6.2 – Determining Acceleration Due to Gravity Revision Notes for Leaving Cert Physics

6.2 – Determining Acceleration Due to Gravity

Purpose of the experiment

This experiment aims to determine the acceleration due to gravity (g) using a free-fall apparatus and a scaler timer. You'll also verify that the equation $g = 2s/t²$ gives an accurate value for g and demonstrates that this equation correctly models free fall near Earth's surface.

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This experiment provides hands-on experience with one of the most fundamental constants in physics while teaching important experimental techniques like timing measurements and graphical analysis.

Theory behind the experiment

When an object falls freely under gravity, it follows the kinematic equation:

$s = \frac{1}{2}gt^2$

Where:

s = distance fallen (m)
g = acceleration due to gravity (m/s²)
t = time taken to fall (s)

Rearranging this equation to find g:

$g = \frac{2s}{t^2}$

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Why we rearrange the equation:

By rearranging the kinematic equation to solve for g, we can directly calculate the acceleration due to gravity from our measured values of distance and time, making it the dependent variable in our experiment.

The accepted value for acceleration due to gravity is approximately 9.8 m s⁻¹.

Equipment required

Free-fall apparatus
Scaler timer and connecting leads
Retort stand and clamp
Metre stick

Experimental method

Setting up the experiment

Equipment setup: Position the equipment as shown in the apparatus diagram. Ensure the switch is in position A and the electromagnet is holding the ball in place.
Timer connection: Connect the timer correctly following the manual or your teacher's instructions. Check that the timer switches on and off when it should.

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Critical Setup Check:

Before proceeding with measurements, always verify that your timer is working correctly by testing the switch mechanism. A malfunctioning timer will render all your data useless.

Conducting the measurements

Distance measurement: Measure the distance from the bottom of the ball (when held in position by the electromagnet) to the top of the trapdoor. Record this value as your distance s.
Taking timing measurements: Set the timer to zero. Flick the switch to B, simultaneously releasing the ball and starting the timer. When the ball strikes the trapdoor, the timer stops. Record the timing reading.
Repeat measurements: Reset the timer and repeat step 4 at least three times. Use the smallest time value you recorded, as errors in measurement are likely to result in larger values.
Multiple distances: Repeat the entire procedure five or six times, each time with a different value for distance s.

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Why Multiple Measurements Matter:

Taking multiple readings for each distance allows you to identify and minimise random errors. The smallest time reading is typically the most accurate since timing errors usually result in overestimation.

Data collection and analysis

Recording your results

Create a data table with the following columns:

Distance fallen, s (m)	t₁ (s)	t₂ (s)	t₃ (s)	t₄ (s)	Smallest time, t (s)	Acceleration due to gravity, g (m·s⁻²)

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Sample Calculation:

If a ball falls a distance of s = 0.80 m in time t = 0.404 s:

Step 1: Apply the formula $g = \frac{2s}{t^2}$

Step 2: Substitute the values $g = \frac{2 \times 0.80}{(0.404)^2}$

Step 3: Calculate $g = \frac{1.60}{0.163} = 9.82$ m s⁻²

This result is very close to the accepted value of 9.8 m s⁻¹.

Calculating acceleration due to gravity

Calculate g values: Complete a table of results and calculate the average value of g using the formula $g = 2s/t²$ .
Graphical analysis: Plot a graph of s (on the y-axis) against t² (on the x-axis). Draw the best straight line that fits the points and measure its slope. The acceleration due to gravity is given by: $g = 2 \times \text{slope of graph}$ . Compare this value with your calculated average from step 7.

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Understanding the Graph:

The relationship s = ½gt² can be written as s = (g/2)t². This is in the form y = mx, where the slope m = g/2. Therefore, g = 2 × slope, giving you an independent method to verify your calculated results.

Verification of results

Given that acceleration due to gravity is 9.8 m s⁻¹, your results should verify that the equation $g = 2s/t²$ accurately models free fall near Earth's surface.

Sources of error

Understanding potential errors helps improve your experimental accuracy:

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Critical Error Sources:

Object selection: Ensure the object that falls is a small, smooth, dense metal sphere. This minimises the effect of air resistance, allowing you to use the equation $g = 2s/t²$ with negligible error.
Timing accuracy: For each fall of a given height, use the smallest value of time t. Errors in measuring time tend to overestimate its value.
Distance measurement: Take great care when measuring the value of s. Avoid parallax error by ensuring your eye is level with the measurement points.

Key exam questions

Why should the smallest time for a falling distance be used in calculations to find g?
State the effect on the final result if the time value used in the calculation is too large.
Prove that the slope of the graph multiplied by two gives the value of g.
Why is the formula $g = 2s/t²$ used rather than $s = \frac{1}{2}gt²$ in this experiment?
List two disadvantages of only allowing the metal ball to fall through a small distance (say 40 cm) in this experiment.
Give two ways of minimising the effect of air resistance in this experiment.

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Key Points to Remember:

The key formula for this experiment is g = 2s/t²
Always use the smallest recorded time to minimise measurement errors
A dense, smooth metal sphere reduces air resistance effects significantly
The accepted value for g is approximately 9.8 m s⁻¹
Graphical analysis provides an independent method to verify your calculated results
Multiple measurements at different distances improve the reliability of your results

6.2 – Determining Acceleration Due to Gravity (Leaving Cert Physics): Revision Notes