Measure density with volume changes and graphs
You will be able to: Determine density from displacement measurements and a mass–volume graph.
How can you find the volume of an irregular object?
An irregular solid does not fit a simple length-times-width-times-height calculation. Lower it completely into a graduated cylinder and the water level rises. That rise measures displaced water volume, equal to the solid’s submerged volume.
A useful starting point: Fluids and density: compare equal volumes →
Words and symbols before equations
- Displaced volume
- Volume of fluid pushed aside by the object.
- Meniscus
- Curved liquid surface; read its appropriate reference level at eye height.
- Tare
- Zero a balance with the empty container in place.
- Slope of m versus V
- Mass change divided by volume change; density for samples of the same uniform material.
What this picture assumes
Synthetic uniform-material samples at volumes 1, 2, 3 and 4 L. Masses are calculated exactly, not measured. No container mass or measurement noise; an origin intercept is the ideal prediction.
Connect the picture to the physics
Measure mass with a balance. Record initial fluid volume V_i, submerge the object without trapped air, and record final volume V_f. For a fully submerged nonporous object, V_object=V_f−V_i. It must not dissolve or absorb the liquid; a floating object needs an additional method that accounts for the submerging tool.
Use consistent units. Milliliters and cubic centimeters are equal volumes: 1 mL=1 cm³=10⁻⁶ m³. Subtract the volume readings before taking the density ratio. Avoid confusing total cylinder volume with object volume.
For several samples of the same material, plot m vertically and V horizontally. The slope is ρ. A nonzero intercept can reveal an unremoved container mass; examine it instead of forcing a line through zero. Repeat measurements and record instrument resolution and scatter. Synthetic exact points illustrate a model, not the accuracy of a real experiment.
A worked example, step by step
A 120 g solid raises water from 50 mL to 90 mL. Find its average density.
- Displaced volume=90−50=40 mL=40×10⁻⁶ m³.
- Mass=120 g=0.120 kg.
- ρ=0.120/(40×10⁻⁶)=3000 kg/m³.
- Using 90 mL would incorrectly include the original water volume.
Use the increase in volume, not the final total reading. Trapped bubbles falsely enlarge the measured object volume.
Does an unremoved container mass change the ideal slope of total mass versus sample volume?
Compare with an explanation
A constant container mass shifts the intercept; the slope remains the liquid density. A single total-mass/volume ratio would be biased.
Predict. Change one thing. Explain.
Change the material density used for synthetic samples. Predict the slope of mass versus volume. The axes use kg and liters, so multiply a slope in kg/L by 1000 to express density in kg/m³.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Synthetic slope=1 kg/L = 1000 kg/m³. Samples: 1 L → 1 kg; 2 L → 2 kg; 3 L → 3 kg; 4 L → 4 kg. Equal material density produces one straight line.
Synthetic uniform-material samples at volumes 1, 2, 3 and 4 L. Masses are calculated exactly, not measured. No container mass or measurement noise; an origin intercept is the ideal prediction.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant force, motion or energy relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA solid of mass 0.20 kg raises water from 60 to 100 mL. (a) Find volume in m³. (b) Find density. (c) Predict the effect of trapped bubbles on the inferred density. (d) Name one improvement to the measurement.
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Compare with the answer and four-point rubric
- 1 point: 40 mL=0.000040 m³.
- 1 point: ρ=5000 kg/m³.
- 1 point: Bubbles increase measured displaced volume, making the inferred density too low.
- 1 point: Remove bubbles, read at eye height, use an appropriately graduated cylinder, and repeat; any one justified improvement earns the point.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What does displacement measure for a fully submerged solid?
Its volume, under the stated conditions.
RECALL 2What does mass–volume slope give?
Density, with units matching the axes.
RECALL 3Why read the liquid level at eye height?
To reduce viewing-angle error.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Measure density with volume changes and graphs
- V_object=V_f−V_i when fully submerged and without absorption or trapped air.
- Slope of m vs V=ρ; 1 mL=10⁻⁶ m³.
Remember: Use the increase in volume, not the final total reading. Trapped bubbles falsely enlarge the measured object volume.
Conditions: Synthetic uniform-material samples at volumes 1, 2, 3 and 4 L. Masses are calculated exactly, not measured. No container mass or measurement noise; an origin intercept is the ideal prediction.
Refresh Kid · Unit 8 · Objectives 8.1.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 8.1, objectives 8.1.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Fall-2026 corrections also checked. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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