Understanding Density Calculation

Density is a fundamental property of matter that tells us how much mass is packed into a certain volume. It is defined as the ratio of mass to volume.

The formula for density (\(\rho\)) is:

\[\rho = \frac{m}{V}\]

where:

• \(m\) is the mass of the object
• \(V\) is the volume of the object

In this problem, we are given the dimensions of a piece of wood and its weight. We need to find its density in kilograms per cubic meter (kg/m³).

Calculating the Volume of the Wood

The piece of wood is rectangular, so its volume can be calculated by multiplying its length, width, and height. The dimensions are given in centimeters (cm), but we need the volume in cubic meters (m³). We must first convert the dimensions from cm to meters (m).

Conversion factor: 1 cm = 0.01 m

• Length (\(l\)) = 6 cm = \(6 \times 0.01\) m = 0.06 m
• Width (\(w\)) = 8 cm = \(8 \times 0.01\) m = 0.08 m
• Height (\(h\)) = 5 cm = \(5 \times 0.01\) m = 0.05 m

Now, calculate the volume (\(V\)):

\[V = l \times w \times h\]

\[V = 0.06 \text{ m} \times 0.08 \text{ m} \times 0.05 \text{ m}\]

\[V = (6 \times 10^{-2}) \text{ m} \times (8 \times 10^{-2}) \text{ m} \times (5 \times 10^{-2}) \text{ m}\]

\[V = (6 \times 8 \times 5) \times 10^{(-2 - 2 - 2)} \text{ m}^3\]

\[V = 240 \times 10^{-6} \text{ m}^3\]

\[V = 2.4 \times 10^{-4} \text{ m}^3\]

So, the volume of the piece of wood is \(2.4 \times 10^{-4}\) m³.

Calculating the Mass of the Wood from Weight

We are given the weight of the wood, which is 1.92 N. Weight is the force exerted on an object due to gravity. It is related to mass by the formula:

\[\text{Weight} (W) = \text{mass} (m) \times \text{acceleration due to gravity} (g)\]

We are given \(W = 1.92\) N and \(g = 10\) m/s². We can rearrange the formula to find the mass (\(m\)):

\[m = \frac{W}{g}\]

\[m = \frac{1.92 \text{ N}}{10 \text{ m/s}^2}\]

\[m = 0.192 \text{ kg}\]

Thus, the mass of the piece of wood is 0.192 kg.

Calculating the Density of the Wood

Now that we have the mass (m = 0.192 kg) and the volume (V = \(2.4 \times 10^{-4}\) m³), we can calculate the density (\(\rho\)) using the density formula:

\[\rho = \frac{m}{V}\]

\[\rho = \frac{0.192 \text{ kg}}{2.4 \times 10^{-4} \text{ m}^3}\]

To make the calculation easier, we can write 0.192 as \(192 \times 10^{-3}\) and \(2.4 \times 10^{-4}\) as \(24 \times 10^{-5}\) (or keep it as is and simplify).

\[\rho = \frac{192 \times 10^{-3}}{2.4 \times 10^{-4}} \text{ kg/m}^3\]

Divide the numerical parts:

\[\frac{192}{2.4} = \frac{1920}{24} = 80\]

Combine the powers of 10:

\[10^{-3} / 10^{-4} = 10^{-3 - (-4)} = 10^{-3 + 4} = 10^1\]

So, the density is:

\[\rho = 80 \times 10^1 \text{ kg/m}^3\]

\[\rho = 800 \text{ kg/m}^3\]

The density of the piece of wood is 800 kg/m³.

Summary of Calculations for Density

Final Answer for Wood Density

The calculated density of the piece of wood is 800 kg/m³.

Revision Table: Key Physics Concepts

Additional Information on Density and Measurement

Density is an intensive property, meaning it does not depend on the amount of substance. A small piece of wood from the same type of tree will have the same density as a large piece, assuming it's uniform.

Units are very important in physics calculations. Always ensure that all values are in consistent units (like SI units - meters, kilograms, seconds) before performing calculations. Conversion factors are necessary when units are mixed.

Weight is a force and is measured in Newtons (N). Mass is a measure of inertia and is measured in kilograms (kg). The relationship between them depends on the local acceleration due to gravity (\(g\)). On Earth, \(g\) is approximately 9.8 m/s², but for simpler calculations in problems like this, it is often given as 10 m/s².

Understanding how to calculate volume for different shapes (cubes, cylinders, spheres) and how to convert between different units of length, area, and volume is crucial for solving density problems.