Question

A tin can has a total volume of \(1200 \mathrm{~cm}^{3}\) and a mass of \(130 \mathrm{~g}\). How many grams of lead shot could it carry without sinking in water? The density of lead is \(11.4 \mathrm{~g} / \mathrm{cm}^{3}\).

Short answer

The tin can can carry up to 1070 g of lead shot without sinking in water.

Step-by-step solution

Calculate weight of water displaced by the can

The volume of the tin can is \(1200 \, \mathrm{cm}^{3}\). The weight of this volume of water displaced by the can is equal to the volume multiplied by the density of water, which is \(1 \, \mathrm{g/cm}^{3}\). So the weight is \(1200 \, \mathrm{cm}^{3}\times1 \, \mathrm{g/cm}^{3} = 1200 \, \mathrm{g}\).

Check weight of the empty can

The weight of the empty tin can is given as \(130 \, \mathrm{g}\).

Calculate maximum weight of lead shot

The maximum weight of lead shot the can can carry without sinking is the weight of the water displaced minus the weight of the can. So, \(1200 \, \mathrm{g} - 130 \, \mathrm{g} = 1070 \, \mathrm{g}\).

Calculate the volume of lead shot this mass represents

The density of lead is given as \(11.4 \, \mathrm{g/cm^3}\). So to find the volume of lead shot which weighs 1070 g, we can use the formula: Volume = Mass / Density. \(\frac{1070 \, \mathrm{g}}{11.4 \, \mathrm{g/cm^3}} = 93.859649 \, \mathrm{cm^3}\). Since the can's volume is 1200 cm³, the can can indeed carry this volume of lead shot without sinking.

Key concepts

Archimedes' Principle

Archimedes' principle is a law of physics fundamental to the concept of buoyancy, which is the ability of an object to float or rise when submerged in a fluid. According to this principle, any object, wholly or partially immersed in a fluid, experiences an upward force equal to the weight of the fluid displaced by the object.

Imagine you're taking a bath and you submerge a rubber duck into the water. As soon as the duck is underwater, it pushes away or 'displaces' water around it. Archimedes' principle states that the water pushes back on the duck with a force equal to the weight of the water displaced. If the upward buoyant force is stronger than the weight of the duck, it will float.

In practical problems, like determining whether a tin can will sink or float, you calculate the buoyant force by multiplying the volume of the object (in this case, the can) with the density of the fluid (water) and the acceleration due to gravity. This principle is vital when engineers design ships and submarines or when scientists measure the density of an object by water displacement.

Density

Density is a measure of how much mass is contained in a given volume. It is a fundamental property of materials and is mathematically defined as mass divided by volume. The standard metric unit for density is grams per cubic centimeter (\t\(\mathrm{g/cm^3}\)).

Density plays a critical role in buoyancy and whether an object will sink or float in a fluid. An object with a density greater than the fluid will sink, while one with a lower density will float. For example, the density of lead is high, 11.4 \(\mathrm{g/cm^3}\), meaning that for any given volume, lead is quite heavy.

When solving problems related to buoyancy, knowing the density of the object and the fluid allows us to predict and calculate outcomes. It's also a key factor in material selection across various industries, including construction, manufacturing, and aeronautics, influencing decisions based on weight, stability, and structural integrity.

Mass-Volume Relationship

The mass-volume relationship is an understanding of how the mass of an object relates to its volume and is tightly interconnected with the concept of density. Mathematically, it's expressed by the equation \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \).

This relationship is crucial in calculating how much mass a vessel, like our tin can, can carry without sinking. By rearranging the mass-volume relationship formula (\( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \)), you can find the volume of a certain mass of a substance (like lead shot) if its density is known.

In our problem, using the mass-volume relationship allowed us to calculate the volume of the lead shot that corresponds to the mass it could carry. This helps ascertain if the can will float or sink. Engineering calculations often depend on this relationship to ensure the stability of structures or for the transport of goods, be it in water, air, or space.