Problem 37 - Applied Physics

Question

What is the mass density of a \(50 \overline{0}-\mathrm{g}\) block that displaces \(215 \mathrm{~cm}^{3}\) of water?

Step-by-Step Solution

Answer

The density of the block is approximately 2.33 g/cm³.

Step 1: Understanding the Problem

We need to find the mass density of a block, which is the mass per unit volume. We have a mass of 500 grams and a displaced volume of 215 cm t 3.

Step 2: Writing the Formula for Density

Density is calculated using the formula:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]where the mass is given as 500 grams and the volume is 215 cmt 3.

Step 3: Plugging In Values

Insert the provided values into the density formula:\[ \text{Density} = \frac{500 \text{ grams}}{215 \text{ cm}^3} \]

Step 4: Performing the Division

Divide the mass by the volume:\[ \text{Density} = \frac{500}{215} \approx 2.33 \text{ grams/cm}^3 \]

Step 5: Stating the Result

The density of the block is approximately 2.33 grams per cubic centimeter.

Key Concepts

Mass

Mass refers to the amount of matter contained within an object. It is a fundamental property that dictates how much the object will weigh when subjected to gravity. Mass is typically measured in grams (g) or kilograms (kg) in the metric system. This property is intrinsic and doesn’t change regardless of an object's location. For instance, whether a block is on Earth or the Moon, its mass remains the same.

Understanding mass is crucial when calculating mass density, as it forms the numerator in the density formula. In the exercise, the mass of the block is 500 grams. This tells us it has a certain amount of matter, which is a key component when looking to find its density.

Volume

Volume represents the amount of space that an object occupies. It is measured in cubic centimeters (cm³), liters (L), or for larger objects, cubic meters (m³). When we talk about volume, we refer to the three-dimensional space that an object takes up.

In the exercise, the block displaces 215 cm³ of water, indicating its volume. An easy way to think about volume is through displacement; when you submerge an object in water, it pushes aside the water, showing its volume. This concept is vital for finding mass density because it provides the denominator in the density calculation.

Density Formula

Density is a measure of how much mass is contained within a given volume. The formula to calculate density is:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]This equation helps us understand how tightly packed the mass is within a specified space. A higher density means more mass in a given volume, and vice versa. For example, in our exercise, we divide the mass of 500 grams by the displaced volume of 215 cm³ to find a density of approximately 2.33 grams per cm³. This illustrates that for every cubic centimeter, there is about 2.33 grams of mass.

Unit Conversion

Unit conversion is essential when working with formulas like density, especially if units don't match up. Although the given exercise doesn't require conversion, it's helpful to know this skill. Units for mass might need converting between grams and kilograms, while volume may require conversion between liters, cubic centimeters, or even cubic meters based on the context.

For instance:

• 1 kilogram = 1000 grams
• 1 liter = 1000 cubic centimeters

Being able to convert units ensures that our calculations are accurate and consistent. If you had a mass in kilograms and volume in liters, you’d likely need to convert either mass to grams or volume to cubic centimeters to maintain compatibility in the density formula.