Physics Required Practical: Determination of g

Determining the acceleration due to gravity, g, is a cornerstone experiment in physics that connects fundamental theory with measurable reality. Mastering this required practical is essential not only for exam success but also for developing core experimental skills in precision measurement, data analysis, and uncertainty evaluation. You will learn to use timing technology to capture the brief motion of a falling object and extract a fundamental constant of our universe from your own data.

The Theoretical Foundation: Free-Fall Kinematics

The experiment is built upon the equations of motion for an object under constant acceleration. For an object starting from rest (initial velocity $u=0$), the displacement $s$ is related to the time taken $t$ and the acceleration $a$ by the equation: $$s=\frac{1}{2}a{t}^2$$ In the specific case of an object in free-fall near the Earth's surface, neglecting air resistance, the acceleration $a$ is equal to g. The value of g varies slightly by location but is approximately $9.81{\text{ m s}}^{-2}$. The key to a precise measurement is to rearrange this relationship into a linear form suitable for graphical analysis. By plotting the displacement $s$ (on the y-axis) against the square of the time $t^2$ (on the x-axis), the equation becomes: $$s=(\frac{1}{2}g)t^2$$ This is now in the linear form $y=mx+c$, where the gradient $m$ is equal to $\frac{1}{2}g$. Therefore, determining the gradient of a line of best fit through your $s$ vs $t^2$ data allows you to calculate g using $g=2m$.

Apparatus Setup: Precision Timing and Release

A reliable experiment depends on eliminating human reaction time from the measurement. The standard setup involves a steel ball bearing, an electromagnetic release mechanism, and a pair of light gates connected to an electronic timer or data logger.

First, the electromagnet is clamped securely to a retort stand. When powered, it holds the steel ball. The circuit includes a switch to cut power to the magnet, initiating the fall without applying any force. Directly below the magnet, you position the first light gate. This gate starts the electronic timer the moment the ball begins its fall. A second light gate is positioned a measurable vertical distance $s$ below the first. As the ball passes this second gate, it stops the timer, recording the time $t$ taken to fall that distance. The distance $s$ is measured precisely using a meter rule or vernier calipers, taking the measurement from the bottom of the ball at the first gate to the bottom of the ball at the second gate to ensure consistency.

Data Collection and Primary Uncertainty Analysis

Your goal is to collect data for time $t$ across a range of distances $s$. You typically start with the second light gate close to the first and then move it down in increments (e.g., 0.200 m), recording the new distance $s$ and the corresponding time $t$ for each position. For each distance, you should repeat the timing several times to calculate a mean time, reducing the impact of random errors.

The major sources of uncertainty you must consider are:

1. Measurement Precision for $s$: Use the smallest division of your ruler (e.g., 1 mm = 0.001 m). The absolute uncertainty might be ±0.001 m. This is most significant for smaller drop distances.
2. Reaction Time (Eliminated): By using the electronic release and light gates, you remove this systematic human error from the timing process. However, the timer itself has a finite resolution (e.g., ±0.001 s), which contributes to uncertainty in $t$.
3. Air Resistance: This is the primary physical limitation. It opposes the weight of the ball, causing the measured acceleration to be less than the true g. Its effect is more pronounced at higher speeds (longer drops). Using a dense, smooth sphere (like steel) minimises this effect.
4. Electromagnetic Release: If the electromagnet retains some residual magnetism or does not release the ball cleanly, it can impart a small initial velocity or delay, introducing a systematic error.

Graphical Analysis and Evaluation

After collecting your data table of $s$ and mean $t$, you compute $t^2$ for each entry. Plotting $s$ against $t^2$ should yield a straight line through the origin if the acceleration is constant. You then draw a line of best fit.

Determining g from the gradient: Calculate the gradient $m$ of your line by choosing two points on the line (not from your data points) that are far apart: $m=\Delta s/\Delta t^2$. Your value for g is then $g=2m$. To evaluate experimental accuracy, compare your value to the accepted local value (e.g., 9.81 m s⁻²). Calculate the percentage difference.

A crucial part of the analysis is using your graph to assess the experiment. A line of best fit that does not pass through the origin suggests a systematic error, such as a consistent delay in the timer start or an incorrect zero for distance. Furthermore, you should consider the percentage uncertainty in your gradient. This can be estimated by drawing worst-acceptable lines (lines of maximum and minimum reasonable gradient through the data points' error bars). The percentage uncertainty in the gradient is: $$\frac{\text{(max gradient - min gradient)}}{2\times \text{best gradient}}\times 100\%$$ Since $g=2m$, the percentage uncertainty in g is the same as that in $m$. This quantitative uncertainty should be reported alongside your value for g.

Common Pitfalls

1. Ignoring the Non-Linear Relationship: Attempting to plot $s$ against $t$ directly and misinterpret the curve. This makes it impossible to determine a single gradient for g.

• Correction: Always transform your data to plot $s$ vs $t^2$. The linear graph is the only reliable way to find g from this experiment.

1. Inconsistent Distance Measurement: Measuring $s$ from the top of the ball at one gate and the bottom at the other, or not accounting for the ball's radius. This introduces a fixed systematic error in all displacement readings.

• Correction: Define clear, consistent points on the ball (e.g., the bottom) from which to measure at both the start and end positions. Use the same points for every measurement.

1. Underestimating Air Resistance: Assuming it is negligible without justification. If your value for g is consistently and significantly below 9.81 m s⁻², air resistance is the most likely cause.

• Correction: Use a dense, smooth, spherical object. Acknowledge air resistance as the main limitation in your evaluation. Note that its effect would cause your $s$ vs $t^2$ graph to curve downwards at larger $t^2$ values (as acceleration decreases).

1. Poor Graphical Technique: Drawing a line of best fit that connects the dots or forcing it through the origin when the data doesn't support it.

• Correction: Draw a single straight line with roughly an equal number of points above and below it. Only force it through (0,0) if you are certain there is no systematic error in your distance measurement.

Summary

• The acceleration due to gravity g is determined by analysing the linear relationship between displacement $s$ and the square of fall time $t^2$, derived from $s=\frac{1}{2}g{t}^2$.
• Key apparatus includes an electromagnetic release and light gates to eliminate human reaction time and accurately measure short time intervals.
• Major sources of uncertainty include measurement precision for distance, timer resolution, and the systematic effect of air resistance, which is minimised using a dense sphere.
• Data is processed by plotting an $s$ vs $t^2$ graph, where the gradient $m$ gives g via $g=2m$.
• Experimental accuracy is evaluated by comparing the calculated g to the accepted value and considering percentage difference, while the reliability of the result is quantified by analysing the percentage uncertainty in the graph's gradient.