Graph Work and Gradient Rules

Core Rules for Graph Construction

• Rule 1: Scale Selection

  • Choose the largest convenient scale for the axes.

  • Ensure all recorded readings will fit on the graph paper before finalizing the scale for each axis.

• Rule 2: Axis Labeling

  • Every axis must be labeled clearly.

  • Labels must indicate what specifically is being measured.

  • Labels must include the units of measurement being used.

• Rule 3: Plotting Data Points

  • Points should be plotted using small x’s ($x$).

  • Alternatively, points can be plotted as dots with circles around them.

• Rule 4: Drawing the Best Fit Line

  • Draw the best line possible through the data points; this is referred to as a "best fit line."

  • A primary decision is whether the line must pass through the origin ($0,0$).

  • If points do not align to form a perfect straight line, a "best fit line" is necessary.

  • Technique for Drawing:

    1. Balance the ruler so that it passes through as many points as possible.

    2. Ensure that the same number of points are located above the best fit line as below the line.

    3. Ensure that points not lying directly on the line are approximately the same distance from the line.

  • Calculating Gradients:

    • Always choose new points that lie directly on the drawn line to calculate the gradient, rather than using the original data points from the table if they do not sit on the line.

  • Linear vs. Non-linear Data:

    • Draw a straight line or a smooth curve depending on which goes closest to the majority of points.

• Rule 5: Formatting Details

  • Always write a formal title for the graph.

• Rule 6: Mathematical Formula for Gradient

  • The gradient represents the change in $y$ divided by the change in $x$.

  • $Gradient = \frac{y_2 - y_1}{x_2 - x_1}$

Question 1: Mass and Volume Relationship

• Data Table:

  • $Mass / (g)$: $5$, $15$, $25$, $30$, $60$

  • $Volume / (cm^3)$: $1$, $3$, $5$, $6$, $12$

• Graph Plotting Instructions:

  • Plot $Mass$ on the $y-axis$.

  • Plot $Volume$ on the $x-axis$.

  • Scale for $y-axis$: $2\,cm$ represents $5\,g$.

  • Scale for $x-axis$: $1\,cm$ represents $1\,cm^3$.

• Post-Plotting Analysis:

  • Determine the gradient of the resulting line.

  • State the specific units related to the gradient.

  • Identify what the gradient represents physically (the relationship between mass and volume).

Question 2: Mass and Density Correlation

• Data Table:

  • $Mass / (kg)$: $1$, $2$, $5$, $6$, $12$

  • $Density / (kg/m^3)$: $10$, $20$, $50$, $60$, $120$

• Graph Plotting Instructions:

  • Plot $Mass$ on the $y-axis$.

  • Plot $Density$ on the $x-axis$.

  • Scale for $y-axis$: $2\,cm$ represents $1\,kg$.

  • Scale for $x-axis$: $1\,cm$ represents $10\,kg/m^3$.

• Post-Plotting Analysis:

  • Determine the gradient and the units of the gradient.

  • Identify what physical property or relationship the gradient represents.

Question 3: Mass and Volume Practice with Best Fit Lines

• Note: Best fit lines should be applied to all following practice questions.

• Data Table:

  • $Mass / (g)$: $5$, $13$, $17$, $21$, $25$, $29$, $40$

  • $Volume / (cm^3)$: $0$, $10$, $15$, $20$, $25$, $30$, $40$

• Graph Plotting Instructions:

  • Plot $Mass$ on the $y-axis$.

  • Plot $Volume$ on the $x-axis$.

  • Scale for $y-axis$: $1\,cm$ represents $2\,g$.

  • Scale for $x-axis$: $2\,cm$ represents $5\,cm^3$.

• Post-Plotting Analysis:

  • Determine the gradient and state the units of the gradient.

Question 4: Force and Mass Experiments

• Data Table:

  • $Force / (N)$: $0$, $5$, $10$, $16$, $20$, $24$

  • $Mass / (kg)$: $0$, $1$, $2$, $3$, $4$, $5$

• Graph Plotting Instructions:

  • Plot $Force$ on the $y-axis$.

  • Plot $Mass$ on the $x-axis$.

  • Scale for $y-axis$: $1\,cm$ represents $1\,N$.

  • Scale for $x-axis$: $2\,cm$ represents $1\,kg$.

• Post-Plotting Analysis:

  • Calculate the gradient and determine its units.

  • Practical Calculation: If an object made of the same material used in this experiment has a mass of $3.5\,kg$, determine the force acting on it based on your graph.

Question 5: Supplemental Density and Mass Analysis

• Data Table:

  • $Mass / (kg)$: $1$, $2$, $5$, $6$, $12$

  • $Density / (kg/m^3)$: $10$, $20$, $50$, $60$, $120$

• Graph Plotting Instructions:

  • Plot $Mass$ on the $y-axis$ and $Density$ on the $x-axis$.

  • Scale for $y-axis$: $2\,cm$ represents $1\,kg$.

  • Scale for $x-axis$: $1\,cm$ represents $10\,kg/m^3$.

• Calculations:

  • Determine the gradient using the formula:

  • $Gradient = \frac{y_2 - y_1}{x_2 - x_1}$

  • State the units of the gradient and identify what the gradient represents.

Attribution

• The material provided in these notes is based on the work of T. Harding, A. Lovell, and D. Whitehall.