = k\frac{|q_1 q_2|}{r^2}$ where $k \approx 9 \times 10^9\,Nm^2/C^2$ . The force is attractive for unlike charges and repulsive for like charges.

Electric Field ( $E$ ) is the force per unit charge ( $E = F/q$ ). The field from a point charge is $E = k\frac{q}{r^2}$ . Field lines move away from positive charges and toward negative charges.

Current Electricity and Ohm's Law

Electric Current ( $I$ ) is the rate of flow of charge: $I = \frac{\Delta Q}{\Delta t}$ Measured in Amperes ( $A$ ). Potential Difference (Voltage, $V$ ) is the work done to move unit charge: $V = \frac{W}{Q}$ Measured in Volts ( $V$ ).

Ohm's Law states $V = IR$ . Resistance ( $R$ ) depends on the material's properties: $R = \rho \frac{L}{A}$ where $\rho$ is resistivity ( $\Omega m$ ), $L$ is length, and $A$ is cross-sectional area. Conductors have low resistivity; insulators have high resistivity.

Resistors in Circuits

Series Connection: Resistors are connected end-to-end. Current is constant throughout. $V_{total} = V_1 + V_2 + V_3$ $R_{eq} = R_1 + R_2 + R_3$
Parallel Connection: Resistors share common points. Voltage is constant across branches. $I_{total} = I_1 + I_2 + I_3$ $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

Measurement devices:

Ammeter: Measures current; must be connected in series; has very low resistance.
Voltmeter: Measures potential difference; must be connected in parallel; has very high resistance.

Magnetism and Magnetic Fields

Magnets have two poles: North ( $N$ ) and South ( $S$ ). Like poles repel; unlike poles attract. Magnetic monopoles do not exist. Magnetic field lines form closed loops moving $N$ to $S$ outside and $S$ to $N$ inside.

Earth's magnetic field acts like a giant bar magnet, with its magnetic South pole near the geographic North pole. This allows a compass to find direction.

Electromagnetism shows that moving charges create magnetic fields. The field $B$ around a long straight wire is: $B = \frac{\mu_0 I}{2\pi d}$ where $\mu_0 = 4\pi \times 10^{-7}\,Tm/A$ . Direction is determined by the right-hand rule (thumb along current, fingers curl along field).

Magnetic Force:

On a moving charge: $F = qvB\sin(\theta)$
On a current-carrying wire: $F = ILB\sin(\theta)$
Between parallel wires: Attractive if currents are in the same direction; repulsive if opposite.

Electromagnetic Waves and Optics

Electromagnetic (EM) waves are transverse waves of oscillating electric and magnetic fields. They travel in a vacuum at speed $c \approx 3.00 \times 10^8\,m/s$ . Relationship: $c = \lambda f$ .

The EM Spectrum includes (longer to shorter $\lambda$ ): Radio waves, Microwaves, Infrared, Visible light (VIBGYOR), Ultraviolet, X-rays, and Gamma rays.

Geometrical Optics Laws:

Reflection: $\theta_{incidence} = \theta_{reflection}$ .
Refraction (Snell's Law): $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$ , where refractive index $n = c/v$ .

Total Internal Reflection occurs when light moves from a denser to a rarer medium and the angle of incidence exceeds the critical angle $\theta_c = \sin^{-1}(\frac{n_2}{n_1})$ . Application: Fiber optics.

Mirror and Lens Formula: $\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$ Magnification $m = \frac{h'}{h} = -\frac{v}{u}$ .

The Human Eye focuses light onto the retina. Defects include Myopia (corrected with concave lenses) and Hypermetropia (corrected with convex lenses). The least distance of distinct vision is $25\,cm$ .

Colors of light:

Primary additive colors: Red, Green, Blue. Red + Green + Blue = White light.
Secondary colors: Cyan (G+B), Magenta (R+B), Yellow (R+G).

In the study of physics and science, precise measurements are the foundation for discovering new phenomena and developing consistent theories. Any number used to quantitatively describe a physical phenomenon is termed a physical quantity. These quantities are fundamentally categorized into two groups: scalars and vectors. Scalar quantities are physical quantities that are completely specified by a single numerical value together with an appropriate unit of measurement. Examples include mass (e.g., $12.21\,kg$ ), length (e.g., $1.42\,m$ ), density (e.g., $1000\,kg/m^3$ ), time, distance, speed, volume, temperature, energy, and power. These descriptions require no additional information beyond magnitude. Vector quantities are physical quantities that require both magnitude and direction for a complete description. For example, stating a train's velocity is $100\,km/h$ is incomplete without specifying the direction of travel. Other examples include force, displacement, acceleration, momentum, impulse, weight, and electric field strength. Vectors are represented algebraically using bold letters such as A or a letter with an arrow over it, such as $\vec{A}$ . The magnitude of a vector is designated as $A$ or by using absolute value notation, $|\vec{A}|$ . Geometrically, a vector is represented by an arrow-tipped line segment. The initial point is the "tail," and the final point is the "head." Specific types of vectors include: 1. Zero (Null) Vector: A vector with zero magnitude and no specific direction. 2. Unit Vector: A vector with a magnitude of exactly one. 3. Equal Vectors: Vectors sharing the same magnitude and direction. 4. Negative of a Vector: A vector with the same magnitude but an opposite direction compared to the original vector. Unlike scalars, vectors cannot be added using simple arithmetic unless they are in the same direction. Only vectors of the same nature (e.g., two forces or two velocities) can be summed.