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

9.1 – Determining Acceleration Due to Gravity

Experimental aim

This experiment has two main objectives that work together to help us understand how gravity affects pressure in fluids:

a) Primary aim: To determine the acceleration due to gravity (g) using the hydrostatic pressure formula:

$g = \frac{P}{\rho h}$

This formula comes from the fundamental relationship F = hρg, where force is related to height, density, and gravity.

b) Secondary aim: To verify that this equation gives an accurate value for g, proving that the equation correctly models how pressure behaves in liquids near Earth's surface.

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The hydrostatic pressure formula is derived from the principle that pressure in a fluid increases with depth due to the weight of the fluid column above. This fundamental relationship allows us to use simple pressure measurements to determine one of physics' most important constants.

Summary of method

The experiment works by measuring how pressure changes with depth in a liquid. Here's the basic principle:

When you submerge a pressure sensor deeper into water, the pressure increases because there's more water pressing down from above. This relationship between pressure and depth is directly connected to gravitational acceleration. By measuring pressure at different depths and knowing the water's density, you can calculate g.

The beauty of this method is that it uses everyday materials to measure one of the most fundamental constants in physics!

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This experimental approach is particularly elegant because it demonstrates the direct connection between a measurable quantity (pressure) and a fundamental constant (gravitational acceleration), making abstract physics concepts tangible and observable.

Equipment required

You'll need the following apparatus for this experiment:

Absolute pressure sensor - this measures the total pressure at any depth
Plastic tubing - connects the sensor to the measurement point underwater
Metre stick - for measuring depth accurately
Graduated cylinder - holds the liquid (usually water) at known depths
Data logger and computer - records and processes the pressure measurements
Water - the liquid medium for pressure measurement

Detailed experimental method

Follow these steps carefully to ensure accurate results:

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Step-by-Step Procedure:

Step 1: Initial setup Connect the pressure sensor and tube to the metre stick. Measure and record the atmospheric pressure when the tube bottom is not immersed in water (h = 0). This gives you the baseline atmospheric pressure P₀.

Step 2: Prepare the measurement system Place the bottom of the tubing at a fixed depth (e.g. 5 cm) in a graduated cylinder filled with liquid of known density ρ (usually water).

Step 3: Take measurements

Measure the depth h of the bottom of the tubing below the liquid surface
Record the pressure P_abs given by the data logger at this depth

Step 4: Repeat for multiple depths Repeat this procedure for a series of increasing depths (e.g. 5 cm intervals), recording all pressure values in a data table.

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Ensure consistent measurement intervals and maintain the same liquid temperature throughout the experiment, as density can vary with temperature changes.

Handling the data

You can analyse your results using two different methods:

Method 1: Direct calculation

For each pair of values, calculate g using the formula:

$g = \frac{(P_{abs} - P_0)}{h \rho}$

Where:

P_abs is the absolute pressure at depth h
P₀ is atmospheric pressure
h is the depth below the surface
ρ is the density of the liquid

Calculate the average value of g, ignoring any clear outliers. The result should be close to 9.8 m s⁻², verifying that the equation accurately models pressure in liquids.

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Worked Example: Direct Calculation

If P_abs = 102,500 Pa, P₀ = 101,325 Pa, h = 0.12 m, and ρ = 1000 kg/m³:

Step 1: Calculate pressure difference P_abs - P₀ = 102,500 - 101,325 = 1,175 Pa

Step 2: Apply the formula g = (1,175) / (0.12 × 1000) = 1,175 / 120 = 9.79 m/s²

Method 2: Graphical analysis

Plot a graph of P_abs against h (or use software to do this automatically).

Draw the line of best fit and measure its slope. Since the relationship is: $P_{abs} = P_0 + \rho g h$

The slope equals ρg, so: $g = \frac{\text{slope}}{\rho}$

Again, your result should be close to 9.8 m s⁻², confirming the equation's accuracy.

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Worked Example: Graphical Analysis

If the slope of your P_abs vs h graph is 9,750 Pa/m and ρ = 1000 kg/m³:

g = slope / ρ = 9,750 / 1000 = 9.75 m/s²

Explanation of the graph

The relationship you're investigating is linear. When you plot absolute pressure (P_abs) against depth (h), you should get a straight line because:

P_abs = P₀ + ρgh
This has the form y = c + mx, where the slope m = ρg

The fact that this relationship is linear proves that pressure increases uniformly with depth, exactly as predicted by hydrostatic pressure theory.

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The linearity of this relationship is a powerful confirmation of our theoretical understanding. Any significant deviation from linearity would suggest either experimental error or that our theoretical model needs refinement.

Sources of error

Be aware of these potential sources of experimental error:

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Critical Measurement Errors to Avoid:

1. Tube positioning: Make sure the open entrance at the bottom of the tube is horizontal when submerged in water. Any angle will affect the pressure reading.

2. Meniscus reading: Read the bottom of the water meniscus in the graduated cylinder when measuring depth, not the top or middle.

3. Air bubbles: If water enters the end of the plastic tube, measure the distance from the top of this water column to the meniscus in the graduated cylinder for accurate depth measurement.

4. Parallax error: Avoid parallax when reading the metre stick by ensuring your eye is level with the measurement point.

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Key Points to Remember:

The formula g = P/(ρh) allows you to calculate gravitational acceleration from pressure measurements
Pressure increases linearly with depth in a fluid due to the weight of liquid above
Your calculated value of g should be approximately 9.8 m s⁻² if the experiment is performed correctly
Both direct calculation and graphical methods should give similar results
Careful measurement technique is essential to minimise experimental errors

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