Experiment 108: Archimedes’ Principle Study Guide

Grading Rubric and Course Identification

• Total Points Available: $100$
• Grade Components Breakdown:
  • Data Tables and Graphs: $20\,\text{points}$
  • Sample Computations: $5\,\text{points}$
  • Guide Questions: $10\,\text{points}$
  • Analysis: $10\,\text{points}$
  • Conclusion: $10\,\text{points}$
  • Presentation: $5\,\text{points}$
  • Group Report Grade: $40\,\text{points}$
• Course Details:
  • Experiment Number: $108$
  • Experiment Title: Archimedes’ Principle
  • Course Code: PHY104L

Introduction to Archimedes' Principle

• The Buoyant Force: When a body is immersed in a fluid, it appears to become lighter because the fluid exerts an upward force on the body. This force supports either the entire weight or a part of the weight of the object and is formally called the buoyant force.
• Archimedes' Discovery: Archimedes discovered that the difference between the weight of a body measured in air and its weight when measured in a liquid is equivalent to the weight of the liquid displaced by the body.
• Core Principle: This apparent loss of weight is exactly equal to the weight of the displaced liquid.
• Displaced Liquid Volume: The volume of the displaced liquid is equal to the volume of the portion of the body that is below the liquid's surface.
• Experimental Method: This lab utilizes the "loss of weight method" to determine the density and specific gravity of unknown solid and liquid samples.

Experimental Objectives

1. To employ Archimedes’ Principle to determine the density or specific gravity ($SG$) of a solid that sinks in water (denser than water).
2. To employ Archimedes’ Principle to determine the density or specific gravity ($SG$) of a solid that floats in water (less dense than water).
3. To employ Archimedes’ Principle to determine the density or specific gravity ($SG$) of a known liquid.
4. To employ Archimedes’ Principle to determine the density or specific gravity ($SG$) of an unknown liquid.

Theoretical Framework

• Density ($\rho$): An intrinsic physical property defined as mass per unit volume. While sensitive to temperature and pressure, it is considered constant if changes in those variables are minimal.
  • Equation 1: $\rho = \frac{m}{V}$
  • Units: $kg/m^3$ (MKS units) or $g/cm^3$ (cgs units).
• Specific Gravity ($SG$): Also known as relative density. It is a unitless comparison of a material's density to the density of water at its densest point ($4^\circ\text{C}$).
  • Equation 2: $SG = \frac{\rho_{\text{substance}}}{\rho_{\text{water}}}$
• Weight in Air ($W_A$): The weight of an object in air is given by gravity acting on its mass ($W = mg$). Using Equation 1, mass is density times volume ($m = \rho V$).
  • Equation 3: $W_A = \rho V g$
• Buoyant Force ($F_B$): When submerged, the object loses weight equal to the buoyant force. This is the difference between weight in air ($W_A$) and weight in liquid ($W_L$).
  • Equation 4 (Weight Loss): $F_B = W_A - W_L$
• Archimedes' Law for Buoyancy: This force is also equal to the weight of the displaced liquid ($W_{\text{displaced liquid}} = \rho_{\text{liquid}} V g$).

Density and Specific Gravity Table

Specific Gravity Calculations

Solid Denser than Water

• If a solid sinks, its specific gravity is the ratio of its weight in air to its weight loss in water ($W_A - W_W$).
• Equation 5: $SG = \frac{W_A}{W_A - W_W}$

Solid Less Dense than Water (Floating)

• Because the solid floats, a sinker must be used to submerge it completely. The specific gravity is calculated by comparing the weight in air to the change in weight when the solid is moved from above the water to below the water while the sinker remains submerged.
• Equation 6: $SG = \frac{W_A}{W_{CA-SW} - W_{CW-SW}}$
  • $W_A$: Weight of the solid in air.
  • $W_{CA-SW}$: Weight of the solid in air while the sinker is in water.
  • $W_{CW-SW}$: Weight of both solid and sinker in water.

Liquids

• The specific gravity of a liquid is the ratio of the buoyant force exerted by that liquid on a solid to the buoyant force exerted by water on the same solid.
• Equation 7: $SG = \frac{F_B \text{ of liquid}}{F_B \text{ of water}} = \frac{W_A - W_L}{W_A - W_W}$

Apparatus and Equipment

• Hardware:
  • 1 Digital weighing scale
  • 1 Iron stand
  • 2 Beakers
  • 1 250-mL graduated cylinder
  • 50 cm String
• Samples:
  • 1 Brass metal
  • 1 Aluminum metal
  • 1 Candle
  • 1 Cork
• Liquid Reagents:
  • 100 mL 10% Salt solution
  • 100 mL Unknown liquid

Safety Precautions and Equipment Care

• Scale Stability: Ensure the digital weighing scale is secured properly when mounting it on the iron stand.
• Chemical Safety: Do not smell the unknown liquid to avoid inhaling potentially harmful fumes.

Procedures

Part A: Solid Denser than Water

1. Measure weight in air ($W_A$) for the metal sample (Aluminum and Brass).
2. Mount the scale on the iron stand and suspend the metal via a string from the bottom hook.
3. Submerge the metal completely in a beaker of water and record weight in water ($W_W$).
4. Calculate $SG$ using Equation 5 and determine the percentage error against accepted values.

Part B: Solid Less Dense than Water

1. Measure weight of the solid (Candle or Cork) in air ($W_A$).
2. Tie a string to the scale with the floating solid and a sinker (metal sample) below it.
3. Record weight ($W_{CA-SW}$) where the sinker is in water but the candle/cork is above water.
4. Record weight ($W_{CW-SW}$) where both the solid and sinker are submerged.
5. Calculate $SG$ using Equation 6 and determine percent error.

Part C & D: Known and Unknown Liquids

1. Measure metal sample weight in air ($W_A$).
2. Measure weight submerged in water ($W_W$).
3. Measure weight submerged in the test liquid ($W_L$).
   • How to make 10% salt solution: Dissolve $10\,g$ of salt in $80\,mL$ of water. Once dissolved, add water to reach the $100\,mL$ level.
4. Calculate $SG$ using Equation 7. For the unknown liquid, use the obtained $SG$ to research its identity online.

Guide Questions

1. Density Comparison Procedure: Given two beakers (half-filled with water and unknown liquid) and a single piece of metal:
   • Measure the weight of the metal in air ($W_A$).
   • Submerge it in water and measure the buoyant force ($F_{B\,water} = W_A - W_W$).
   • Submerge it in the unknown liquid and measure the buoyant force ($F_{B\,liq} = W_A - W_L$).
   • If $F_{B\,liq} > F_{B\,water}$, the unknown liquid is denser than water.
2. Metal Density Comparison: Given two metals (A and B) of equal mass and a beaker of water:
   • Calculate the buoyant force for both while submerged ($F_B = W_A - W_W$).
   • The metal with the larger buoyant force has a larger volume.
   • Since $\rho = m/V$ and masses are equal, the metal with the smaller buoyant force (smaller volume) has the greater density.
3. Sources of Error: Potential sources include air bubbles clinging to the submerged object, contact between the submerged object and the beaker walls/bottom, and scale calibration errors.