Basic Physics

• Learners should be able to demonstrate and apply their knowledge and understanding of:

• (a) The 6 Essential Base SI Units

• The SI (International System of Units) consists of 7 base units, but six are most commonly used in physics. These are:

• (b) Representing Units in Terms of Base SI Units and Prefixes

• ⇒ Representing Derived Units:
• Many physical quantities are represented as combinations of the base units. For example:
• – Force (F):
• [math] F = ma[/math]
• Unit of [math]Force = kgms^{-2} = N[/math]
• – Energy (E):
• [math]E = Force . Displacement [/math]
• Unit of  [math]Energy = kgm^2 s^{-2}[/math]
• ⇒ Prefixes:
• The standard prefixes are extensions to the base SI units to simplify the quoting of measurements.
• Prefixes are short one-to three-syllable additions that are attached to the beginnings of words to slightly change their meaning.
• Or
• A prefix is a letter or group of letters that appears at the beginning of a word to change its meaning.
• SI prefixes are used to express large or small quantities:
• – kilo- (k) = [math]10^{3}[/math]
• – milli- (m)= [math]10^{-3}[/math]
• – micro- (μ)= [math]10^{-6}[/math]
• – mega- (M)= [math]10^{6}[/math]
• – nano- (n)=[math]10^{-9}[/math]
• Example: 1 mm= [math]10^{-3}[/math]
• 1μA = [math]10^{-6}[/math]A.

• (c) Checking Equations for Homogeneity Using Units

• ⇒ Homogeneity Principle:
• An equation is dimensionally consistent (homogeneous) if the dimensions on both sides of the equation are the same.
• Example 1:
• Kinematic equation: [math] s = ut + \frac{1}{2} at^2[/math]
• [math]- \text{s: [L]} \\
  – \text{ut: [L/T]} \cdot \text{[T]} \\
  – \frac{1}{2} a t^2: \text{[L/T}^2] \cdot \text{[T}^2] \\
  – \text{Both sides: [L]}[/math]
• Example 2:
• Energy:[math]E = \frac{1}{2} mv^2 [/math]
• – m: [M],[math] v^2: [L^2/T^2] [/math]
• – E: [M] [math] \cdot [L^2/T^2][/math].
• Usefulness:
• Helps verify correctness of equations.
• Does not verify constants without units.

• (d) Scalar and Vector Quantities

• ⇒ Scalar Quantities:
• Defined by magnitude only.
• Examples: speed, time, mass, energy, power, density.
• ⇒ Vector Quantities:
• Defined by magnitude and direction.
• Examples: displacement, velocity, acceleration, force, momentum.
• Examples:
• Displacement: Vector, includes direction.
• Speed: Scalar, magnitude of velocity.
• Density: Scalar, no direction associated.
• Force: Vector, acts in a specific direction.

• (e) Addition and Subtraction of Coplanar Vectors

• ⇒ Graphical Method:
• Place vectors tail-to-head.
• Resultant vector is the line from the tail of the first vector to the head of the last vector.
• 
• Figure 1 Heat to tail rule
• ⇒ Mathematical Method (Two Perpendicular Vectors):
• Use Pythagoras’ theorem for magnitude:
• [math]R = \sqrt{A^2 + B^2} [/math]
• Use trigonometry for angle:
• [math]tanθ = \frac{B}{A}[/math]
• Example:
• Add vectors A=3 m east and B=4 m north:
• [math]R = \sqrt{A^2 + B^2} \\
  R = \sqrt{(3)^2 + (4)^2} \\
  R = 5 \, \text{m} [/math]
• Angle:
• [math]\tan{\theta} = \frac{B}{A} \\
  \tan{\theta} = \frac{4}{3} \\
  \theta = \tan^{-1}{\left(\frac{4}{3}\right)} \\
  \theta \approx 53.1^\circ \, \text{N of E} [/math]

• (f) Resolving a Vector into Two Perpendicular Components

• A vector can be split into two perpendicular components along x– and y-axes.
• Method:
• For a vector R at an angle θ:
• – Horizontal component:
• [math]R_x=R cos⁡θ[/math]
• – Vertical component:
• [math]R_y = R sin⁡θ [/math]
• 
• Figure 2 Two perpendicular components
• Example:
• A force F=10 N acts at to the horizontal.
• [math]F_x = F \cos{\theta} \\
  F_x = 10 \cos{30^\circ} \\
  F_x = (10)(0.866) \\
  F_x = 8.66 \, \text{N} \\
  F_y = F \sin{\theta} \\
  F_y = 10 \sin{30^\circ} \\
  F_y = (10)(0.5) \\
  F_y = 5 \, \text{N} [/math]
• Applications:
• Decomposing forces in mechanics.
• Calculating work done when force is not parallel to displacement.

• g) Concept of Density and the Equation: ​

• ⇒ Definition of Density:
• Density (ρ) is defined as the mass (m) of a substance per unit volume (V).
• Mathematically:
• [math]ρ = \frac{m}{V}[/math]
• Where:
• – ρ: Density in [math]kg/m^3 or g/cm^3[/math] ,
• – m: Mass in kg or g,
• – V: Volume in [math]m^3 \text{or} cm^3[/math] .
• ⇒ How to Use the Equation:
• To calculate density:
• – Rearrange to
• [math]ρ = \frac{m}{V}[/math]
• – Substitute known values of m and V.
• To calculate mass:
• – Rearrange to
• [math]m = ρV [/math]
• To calculate volume:
• – Rearrange to
• [math]V = \frac{m}{ρ}[/math]
• Example:
• A block has a mass of 2kg and a volume of 001 [math]m^3[/math]. Find its density:
• [math]\rho = \frac{m}{V} \\
  \rho = \frac{2}{0.001} \\
  \rho = 2000 \, \text{kg/m}^3 [/math]

• h) Turning Effect of a Force (Moment)

• ⇒ Definition:
• The turning effect of a force is called the moment. It measures the tendency of a force to rotate an object about a pivot or axis.
• 
• Figure 3 Moment of a force
• ⇒ Equation for Moment:
• Moment = Force × Perpendicular Distance from Pivot
• Where:
• – Force (F): in N,
• – Distance (d): perpendicular distance in m.
• Direction of Rotation:
• – Clockwise (CW) or Counterclockwise (CCW).
• – Larger forces or longer distances produce greater moments.
• Example:
• A 10 N force acts 5 m from the pivot. Calculate the moment:
• Moment = Force × Perpendicular Distance from Pivot
• Moment = 10 × 0.5
• Moment = 5 Nm

• i)  Principle of Moments

• ⇒ Definition:
• The principle of moments states:
• – For a body in equilibrium, the sum of clockwise moments equals the sum of counter clockwise moments
• ⇒ Applications:
• Used to solve problems involving levers, seesaws, or beams.
• Example:
• A beam is balanced on a pivot. A 20 N force acts 2 m from the pivot on one side. On the other side, a force acts 1 m Find the second force (F):
• Clockwise moment = Counterclockwise moment
• 20 × 2 = F × 1
• F = 40 N

• j) Centre of Gravity

• ⇒ Definition:
• The center of gravity (CoG) is the point at which the entire weight of a body appears to act.
• ⇒ CoG in Uniform Objects:
• – Cylinder: Midpoint of the central axis.
• – Sphere: Geometric center.
• – Cuboid (Beam): Center of the object.
• 
• Figure 4 Center of gravity
• ⇒ Stability:
• An object is stable if its CoG is low and lies within the base of support.
• If CoG moves outside the base of support, the object tips over.

• k)   Equilibrium Conditions

• For a Body to Be in Equilibrium:
• – Resultant Force is Zero:
• ∑F=0
•  This ensures no linear acceleration.
• – Net Moment is Zero:
• ∑Clockwise Moments = ∑Counterclockwise Moments
• This ensures no rotational acceleration.
• ⇒ Example of Equilibrium Calculation
• Problem:
• A uniform beam of weight 100 N and length 4 m is supported by two pivots. A 200 N load is placed 1 m from the left pivot. Find the reactions at the two supports.
• Solution:
• Let and ​ be the reactions at the left and right pivots, respectively.
• Equilibrium conditions:
• [math]\sum F = 0 \\
  R_1 + R_2 = 100 + 200 = 300 \, \text{N} [/math]
• Taking moments about the left pivot:
• [math]R_2 \times 4 = 200 \times 1 + 100 \times 2 \\
  R_2 = 100 \, \text{N} [/math]
• Substitute into [math]R_1 + R_2 = 300[/math]:
• [math]R_1 = 300 – 100 \\
  R_1 = 200 \, \text{N} [/math]
• Result:
• [math]- R_1 = 200 \, \text{N} \\
  – R_2 = 100 \, \text{N}[/math]

Specified Practical Work

• 1. Measurement of the Density of Solids

• ⇒ Objective:
• To measure the density of different solid objects (regular or irregular) by determining their mass and volume.
• ⇒ Apparatus:
• Digital balance or weighing scale
• Ruler (for regular solids like cubes, cuboids, or cylinders)
• Measuring cylinder (for irregular solids)
• Overflow can (for larger irregular solids)
• Thread (for suspending objects, if needed)
• Water (to determine volume using displacement)
• Calculator
• 
• Figure 5 Measurement of density
• ⇒ Method for Regular Solids:
• Measure the Mass:
• – Use a digital balance to measure the mass (mmm) of the solid.
• – Record the mass in kilograms (kg) or grams (g).
• Measure the Dimensions:
• – Use a ruler to measure the dimensions of the solid.
• – For a cube or cuboid, measure length (l), width (w), and height (h).
• – For a cylinder, measure the radius (r) and height (h).
• Calculate the Volume (V):
• – For a cuboid:[math]V = l⋅w⋅h[/math],
• – For a cylinder:[math]V = πr^2 h[/math]
• – Ensure units are consistent (e.g., m³).
• Calculate the Density (ρ):
• – Use the formula [math]ρ = \frac{m}{V} [/math].
• – Record the density in [math]kg/m^3 \text{or}  g/cm^3[/math].
• ⇒ Method for Irregular Solids:
• Measure the Mass:
• – Use a digital balance to measure and record the mass (m).
• Measure the Volume Using Displacement:
• – Fill a measuring cylinder partially with water and record the initial water level ( [math]V_1[/math]).
• – Carefully submerge the irregular solid in the water.
• – Record the final water level ( [math]V_2[/math]).
• – The volume of the object is [math]V = V_2 – V_1[/math].
• Alternative for Larger Objects:
• – Use an overflow can filled with water to the spout level.
• – Submerge the solid, collect the displaced water, and measure its volume with a measuring cylinder.
• Calculate the Density (ρ):
• – Use
• [math]\rho = \frac{m}{V} [/math]
• ⇒ Precautions:
• Ensure the balance is calibrated.
• Avoid parallax error when reading measurements.
• Ensure the object is completely submerged for accurate volume measurement.
• ⇒ Example Calculation:
• A cube with sides 05 m has a mass of 1 kg.
• [math]V = 0.05 \times 0.05 \times 0.05 \\
  V = 0.000125 \, \text{m}^3 \\
  \rho = \frac{m}{V} \\
  \rho = \frac{1}{0.000125} \\
  \rho = 8000 \, \text{kg/m}^3 [/math]

• 2. Determination of Unknown Masses Using the Principle of Moments

• ⇒ Objective:
• To determine the mass of an unknown object by applying the principle of moments.
• ⇒ Apparatus:
• – Meter rule (or beam)
• – Fulcrum or pivot
• – Known masses (calibrated weights)
• – Unknown mass
• – Thread (for suspending objects)
• – Clamp and stand (optional)
• 
• Figure 6 Determine the mass using the principle of moment
• ⇒ Theory:
• The principle of moments states:
• Clockwise Moment = Counterclockwise Moment
• The moment of a force is given by:
• Moment = Force × Perpendicular Distance from Pivot.
• ⇒ Method:
• Set Up the Apparatus:
• – Place the meter rule (or beam) on a pivot such that it can rotate freely.
• – Ensure the pivot is stable.
• Place the Known Mass:
• – Suspend a known mass ([math]m_1[/math] ​) at a measured distance ([math]d_1[/math] ) from the pivot.
• Place the Unknown Mass:
• – Suspend the unknown mass ([math]m_2[/math] ) at a measured distance ( [math]d_2[/math]) from the pivot.
• Achieve Balance:
• – Adjust the position of the masses until the beam is balanced (level).
• Apply the Principle of Moments:
• – For equilibrium:
• [math]m_1 g \times d_1 = m_2 g \times d_2 \\
  m_2 = \frac{m_1 \times d_1}{d_2}[/math]
• Example Calculation:
• Known mass:[math]m_1 = 2 kg[/math] , distance: [math]d_1 = 0.3m[/math],
• Unknown mass distance:[math]d_2 = 0.2m[/math]
• [math]m_2 = \frac{(m_1 \times d_1)}{d_2} \\
  m_2 = \frac{(2 \times 0.3)}{0.2} \\
  m_2 = 3 \, \text{kg}[/math]
• Precautions:
• – Ensure the beam is level when balanced.
• – Avoid air currents or vibrations that may disrupt the setup
• – Measure distances from the pivot accurately.
• Applications:
• – Used in weighing scales and lever-based systems.