Independent and Dependent Variables

• Independent variable: The quantity you change deliberately in an experiment (plotted on the x-axis)
• Dependent variable: The quantity that changes in response (plotted on the y-axis)

For example, when studying how a spring stretches, you change the force applied (independent variable, x-axis) and measure the extension (dependent variable, y-axis).

Types of Relationships

1. Direct Proportion (y ∝ x)

• When one variable doubles, the other doubles
• Graph shape: Straight line passing through the origin
• Example: Extension of a spring vs applied force (Hooke's Law)

2. Inverse Proportion (y ∝ 1/x)

• When one variable doubles, the other halves
• Graph shape: Hyperbola (curved line)
• Example: Pressure vs volume in a fixed amount of gas

3. Square Relationship (y ∝ x²)

• When one variable doubles, the other becomes four times larger
• Graph shape: Parabolic curve
• Example: Distance vs time² for a falling object

1. Plot the data correctly with labeled axes and appropriate scales
2. Identify the shape of the graph line
3. Test for direct proportion by checking if the line passes through the origin
4. Test for inverse proportion by plotting y against 1/x — this should give a straight line through the origin
5. Test for square relationships by plotting y against x² — this should give a straight line through the origin
6. Calculate the slope of the straight-line graph to find the constant of proportionality

Experiment: Hooke's Law

A spring was hung from a retort stand. Different masses were attached, and the extension of the spring was measured. The results are shown in the table:

Step 1: Plot the graph

• Plot force (N) on the x-axis
• Plot extension (cm) on the y-axis
• Draw the best straight line through the points

Step 2: Analyze the graph

The line is straight and passes through the origin (0,0). This tells us there is a direct proportion relationship between force and extension.

Step 3: Find the mathematical relationship

Calculate the slope (gradient) of the line:

$\text{slope} = \frac{\text{change in } y}{\text{change in } x} = \frac{5.0 - 1.0}{2.5 - 0.5} = \frac{4.0}{2.0} = 2.0$

This slope represents the spring constant (k). The mathematical equation is:

$F = k \times e$

Where:

• F = force in newtons (N)
• e = extension in centimeters (cm)
• k = spring constant = 2.0 N/cm

So the relationship is: F = 2e or F ∝ e

This is Hooke's Law: the extension of a spring is directly proportional to the applied force, provided the elastic limit is not exceeded.

• Graphs help us visualize how physical quantities relate to each other
• Straight lines through the origin indicate direct proportion
• Curved graphs may indicate inverse or square relationships
• By re-plotting data in different ways (like y vs 1/x or y vs x²), we can identify the exact mathematical relationship
• The slope of a straight-line graph gives us the constant of proportionality

In Tanzania, mechanics and engineers use graph analysis when testing materials for construction. For example, when building a bridge in Dar es Salaam, engineers plot the stress versus strain graph for steel beams to determine how much load the structure can safely carry. By finding the relationship from the graph, they calculate the maximum weight the bridge can support, ensuring it is strong enough for vehicles like trucks carrying goods from the port to inland regions while remaining within safe limits.