# Methods to Increase Accuracy ## Overview - Accuracy of an experiment can be significantly increased by repeating measurements and using mean values - Methods that reduce **systematic errors** lead to increased accuracy - To reduce uncertainty in experimental results, modifications to the method may be necessary: - Timing over multiple oscillations - Using a fiducial marker - Using a set square or plumb line

Systematic errors are consistent, repeatable errors that affect all measurements in the same way

## Using Multiple Oscillations - Uncertainty in a measurement of periodic time can be reduced by: - Measuring many oscillations to calculate the average time for one oscillation - Increasing the total time measured for multiple swings ### Why This Works - When timing a single swing of a pendulum, reaction time introduces significant uncertainty - By timing 10 (or more) complete oscillations and dividing by 10, the **relative** effect of reaction time becomes much smaller - Think of it like measuring ingredients: measuring 100g is more accurate than measuring 10g and multiplying by 10 ### Quantitative Benefit The chart below shows how percentage uncertainty decreases as you time more oscillations. This example assumes a pendulum with a 2.0 s period and a typical human reaction time uncertainty of ±0.2 s:

- Timing 1 oscillation (2.0 s): ±0.2 s uncertainty = **10% uncertainty** - Timing 10 oscillations (20.0 s): ±0.2 s uncertainty = 1% uncertainty in total time, which gives **1% uncertainty** in the calculated period - The percentage uncertainty is reduced by a factor equal to the number of oscillations timed ### Practical Application - Measure the time taken for a pendulum to complete 10 complete oscillations - Divide this time by 10 to determine the time period of one oscillation - This technique works for any periodic motion (springs, sound waves, rotating systems)

One complete oscillation returns the pendulum to its starting position, moving in the same direction

### Example of One Complete Oscillation | Position | Description | |---|---| | Starting point | Pendulum bob at maximum displacement on the left | | Through centre | Bob swings right through equilibrium position | | Maximum right | Bob reaches maximum displacement on the right | | Through centre again | Bob swings left through equilibrium position | | Back to start | Bob returns to starting position (1 complete oscillation) | ## Fiducial Marker

A fiducial marker is a fixed reference point used to improve the accuracy of timing measurements

- A fiducial marker acts as a clear, consistent reference point - Commonly used when measuring the time period of a pendulum with a stopwatch ### How It Improves Accuracy - Make timings by sighting the pendulum **as it passes the fiducial marker** - Sight the pendulum at its **highest speed** (at the centre of oscillation) - The pendulum swings fastest at its lowest point - The pendulum swings slowest at the top of each swing - This reduces parallax error and gives a more precise moment to start/stop timing ### Setting Up a Fiducial Marker - Place a pin and small piece of coloured tape (e.g., blu tack) at the centre of oscillation - Position yourself at eye level with the marker - Start timing as the pendulum bob passes through the marker - Stop timing after the desired number of oscillations, again as it passes through the marker

Always sight the pendulum passing the marker in the same direction to ensure consistency

## Set Squares & Plumb Lines - These tools help ensure experimental setups are properly aligned - Misalignment introduces systematic errors that affect all measurements ### Applications of a Set Square A **set square** can be used to determine whether: 1. An object is vertical 2. Two objects are at right angles to each other 3. Two lines are parallel ### Applications of a Plumb Line - A **plumb line** (a weight suspended on a string) naturally hangs vertically due to gravity - Used to determine if a setup is vertically aligned accurately - Can be used in combination with a set square for precise alignment ### Example Setup Consider setting up apparatus to measure oscillations: | Component | Purpose | Tool Used | |---|---|---| | Clamp stand | Must be vertical | Plumb line alongside stand | | Horizontal ruler | Must be perpendicular to stand | Set square against ruler and stand | | Pendulum string | Must hang vertically when at rest | Plumb line for comparison | | Metre rule | Must be horizontal | Set square to check against vertical reference |

Even small misalignments can introduce systematic errors affecting all measurements equally

### Practical Example: Pendulum Experiment When investigating simple harmonic motion: - Use a **plumb line** alongside the clamp stand to ensure the stand is vertical - Place a **set square** against the horizontal metre rule to confirm it's perpendicular to the stand - This ensures the pendulum bob's position can be accurately measured - Without proper alignment, the measured length of the pendulum would be systematically incorrect ### Why Alignment Matters - If a ruler is tilted by just 5°, measurements could be systematically high by approximately 0.4% - For a 1.00 m pendulum, this introduces an error of about 4 mm - Since period $T \propto \sqrt{L}$, this affects the time period calculations - These errors cannot be reduced by repeating measurements—the setup must be corrected ## Summary Table | Method | What It Reduces | Why It Works | |---|---|---| | Multiple oscillations | Random timing errors | Averages out reaction time over many cycles | | Fiducial marker | Random timing errors | Provides consistent, clear reference point | | Set square | Systematic alignment errors | Ensures angles are exactly 90° | | Plumb line | Systematic alignment errors | Uses gravity to define true vertical |

Combining multiple accuracy-improving methods gives the most reliable results