Internal Structure and Density
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AP Physics 1 › Internal Structure and Density
Two sealed containers have the same outer volume. Container $R$ contains a dense liquid with a small air pocket; container $S$ is completely filled with the same liquid. Which has greater average density?
Container $S$, because it has more mass in the same volume.
They have equal density because the liquid is the same in both.
Container $R$, because it contains both air and liquid.
Cannot be determined without the container material density.
Explanation
This question examines how internal voids affect average density calculations. Density is mass per unit volume (ρ = m/V), and average density considers the total mass and total volume including any internal spaces. Container S, completely filled with liquid, has greater mass than container R (which contains the same liquid plus an air pocket) while both have the same outer volume. Since air has negligible mass compared to the liquid, container S has higher average density: ρ_S > ρ_R. The air pocket in R reduces the total mass without changing volume, lowering the average density. Choice C incorrectly focuses on the liquid being the same, ignoring how the air pocket affects overall density. When calculating average density of composite systems, include all components and spaces in your mass and volume calculations.
A metal cylinder is cut in half perpendicular to its axis, producing two smaller cylinders. Each half has half the mass and half the volume of the original. How does density change?
Density stays the same because mass and volume scale together.
Density halves because mass is halved.
Density doubles because the piece is smaller.
Density becomes zero because the object was cut.
Explanation
This question tests whether density is an intensive or extensive property related to internal structure. Density equals mass divided by volume (ρ = m/V) and describes how tightly atoms or molecules are packed within a material. When the cylinder is cut in half, each piece has half the original mass (m/2) and half the original volume (V/2), giving density ρ = (m/2)/(V/2) = m/V, which equals the original density. The internal atomic structure and packing remain unchanged by the cutting process, so density stays constant. Choice A incorrectly assumes smaller size means higher density without considering proportional mass reduction. The key principle is that density is an intensive property: it depends on material composition and structure, not on the amount of material.
A uniform cube $Q$ and a uniform cube $R$ have equal mass. Cube $Q$ has greater side length. Which statement is correct?
Cube $R$ is denser because the same mass occupies less volume.
They have equal density because their masses match.
Cube $Q$ must be denser because it has greater volume.
Cube $Q$ is denser because it is larger.
Explanation
This question examines density when cubes have equal mass but different side lengths and volumes. Density equals mass divided by volume (ρ = m/V), and cube volume equals side length cubed. Cube R has the same mass as cube Q but smaller side length (thus smaller volume), meaning R's matter is more concentrated and has higher density. Option A incorrectly suggests the larger cube is denser, which violates the inverse relationship between volume and density when mass is held constant. For equal-mass cubes, smaller side length always indicates greater density.
A solid object is drilled to remove some material, decreasing its mass and volume proportionally. Which statement about density is correct?
Density increases because mass decreases.
Density decreases because volume decreases.
Density becomes unpredictable because the shape changed.
Density stays the same because the material is unchanged.
Explanation
This question examines how proportional removal of material affects density. Density is mass per unit volume (ρ = m/V), and when mass and volume decrease by the same proportion, their ratio remains constant. If drilling removes material uniformly, both mass and volume decrease proportionally, keeping density unchanged because the material composition remains the same. Option A incorrectly suggests density increases when mass decreases, ignoring that volume also decreases proportionally. When mass and volume change proportionally for the same material, density remains constant.
Two objects have equal volume. Object $X$ has greater mass than object $Y$. Which statement about density is correct?
Object $Y$ is denser because it is lighter.
They have equal density because their volumes match.
Object $Y$ must have greater volume.
Object $X$ is denser because it has more mass per unit volume.
Explanation
This question tests density understanding when objects have equal volumes but different masses. Density is defined as mass per unit volume (ρ = m/V), so with identical volumes, the object with greater mass has higher density. Object X contains more mass than object Y in the same volume, indicating more matter per unit volume and higher density. Option A incorrectly states that the lighter object is denser, which directly contradicts the density formula. When volumes are equal, always identify which object has greater mass to determine higher density.
Two objects have the same mass and are made of different materials. Object 1 has smaller volume. What is supported?
Density cannot be compared without knowing their weights in air.
Object 2 has greater density because it is larger.
Object 1 has greater density than object 2.
They have equal density because masses match.
Explanation
This question tests density comparison for equal-mass objects made of different materials. Density equals mass divided by volume (ρ = m/V), so when masses are equal, the object with smaller volume has higher density. Object 1 has the same mass as object 2 but occupies less space, indicating that object 1 is made of denser material with tighter atomic packing. Option B incorrectly suggests the larger object is denser, which violates the inverse relationship between volume and density. For equal-mass objects, smaller volume always indicates denser material and higher density.
Two identical-volume containers are filled with different gases and sealed. Container 1 has greater mass. Which conclusion is supported?
Gas in container 2 is denser because it is lighter.
Gas in container 1 has greater density.
Container 2 must have larger volume.
Both gases have equal density because both are gases.
Explanation
This question applies density concepts to gases in equal-volume containers with different masses. Density equals mass divided by volume $(\rho = m / V)$, so when volumes are identical, the gas with greater mass has higher density. Container 1 has more mass than container 2 in the same volume, indicating that the gas in container 1 is denser than the gas in container 2. Option A incorrectly claims the lighter gas is denser, which contradicts the fundamental relationship between mass and density. For equal-volume containers, greater mass always indicates higher gas density.
A uniform material sample is compressed so its volume decreases while its mass stays the same. What happens to its density?
Density increases because the same mass occupies less volume.
Density stays the same because mass is unchanged.
Density becomes zero because volume changes.
Density decreases because volume decreases.
Explanation
This question examines how compression affects density when mass remains constant. Density equals mass divided by volume (ρ = m/V), so when mass stays the same but volume decreases due to compression, density must increase. The same amount of matter now occupies less space, resulting in higher concentration and greater density. Option A incorrectly suggests density decreases when volume decreases, which contradicts the inverse relationship between volume and density. When mass is constant, compressing material (reducing volume) always increases density.
A sealed cube of side length $L$ is filled with liquid. Another sealed cube of side length $L$ is filled with a different liquid. The second cube has greater mass. What follows?
The first liquid is denser because it has less mass.
Density cannot be compared for liquids.
Both liquids have equal density because the cubes match.
The second liquid is denser because equal volumes have greater mass.
Explanation
This question applies density concepts to liquids using equal-volume cubes with different masses. Density equals mass divided by volume (ρ = m/V), so when cube volumes are identical due to equal side lengths, the liquid with greater mass has higher density. The second cube has more mass than the first cube in the same volume, indicating that the second liquid is denser than the first liquid. Option A incorrectly claims the first liquid is denser despite having less mass, which contradicts the density relationship. For equal volumes of liquids, greater mass always indicates higher density.
A solid ball and a hollow ball have the same outer radius. The solid ball has greater mass. What can be inferred?
The solid ball has greater average density.
Both have the same density because their sizes match.
The hollow ball must have greater mass because it has more volume.
The hollow ball has greater density because it contains air.
Explanation
This question examines density comparison between solid and hollow objects with the same outer dimensions. Density is mass per unit volume $(ρ = m/V)$, and with equal outer radii, both balls have the same external volume. The solid ball has greater mass because it contains material throughout its volume, while the hollow ball contains air, resulting in higher average density for the solid ball. Option A incorrectly suggests the hollow ball is denser because it contains air, but air has much lower density than solid materials. Solid objects are always denser than hollow versions with the same outer dimensions.