1.2 - Scientific Investigation

Interpreting Graph Shapes

Graphs of investigation data show the relationship between two physical quantities.

Shape of graph of $y$ against $x$	Interpretation
Straight line through the origin with a positive gradient	$y$ is directly proportional to $x$: $y\propto x$, $y=mx$.
Straight line with a positive gradient, not through the origin	$y$ increases linearly with $x$: $y=mx+c$.
Straight line with a negative gradient, not through the origin	$y$ decreases linearly with $x$: $y=mx+c$, with $m<0$.
Curve through the origin with a positive gradient	$y$ increases with $x$.
Curve not through the origin with a negative gradient	$y$ decreases with $x$.
Inverse-proportion curve with a negative gradient that does not intersect either axis	$y$ is inversely proportional to $x$: $y\propto 1/x$. A graph of $y$ against $1/x$ is a straight line through the origin with a positive gradient.

• A straight line that increases but does not pass through the origin indicates a linear increase rather than direct proportionality.
• For a straight line, gradient $m$ and $y$-intercept $c$ give the equation $y=mx+c$.

Analysing Graphs

1. Relationship: interpret the graph shape to describe how the two variables are related.
2. Gradient: calculate $\Delta y/\Delta x$ and identify the physical quantity represented by the gradient.
3. Area under the graph: use the relevant geometric area formula, including rectangle and triangle areas, to determine the physical quantity represented.
4. Interpolation: read a value within the range of the graph; use a known $x$ to obtain $y$, or a known $y$ to obtain $x$.
5. Extrapolation: extend the graph beyond the experimental data to predict the corresponding $x$ or $y$ value.

Force–Extension Graph of a Spring

• Arrangement: a spring hangs from a retort stand with slotted weights attached. Extension, $x$, is the increase from its initial length.
• The relationship is $F=kx$, where $F$ is force, $x$ is extension and $k$ is the spring constant.
• For a graph of $F$ against $x$, the gradient represents $k$.
• The area under the force–extension graph represents the work done in stretching the spring.
• The supplied data pairs, written as $(x/\text{cm},F/\text{N})$, are $(0.8,0.5)$, $(1.6,1.0)$, $(2.4,1.5)$, $(3.2,2.0)$, $(4.0,2.5)$, $(4.8,3.0)$, $(5.6,3.5)$ and $(6.4,4.0)$.

Experiment: Length and Period of a Simple Pendulum

• Inference: the period of oscillation depends on the pendulum’s length.
• Hypothesis: a longer pendulum has a longer period of oscillation.
• Aim: investigate the relationship between pendulum length, $l$, and period of oscillation, $T$.

Variable	Quantity
Manipulated	Length of pendulum, $l$
Responding	Period of oscillation, $T$
Constant	Mass of pendulum bob

• Apparatus: retort stand, protractor, pendulum bob, stopwatch, metre rule and G-clamp.
• Materials: 100 cm thread and two small pieces of plywood.
• Arrangement: suspend the bob by the thread from the retort stand, with the thread held between the plywood pieces. Secure the stand using the G-clamp. Pendulum length extends from the suspension point to the centre of the bob.

Pendulum Procedure and Results

1. Set up the pendulum apparatus and adjust $l$ to 20.0 cm.

2. Displace the bob by an angle less than $10^{\circ }$ from the vertical, then release it to oscillate.

3. Measure the time, $t_1$, for 20 complete oscillations.

4. Repeat the timing and record $t_2$.

5. Calculate the mean time and the period, then calculate $T^2$:

   $$t_{\text{average}}=\frac{t_1+t_2}{2},T=\frac{t_{\text{average}}}{20}$$

6. Repeat the measurements and calculations for $l=$ 30.0, 40.0, 50.0, 60.0 and 70.0 cm.

7. Record the results for each length, including both timings, their mean, $T$ and $T^2$.

Recorded or calculated quantity	Unit
Pendulum length, $l$	cm
Time for 20 oscillations: $t_1$, $t_2$, $t_{\text{average}}$	s
Period, $T$	s
Square of period, $T^2$	${\text{s}}^2$

Pendulum Analysis and Experimental Report

• Data analysis uses separate graphs of $T$ against $l$ and $T^2$ against $l$ to establish the shape and relationship in each case.
• The gradient, $m$, of $T^2$ against $l$ is expressed in S.I. units. Here, $g$ is the Earth’s gravitational acceleration:

• Comparing with $y=mx$ gives $m=4{\pi }^2/g$, hence $g=4{\pi }^2/m$. Use $l$ in metres to obtain $g$ in ${\text{m s}}^{-2}$.
• The standard value for comparison is $g=9.81{\text{m s}}^{-2}$.
• A complete report records the inference, hypothesis, aim, variables, apparatus, materials, procedure, results, data analysis, conclusion and discussion.
• The conclusion follows from the measured results and the graph relationships.