Lab 06: Archimedes' Principle and Specific Gravity

Lab 06: Archimedes' Principle and Specific Gravity

Learning Objectives

Define buoyant force, apparent weight, density, and specific gravity.
Verify Archimedes' Principle using weight readings in air and in water.
Determine the specific gravity of a solid denser than water using the loss-of-weight method.
Determine the specific gravity of a liquid using apparent weight measurements.
Apply the optional floating solid method and direct displaced-volume method.
Analyze why objects seem lighter in fluids and why floating occurs.
Identify common sources of error in buoyancy and specific-gravity experiments.

Archimedes' Principle explains why objects appear lighter in water and why floating objects displace fluid. When an object is partially or fully immersed in a fluid, the fluid exerts an upward buoyant force on the object. This experiment uses apparent weight readings and displaced volume to determine buoyant force, density, and specific gravity.

Target Learning Outcome

TLO 6: Apply Archimedes' Principle to determine buoyant force, apparent weight, and specific gravity of solids and liquids.

I. Discussion of Theory

Fluid Pressure

Fluid pressure is the force per unit area exerted by a fluid. In a static fluid, pressure increases with depth due to the weight of the fluid above.

Buoyant Force

The buoyant force is the upward force exerted by a fluid on an object that is partially or completely immersed in it. This upward force occurs because fluid pressure is greater at greater depth, so the upward pressure force on the bottom of an object is larger than the downward pressure force on the top.

Archimedes' Principle

Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.

Archimedes' Principle

The buoyant force equals the weight of the displaced fluid.

F_b = W_{\text{fluid displaced}} = \rho_f V_d g

Variables

Symbol	Description	Unit
$F_b$	buoyant force	N
$W_{\text{fluid displaced}}$	weight of fluid displaced by the object	N
$\rho_f$	density of the fluid	$kg/m^3$
$V_d$	volume of displaced fluid	$m^3$
$g$	acceleration due to gravity	$m/s^2$

Apparent Weight

Apparent weight is the weight reading of an object while it is immersed in a fluid. It is less than the true weight because the buoyant force acts upward.

Apparent weight

The apparent weight is the true weight minus the buoyant force.

W_{\text{apparent}} = W_{\text{air}} - F_b

F_b = W_{\text{air}} - W_{\text{fluid}}

Variables

Symbol	Description	Unit
$W_{\text{air}}$	weight of the object in air	N
$W_{\text{fluid}}$	apparent weight of the object while immersed in fluid	N
$F_b$	buoyant force	N

Density

Density is the mass of a substance per unit volume.

Specific Gravity

Specific gravity is the ratio of the density of a substance to the density of water. It has no unit because it is a ratio of two densities.

Specific gravity by density ratio

SG = \frac{\rho_{\text{object}}}{\rho_{\text{water}}}

Specific gravity of a solid denser than water

For a solid fully submerged in water, the loss of weight equals the buoyant force.

SG_{\text{solid}} = \frac{W_{\text{air}}}{W_{\text{air}} - W_{\text{water}}}

Variables

Symbol	Description	Unit
$W_{\text{air}}$	weight of solid in air	N
$W_{\text{water}}$	apparent weight of solid in water	N

Specific gravity of a liquid

Using the same solid immersed in water and in the test liquid, the liquid's specific gravity can be found from loss of weight readings.

SG_{\text{liquid}} = \frac{W_{\text{air}} - W_{\text{liquid}}}{W_{\text{air}} - W_{\text{water}}}

Variables

Symbol	Description	Unit
$W_{\text{liquid}}$	apparent weight of the solid in the test liquid	N

Specific gravity of a floating solid (Auxiliary Weight Method)

Using a sinker to fully submerge a lighter solid.

SG_{\text{floating solid}} = \frac{W_{\text{s, air}}}{W_{\text{s, air}} + W_{\text{a, water}} - W_{\text{s+a, water}}}

Variables

Symbol	Description	Unit
$W_{\text{s, air}}$	weight of floating solid in air	N
$W_{\text{a, water}}$	apparent weight of auxiliary weight alone in water	N
$W_{\text{s+a, water}}$	apparent weight of solid plus auxiliary weight in water	N

Direct volume density method

\rho_{\text{object}} = \frac{m_{\text{object}}}{V_d}

Variables

Symbol	Description	Unit
$m_{\text{object}}$	mass of the object	kg
$V_d$	displaced volume	$m^3$

Floating condition

A floating object is in vertical equilibrium. Its buoyant force equals its weight:

F_b = W

If $SG > 1$ , the object is generally denser than water and tends to sink. If $SG < 1$ , the object is generally less dense than water and tends to float.

II. Equipment / Materials Needed

Equipment or material	Purpose
Spring balance or force sensor	Measures true and apparent weight.
Solid sample denser than water	Object used for loss-of-weight method.
Solid sample less dense than water	Object used for floating solid method.
Beaker or overflow can	Holds water or test liquid; overflow can measures displaced fluid.
Water	Reference fluid for specific gravity.
Test liquid	Liquid whose specific gravity may be determined.
Thread or thin string	Suspends the object from the balance.
Auxiliary weight (sinker)	Helps fully immerse a floating object.
Graduated cylinder	Optional measurement of displaced volume.
Thermometer	Optional check of water temperature, since density depends slightly on temperature.

Safety and Setup Reminders

Do not let the suspended object touch the bottom or sides of the container during weighing.
Ensure the spring balance is zeroed before taking measurements.
Keep electrical sensors away from spilled liquid, and wipe wet apparatus before storage.
When handling test liquids (e.g., alcohol, saltwater), clean the solid sample between measurements.

III. Diagram of Setup

Apparent weight measurement

        spring balance
             |
             |
          object
      _______________
     |               |
     |     water     |
     |_______________|

The object must be fully submerged and freely suspended.
It must not touch the container.

Forces on a submerged object

             Fb upward
                ↑
            +-------+
            | solid |
            +-------+
                ↓
             W downward

The scale reads the apparent weight:
W_apparent = W_air - Fb

Optional floating solid with auxiliary weight

        spring balance
             |
         [floating]
          [solid]
             |
         (auxiliary)
          (weight)
      _______________
     |               |
     |     water     |
     |_______________|

Both objects are fully submerged to measure W_{s+a, water}.

Optional hydrometer application

             |  |
             |  | reading scale
           __|__|__
          |        |
          | liquid |
          |        |
          |________|

IV. Procedure

A. Specific gravity of a solid denser than water

STEP-BY-STEP

Tie the solid sample securely with a thin thread.
Hang the object from the spring balance and record its weight in air as $W_{\text{air}}$ .
Fill the beaker with enough water to fully submerge the sample.
Lower the sample into the water while it remains suspended from the spring balance.
Make sure the sample is fully submerged and not touching the bottom or sides of the container.
Record the apparent weight in water as $W_{\text{water}}$ .
Repeat the measurement for at least three trials.
Compute the buoyant force using $F_b = W_{\text{air}} - W_{\text{water}}$ .
Compute the specific gravity using $SG = \frac{W_{\text{air}}}{W_{\text{air}} - W_{\text{water}}}$ .

B. Specific gravity of a liquid

STEP-BY-STEP

Use the same solid sample from Part A.
Record its weight in air as $W_{\text{air}}$ if not already recorded.
Record its apparent weight in water as $W_{\text{water}}$ .
Pour the test liquid into a clean container.
Fully submerge the solid in the test liquid without letting it touch the container.
Record the apparent weight in the test liquid as $W_{\text{liquid}}$ .
Compute the specific gravity of the liquid using $SG_{\text{liquid}} = \frac{W_{\text{air}} - W_{\text{liquid}}}{W_{\text{air}} - W_{\text{water}}}$ .

C. Optional floating solid method

STEP-BY-STEP

Record the weight of the floating solid in air as $W_{\text{s, air}}$ .
Attach an auxiliary weight (sinker) to the spring balance.
Lower the auxiliary weight into water so it is completely submerged, but leave the floating solid in air. Measure the combined weight as $W_{\text{a, water}} + W_{\text{s, air}}$ . Subtract $W_{\text{s, air}}$ to find $W_{\text{a, water}}$ .
Attach both the floating solid and auxiliary weight together.
Lower both into the water until completely submerged.
Record the apparent weight of the floating solid plus auxiliary weight in water as $W_{\text{s+a, water}}$ .
Compute specific gravity using $SG_{\text{floating solid}} = \frac{W_{\text{s, air}}}{W_{\text{s, air}} + W_{\text{a, water}} - W_{\text{s+a, water}}}$ .

D. Optional displaced-volume method

STEP-BY-STEP

Fill an overflow can with water until it stops dripping.
Place an empty graduated cylinder under the spout.
Gently lower the solid object into the overflow can until fully submerged.
Collect the displaced water in the graduated cylinder.
Record the volume of displaced water as $V_d$ .
Calculate the density and specific gravity using the direct volume density method.

Measurement reminder

For accurate results, remove air bubbles from the object surface before recording apparent weight. Air bubbles increase buoyancy and make the apparent weight too small.

Unit consistency

Ensure consistent units are used throughout the calculations. Do not mix newtons and gram-force or kilogram-force without proper conversion. It is standard to use Newtons (N) for weight and forces.

V. Student Information

Field	Entry
Name
Schedule
Group No.
Date Performed

VI. Expected Trends

Expected trends

Apparent weight is less than true weight due to the upward buoyant force.
A denser liquid gives a larger buoyant force for the same displaced volume.
An object touching the container gives a wrong reading (lower if touching bottom, higher or erratic if touching side due to friction).

VII. Data and Results

Table 6.1. Solid Denser Than Water

Trial	$W_{\text{air}}$ (N)	$W_{\text{water}}$ (N)	$F_b = W_{\text{air}} - W_{\text{water}}$ (N)	$SG_{\text{solid}}$
1
2
3
Average

Table 6.2. Specific Gravity of a Test Liquid

Trial	$W_{\text{air}}$ (N)	$W_{\text{water}}$ (N)	$W_{\text{liquid}}$ (N)	$SG_{\text{liquid}}$
1
2
3
Average

Table 6.3. Optional Floating Solid

Quantity	Description	Unit
$W_{\text{s, air}}$	weight of floating solid in air	N
$W_{\text{a, water}}$	apparent weight of auxiliary weight alone in water	N
$W_{\text{s+a, water}}$	apparent weight of solid plus auxiliary weight in water	N
$F_{b,s}$	buoyant force on floating solid	N
$SG_{\text{floating solid}}$	specific gravity of floating solid	none

Table 6.4. Optional Displaced Volume

Quantity	Value	Unit
Mass of object ( $m$ )		kg
Displaced volume ( $V_d$ )		m $^3$
Density ( $\rho$ )		kg/m $^3$
Specific gravity ( $SG$ )		none

VIII. Computations

Required computations

STEP-BY-STEP

Compute the loss of weight in water for the solid: $W_{\text{air}} - W_{\text{water}}$ .
Interpret this loss of weight as the buoyant force.
Compute $SG_{\text{solid}}$ for each trial.
Average the computed values of $SG_{\text{solid}}$ .
For a test liquid, compute the loss of weight in liquid: $W_{\text{air}} - W_{\text{liquid}}$ .
Compute $SG_{\text{liquid}}$ using the water loss as the reference.
Report all final values with appropriate significant figures.

Sample computation: buoyant force

Suppose an object displaces $0.0002\,\text{m}^3$ of water ( $\rho = 1000\,\text{kg/m}^3$ ). The buoyant force is:

F_b = \rho_f V_d g = (1000\,\text{kg/m}^3)(0.0002\,\text{m}^3)(9.81\,\text{m/s}^2) = 1.962\,\text{N}

Sample computation: solid specific gravity

Suppose a metal sample weighs $4.80\,\text{N}$ in air and $4.20\,\text{N}$ in water.

F_b = W_{\text{air}} - W_{\text{water}} = 4.80 - 4.20 = 0.60\,\text{N}

SG_{\text{solid}} = \frac{W_{\text{air}}}{W_{\text{air}} - W_{\text{water}}} = \frac{4.80}{0.60} = 8.0

Sample computation: liquid specific gravity

Suppose the same solid weighs $4.80\,\text{N}$ in air, $4.20\,\text{N}$ in water, and $4.32\,\text{N}$ in a test liquid.

SG_{\text{liquid}} = \frac{W_{\text{air}} - W_{\text{liquid}}}{W_{\text{air}} - W_{\text{water}}}

SG_{\text{liquid}} = \frac{4.80 - 4.32}{4.80 - 4.20} = \frac{0.48}{0.60} = 0.80

Sample computation: optional direct density method

If an object has a mass of $0.20\,\text{kg}$ and displaces $2.5 \times 10^{-4}\,\text{m}^3$ of water:

\rho_{\text{object}} = \frac{m}{V_d} = \frac{0.20}{2.5 \times 10^{-4}} = 800\,\text{kg/m}^3

Since water density is $1000\,\text{kg/m}^3$ , the specific gravity is $800 / 1000 = 0.8$ .

IX. Error Analysis

Common sources of error

Object touching the bottom or sides of the container.
Air bubbles attached to the submerged object.
Reading the spring balance before the object is steady.
Thread or support material adding extra force effects.
Object not fully submerged.
Water or test liquid spilling during immersion.
Inaccurate zero setting of the balance.
Temperature differences affecting liquid density.

Ways to improve accuracy

Zero the spring balance before taking readings.
Fully submerge the object without contact with the container.
Remove air bubbles before recording apparent weight.
Repeat readings and average the results.
Use the same solid for water and liquid measurements.
Keep the object steady before reading the scale.

X. Observations and Conclusions

Conclusion guide

A strong conclusion should state whether the results support Archimedes' Principle, report the computed specific gravity values, and explain why apparent weight is smaller in water than in air. Mention the most likely source of error and how it affected the result.

XI. Applications

Hydrometer

A hydrometer is a sealed weighted tube used to measure liquid density or specific gravity. It floats higher in denser liquids and lower in less dense liquids because it changes its displaced volume until the buoyant force equals its weight.

Floating ships

A large object can float if it displaces enough fluid so that the buoyant force equals its weight. This is why a hollow steel ship can float even though solid steel is denser than water. The hollow shape increases the displaced volume of water.

Density testing

Archimedes' Principle is widely used in geology, material science, and quality control to test the density and specific gravity of rocks, metals, and manufactured parts.

XII. Lab Report Format

Include the following sections in your lab report

STEP-BY-STEP

Title Page: Experiment title, your name, group members, and date.
Objectives: Briefly state the purpose of the experiment.
Data and Results: Completed Tables 6.1 through 6.4.
Computations: Show one full sample computation for each required formula.
Answers to Questions: Provide clear and concise answers to the post-lab questions.
Conclusion: Summarize the findings based on the conclusion guide.

XIII. Post-Lab Questions

State Archimedes' Principle in your own words.
Why does a submerged object appear lighter in water than in air?
A solid weighs $6.50\,\text{N}$ in air and $5.70\,\text{N}$ when fully submerged in water. Find the buoyant force and the specific gravity of the solid.
A solid weighs $8.00\,\text{N}$ in air, $6.40\,\text{N}$ in water, and $6.80\,\text{N}$ in another liquid. Find the specific gravity of the liquid.
Why must the object not touch the sides or bottom of the container during apparent weight measurement?
How does a hydrometer use the condition $F_b = W$ to compare liquid densities?

XIV. Selected Answer Key

Answer key

Q3: Buoyant force is $0.80\,\text{N}$ ; Specific gravity is $8.125$ .
Q4: Buoyant force in water is $1.60\,\text{N}$ ; Buoyant force in liquid is $1.20\,\text{N}$ ; Specific gravity of the liquid is $0.75$ .

XV. References

Bueche, F. J., & Hecht, E. (1997). Schaum's Outline of Theory and Problems of College Physics (9th ed.). New York: McGraw-Hill.

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