2.1 Key Terms

• Wavelength (λ): Distance between two consecutive crests or troughs (in meters)
• Frequency (f): Number of waves passing a point per second (in Hertz, Hz)
• Period (T): Time for one complete wave (in seconds)
• Amplitude: Maximum displacement from equilibrium position
• Wave speed (v): Distance traveled by the wave per unit time

2.2 Wave Equation

The relationship between speed, wavelength, and frequency: $v = f \times \lambda$

Also, since $T = \frac{1}{f}$: $v = \frac{\lambda}{T}$

Worked Example: A water wave has wavelength 0.5 m and frequency 4 Hz. Calculate its speed. $v = f\lambda = 4 \times 0.5 = 2 \text{ m/s}$

2.3 Types of Waves

• Transverse wave: Particles vibrate perpendicular to wave direction (e.g., light, water waves)
• Longitudinal wave: Particles vibrate parallel to wave direction (e.g., sound waves)

Friction is the force that opposes motion between two surfaces in contact.

Formula: $F = \mu N$ Where:

• $F$ = frictional force (N)
• $\mu$ = coefficient of friction (no unit)
• $N$ = normal reaction force (N)

On a flat surface, $N = mg$ (weight).

Worked Example: A wooden box of mass 20 kg rests on a horizontal floor. The coefficient of friction is 0.3. Calculate the frictional force. ($g = 10 \text{ m/s}^2$)

$N = mg = 20 \times 10 = 200 \text{ N}$ $F = \mu N = 0.3 \times 200 = 60 \text{ N}$

An object is in equilibrium when:

• Net force = 0 (translational equilibrium)
• Net torque = 0 (rotational equilibrium)

For a rigid body to be in equilibrium: $\sum F = 0$ and $\sum \tau = 0$

Principle of Moments: For equilibrium, clockwise moments = anticlockwise moments $F_1 d_1 = F_2 d_2$

Worked Example: A uniform beam of length 4 m weighs 100 N. A 200 N load is placed 1 m from one end. Where must the support be placed for equilibrium?

Taking moments about the support: Let distance from 100 N weight = $d$ $100 \times d = 200 \times 1$ $d = 2 \text{ m}$

Simple machines help us apply force more easily by changing the direction or magnitude of force.

5.1 Key Concepts

• Effort (E): The force applied to the machine
• Load (L): The resistance or weight to be overcome
• Mechanical Advantage (MA): $MA = \frac{\text{Load}}{\text{Effort}}$
• Velocity Ratio (VR): $VR = \frac{\text{Distance moved by effort}}{\text{Distance moved by load}}$
• Efficiency: $\text{Efficiency} = \frac{MA}{VR} \times 100\%$

5.2 Types of Simple Machines

• Lever: Rigid bar that pivots around a fulcrum
• Pulley: Wheel on an axle to change force direction
• Inclined plane: Sloping surface to reduce force needed
• Wheel and axle: Circular wheel connected to a smaller axle
• Screw: Inclined plane wrapped around a cylinder

Worked Example: A pulley system has VR = 4. If the efficiency is 80%, what is the actual MA? $MA = \text{Efficiency} \times VR = 0.8 \times 4 = 3.2$

If this system lifts a load of 800 N, what effort is required? $MA = \frac{L}{E} \Rightarrow E = \frac{L}{MA} = \frac{800}{3.2} = 250 \text{ N}$

In Tanzania, understanding these principles is essential in everyday activities and local industries. For example, when loading sacks of maize or rice at a market like Mwenge or Kariakoo, workers use simple machines such as inclined planes (ramps) or pulley systems to lift heavy loads onto trucks. By applying the concept of mechanical advantage, they can lift a 500 N sack by applying much less than 500 N of effort, making the work easier and faster. Similarly, understanding friction helps explain why motorcycle tires have treads — to increase $\mu$ and prevent slipping on wet or muddy roads common during the rainy season in many parts of Tanzania.