Question

You have a \(1.0-\mathrm{cm}^{3}\) sample of lead and a \(1.0-\mathrm{cm}^{3}\) sample of glass. You drop each in a separate beaker of water. How do the volumes of water that are displaced by the samples compare? Explain.

Short answer

The volumes of water displaced by the lead and glass samples are the same because the volume of water displaced is determined only by the volume of the submerged object, not by its mass or density. Both samples have the same volume (1.0 cm³), which means that they will displace an equal amount of water when submerged in their respective beakers.

Step-by-step solution

Understand the concept of volume displacement

Volume displacement is the amount of a liquid such as water displaced by a solid object when it is fully submerged. In this case, it is the volume of water displaced by the lead and glass samples.

Determine volume of the samples

We are given that the volume of each sample is 1.0 cm³.

Submerge the samples in water

When you drop each sample into a separate beaker of water, the water level will rise due to the submerged samples displacing some of the liquid.

Observe the water displacement

Since both lead and glass samples have the same volume (1.0 cm³), they will displace an equal amount of water, regardless of their density or mass.

Explain the observation

The volumes of water displaced by the lead and glass samples are the same because the volume of water displaced is determined only by the volume of the submerged object, not by its mass or density. Both samples have the same volume (1.0 cm³), which means that they will displace an equal amount of water when submerged in their respective beakers.

Key concepts

Density in Volume Displacement

Density refers to the mass of an object per unit of volume. It is calculated using the formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). Density does not affect the volume of water displaced by an object when it is submerged. Instead, the volume displaced depends solely on the size of the object.
For our exercise, even though lead is much denser than glass, both have the same volume of 1.0 cm³. Density plays no role in the amount of water displaced, which is equal for both samples.

Understanding Mass

Mass is the measure of the amount of matter in an object. Unlike volume, mass does not directly impact volume displacement. In the context of water displacement, it's important to note that a heavier object doesn't displace more water just because it weighs more; it's the volume that counts.
In our example with lead and glass, the mass of lead might be higher, but both have the same volume. Therefore, they displace the same amount of water, which is specifically 1.0 cm³.

Grasping Archimedes' Principle

Archimedes' principle states that the upward buoyant force experienced by a body immersed in a fluid is equal to the weight of the fluid that the body displaces. This principle helps explain why objects float or sink and how much water they displace.
For objects submerged completely in water, like our samples of lead and glass, Archimedes' principle demonstrates that the submerged volume of these solids will displace an equal volume of water, which corresponds to their own volume.

Mechanics of Water Displacement

Water displacement occurs when an object is submerged in a liquid, pushing the liquid out of the way and causing it to rise. This process is used to measure the volume of irregularly shaped objects.
In the task involving lead and glass, once submerged, both displace 1.0 cm³ of water because that is also their volume. This finding highlights how water displacement is solely dependent on the object's volume, not its density or mass.
Key points about water displacement:

• It's determined by the volume of the submerged object.
• Independent of the object's mass or density.
• A practical way to measure volume for various shapes.