7.2.2 Density of Irregular Shaped Object (Displacement Method)

Not all objects have simple geometric shapes. A stone, a piece of metal, or a key cannot have its volume calculated using a simple formula like length × width × height. For these irregularly shaped objects, we need a different method to find their volume. The displacement method is a clever and reliable technique that uses the fact that an object submerged in a fluid displaces a volume of fluid equal to its own volume. Once the volume is known, the density can be calculated.

Definition

The Displacement Method is a technique used to measure the volume of an irregularly shaped solid by submerging it in a liquid (usually water) in a measuring cylinder and measuring the volume of liquid displaced. The volume of the object is equal to the difference between the final volume (with the object submerged) and the initial volume (without the object).

Main Idea

The main idea is that when an object is completely submerged in a fluid, it displaces a volume of fluid equal to its own volume. By measuring the increase in the fluid level in a measuring cylinder, we can determine the volume of the object. Then, by measuring the object's mass with a balance, we can calculate its density using the formula ρ = m/V. This method works for objects that sink in water and do not dissolve or absorb water.

Core Concept

• Find the mass of the irregularly shaped object using a balance.
• Add some water to a measuring cylinder and measure its initial volume (V_i).
• Tie a thread to the object and lower it gently into the measuring cylinder.
• The water level rises. Measure the final volume (V_f).
• Calculate the volume of the object: V = V_f - V_i.
• Calculate the density: ρ = Mass of objectVolume of object.

Example from Text The text provides an example (Example 7.4): "Mass of a rock is 80.52 grams. It was immersed in a measuring cylinder containing water. From the figure, find the initial and final volumes of water. Use this data to find volume of water and density."

• Given: Mass of rock = 80.52 g, Initial volume of water = 18 mL, Final volume of water = 46 mL
• Volume of object: V = 46 - 18 = 28 mL = 28 cm³
• Density:

Detailed Explanation

The displacement method is based on Archimedes' principle, which states that a body immersed in a fluid experiences an upthrust equal to the weight of the fluid displaced. For the purpose of measuring volume, the key fact is that the volume of fluid displaced equals the volume of the submerged object (provided the object is completely submerged and does not dissolve or absorb the fluid).

Why the Displacement Method Works When you lower an irregular object into water in a measuring cylinder, the water level rises. The rise in water level corresponds to the volume of water that has been "pushed out of the way" (displaced) by the object. Since the object is completely submerged, the volume of water displaced is exactly equal to the volume of the object. This is why the displacement method works.

• The object must sink: If the object floats, you need to use a sinker (a heavy object attached to the floating object) to submerge it. The volume of the sinker must be accounted for.
• The object must not dissolve: If the object dissolves in water (e.g., a sugar cube), the method will not work. You would need to use a liquid in which the object is insoluble (e.g., oil or alcohol).
• The object must not absorb water: If the object is porous (e.g., a sponge or a brick), it will absorb water, and the volume reading will be inaccurate. You can seal the object with a thin waterproof coating (e.g., wax or varnish) before submerging it, but you must account for the volume of the coating.
• Read the meniscus correctly: Read the bottom of the meniscus (the curved surface of the water) at eye level to avoid parallax error.
• Use a thread: Tie the object with a thin thread to lower it into the water. The thread should be thin enough that its volume is negligible.

• Works for any shape: It does not matter how irregular the object is; the method works for any shape as long as the object sinks and does not dissolve or absorb water.
• Simple and inexpensive: It requires only a measuring cylinder, a balance, and water.

• Not suitable for floating objects: (Unless a sinker is used.)
• Not suitable for soluble objects: (E.g., sugar, salt.)
• Not suitable for porous objects: (E.g., wood, sponge, unless sealed.)
• Reading errors: The volume reading can be affected by parallax, air bubbles, and the meniscus.

Intermediate Level

In physics, the displacement method is a practical application of Archimedes' principle. It is used to measure the volume of irregularly shaped objects that cannot be measured with a ruler.

The method involves measuring the initial volume of water in a measuring cylinder, submerging the object, and measuring the final volume. The difference between the two readings is the volume of the object. The mass is measured separately using a balance. Density is then calculated as ρ = m/V.

The displacement method is simple and widely used, but it has limitations. It cannot be used for objects that float, dissolve, or absorb water. For floating objects, a sinker can be used. For soluble objects, a different liquid (in which the object is insoluble) must be used. For porous objects, the object must be sealed before submerging.

When performing the experiment, it is important to read the meniscus at eye level, remove any air bubbles from the object, and use a thin thread to minimize additional volume.

Important Points

• - Displacement Method: Measures volume of irregular solids.
• - Principle: The volume of the displaced fluid equals the volume of the submerged object.
• - Steps: Mass → Initial volume → Submerge → Final volume → Volume = Final - Initial → Density = Mass / Volume.
• - Volume of object: V = V_f - V_i.
• - Density: ρ = m/V.
• - Limitations: Object must sink, not dissolve, and not absorb water.
• - Read the meniscus at eye level.

Remember

• - Displacement Method: For irregular solids.
• - Volume of object = Final volume - Initial volume.
• - Density = Mass / Volume.
• - Object must sink.
• - Object must not dissolve or absorb water.
• - Read meniscus at eye level.

Common Mistakes

• - Forgetting to subtract the initial volume: The volume of the object is the difference between the final and initial volumes, not just the final volume.
• - Not removing air bubbles: Air bubbles on the object can increase the volume reading. Tap the cylinder gently to remove them.
• - Using a thick thread: A thick thread can displace a measurable amount of water, leading to an overestimation of the object's volume. Use a thin thread.
• - Not drying the object before weighing: If the object is wet, the mass will be too high. Dry it first.

Quick Revision

The displacement method is used to measure the volume of irregularly shaped solids. It involves measuring the initial volume of water in a measuring cylinder, submerging the object, and measuring the final volume. The volume of the object is the difference between the final and initial volumes (V = V_f - V_i). The mass is measured using a balance. Density is then calculated as ρ = m/V. The method works for objects that sink in water and do not dissolve or absorb water.

60-Second Revision

• Volume = V_f - V_i.
• Density = m/V.
• Object must sink.
• No dissolving or absorbing.
• Read meniscus at eye level.

Frequently Asked Questions (FAQs)

Q1. What is the displacement method? Answer: The displacement method is a technique for measuring the volume of an irregularly shaped solid by submerging it in a liquid and measuring the volume of liquid displaced. The volume of the object equals the increase in the liquid level.

• Q2. Why is the displacement method used?
• Answer: It is used for irregularly shaped objects whose volume cannot be calculated using simple geometric formulas.

• Measure the mass of the object.
• Add water to a measuring cylinder and record the initial volume (V_i).
• Submerge the object in the water.
• Record the final volume (V_f).
• Calculate the volume of the object: V = V_f - V_i.
• Calculate the density: ρ = m/V.

• The object must sink in water.
• The object must not dissolve in water.
• The object must not absorb water (unless sealed).
• Air bubbles can affect the volume reading.

Q5. How do you handle a floating object in the displacement method? Answer: For a floating object, you can use a sinker (a heavy object) to submerge it. The volume of the sinker must be measured separately and subtracted from the total volume.

• Q6. How do you read the volume correctly in a measuring cylinder?
• Answer: Read the bottom of the meniscus (the curved surface of the water) at eye level to avoid parallax error.

• Q7. A stone has a mass of 60 g. The initial volume of water is 20 mL, and the final volume is 40 mL. What is the density of the stone?
• Answer: Volume = 40 - 20 = 20 mL = 20 cm³. Density = 6020 = 3 g/cm³.

Multiple Choice Questions (MCQs)

• Q1. The displacement method is used to measure the volume of:
• A. Liquids
• B. Regularly shaped solids
• C. Irregularly shaped solids
• D. Gases

• Correct Answer: C
• Explanation: The text states that the displacement method is used for irregularly shaped solid objects like stones.

• Q2. In the displacement method, the volume of the object is equal to:
• A. The initial volume of water
• B. The final volume of water
• C. The difference between the final and initial volumes
• D. The sum of the initial and final volumes

• Correct Answer: C
• Explanation: The text states that the volume of the object is found by subtracting the initial volume from the final volume.

• Q3. The displacement method is based on the principle that:
• A. The mass of the object equals the mass of the displaced fluid
• B. The volume of the object equals the volume of the displaced fluid
• C. The density of the object equals the density of the fluid
• D. The weight of the object equals the weight of the displaced fluid

• Correct Answer: B
• Explanation: The volume of the object equals the volume of the fluid displaced.

• Q4. Which of the following objects cannot be measured using the displacement method in water?
• A. A stone
• B. A piece of iron
• C. A sugar cube
• D. A glass marble

• Correct Answer: C
• Explanation: A sugar cube dissolves in water, so the displacement method would not work (the volume reading would be inaccurate).

• Q5. A stone has a mass of 100 g. The initial water volume is 30 mL, and the final volume is 50 mL. The density of the stone is:
• A. 2 g/cm³
• B. 5 g/cm³
• C. 10 g/cm³
• D. 20 g/cm³

• Correct Answer: B
• Explanation: Volume = 50 - 30 = 20 cm³. Density = 10020 = 5 g/cm³.

• Q6. Why is a thin thread used to lower the object into the water?
• A. To make the object sink
• B. To minimize the additional volume displaced by the thread
• C. To prevent the object from dissolving
• D. To make the object easier to see

• Correct Answer: B
• Explanation: A thin thread minimizes the additional volume displaced by the thread itself, reducing error.

• Q7. What should you do if the object is porous (e.g., a sponge)?
• A. Use a larger measuring cylinder
• B. Seal the object with a waterproof coating before submerging
• C. Use hot water instead of cold water
• D. Weigh the object after submerging

• Correct Answer: B
• Explanation: Sealing the object prevents water from being absorbed, which would affect the mass and volume measurements.

• Q8. The meniscus of water in a measuring cylinder should be read:
• A. At the top of the curve
• B. At the bottom of the curve
• C. At the middle of the curve
• D. At any point

• Correct Answer: B
• Explanation: The bottom of the meniscus (the lowest point of the curve) is the correct reading for water.

• Q9. Which of the following is a limitation of the displacement method?
• A. It can only be used for regular solids
• B. It can only be used for objects that sink
• C. It can only be used for liquids
• D. It is very expensive

• Correct Answer: B
• Explanation: The object must sink in the water for the displacement method to work.

• Q10. A rock has a mass of 80.52 g. The initial volume of water is 18 mL, and the final volume is 46 mL. The density of the rock is approximately:
• A. 1.75 g/cm³
• B. 2.88 g/cm³
• C. 3.22 g/cm³
• D. 4.47 g/cm³

• Correct Answer: B
• Explanation: Volume = 46 - 18 = 28 cm³. Density = 80.5228 ≈ 2.88 g/cm³.

Frequently Asked Questions (FAQs)

Q1. What is the difference between the displacement method and the geometric method? Answer: The geometric method calculates volume from length measurements using formulas (for regular solids). The displacement method measures volume by submerging the object in water (for irregular solids).

• Q2. Can the displacement method be used for gases?
• Answer: No. The displacement method is used for solids. Gases are highly compressible and would dissolve or mix with the liquid, making the method unsuitable.

Q3. What is the difference between the displacement method and Archimedes' principle? Answer: The displacement method is a practical application of the fact that a submerged object displaces its own volume of fluid. Archimedes' principle states that the upthrust on a submerged object equals the weight of the fluid displaced. Both are related to buoyancy, but the displacement method focuses on volume measurement.

• Q4. What should you do if the object floats?
• Answer: Use a sinker (a heavy object) to submerge it. Measure the volume of the sinker separately and subtract it from the total volume.

• Q5. Why is it important to dry the object before weighing it?
• Answer: If the object is wet, the water on its surface adds to its mass, giving a higher reading than the actual mass of the object.

• Parallax error when reading the meniscus.
• Air bubbles on the object.
• Water absorption by the object.
• The volume of the thread.
• Incomplete submergence of the object.

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