Problem 85 - Fundamentals of Physics

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 of density \(11.4 \mathrm{~g} / \mathrm{cm}^{3}\) could it carry without sinking in water?

Step-by-Step Solution

Answer

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

Step 1: Understand Buoyancy Principle

A tin can will neither sink nor float if the combined weight (mass) of the can and its contents is equal to the weight of the water displaced by the can. Here, the weight of the displaced water is equal to the volume of the can multiplied by the density of water (1 g/cm³). So, the weight (or maximum mass) that the can could carry without sinking is equal to the water displaced, which is 1200 g.

Step 2: Calculate Maximum Load

Since the total mass of the tin can, its contents, and the lead should not exceed the mass of the water displaced (1200 g), subtract the mass of the can (130 g) from this to find the maximum mass of lead that the can could carry without sinking.\[ \text{Maximum mass of lead} = 1200 \, \text{g} - 130 \, \text{g} = 1070 \, \text{g} \]

Step 3: Confirm Load Capacity with Lead Density

Lead shot has a density of 11.4 g/cm³. To ensure the can can hold this without sinking, we should consider the mass provided equates to a volume less than or equal to the can's capacity if required. However, since densities are required for different checks, the key value is ensuring mass doesn't exceed capabilities (1070 g) assuming capacity remains consistent.

Key Concepts

Understanding Density

Density is a fundamental concept in physics, representing how much mass is contained in a given volume. It is calculated using the formula:

• \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]

For any substance, knowing its density allows us to determine how much mass is present in a specific volume or vice versa.
Dense materials, like lead, have a high amount of mass packed into a small volume, which is why their density values are high.
On the other hand, less dense materials, such as wood or air, have less mass in the same volume of space.
In the context of buoyancy and this exercise, understanding density helps us predict whether an object will sink or float. Objects that are less dense than the fluid they are in will float, while denser objects will sink.
In this exercise, we explore the density of lead shot, which is 11.4 g/cm³. This value is crucial for calculating how much lead can fit into the can before it sinks. Knowing the density of both the can and lead allows us to ensure that the combined mass doesn't surpass the buoyant capacity.

Mass Calculation of Lead Shot

Calculating mass is essential to determine how much of a material can be held without causing an object to sink. In the exercise, we're asked to find out how much lead the tin can hold without sinking.
The tin can displaces water equal to its volume, which is 1200 cm³. The weight of this water is 1200 grams, since the density of water is 1 g/cm³.

• The crucial insight here is that the combined mass of the can and its contents should not exceed this displaced water mass to avoid sinking.
• We first calculate the current mass of the can (130 g).
• Then, we subtract this mass from the total possible displaced mass (1200 g) to find the maximum additional mass the can can carry.

The calculation would be: \[ 1200 \text{ g} - 130 \text{ g} = 1070 \text{ g} \]
This means the tin can can hold up to 1070 grams of lead shot before it risks sinking, helping us to plan accordingly with precision and avoid buoyancy issues.

Why Use Lead Shot

Lead shot is often used because of its high density. High-density items like lead occupy less space for the same mass when compared to less dense materials.
This is significant in scenarios where space is limited but weight is a priority.

• Lead's density, at 11.4 g/cm³, ensures that a small volume of lead can carry a substantial weight.
• This density allows applications in weights, counterbalances, and even in sporting equipment such as ammunition, where space is at a premium, but mass is beneficial.

In this exercise, the use of lead shot in the tin can demonstrates the practical applications of understanding density and buoyancy.
By carrying lead, which is dense, we optimize the mass the can holds within its 1200 cm³ capacity effectively.
This approach showcases how these calculations translate in real-life decisions, ensuring the can stays buoyant while efficiently carrying its maximum load.