Question

In a year, an average American uses about \(3 \times 10^{11} \mathrm{~J}\) of energy. How much mass does this translate to via \(E=m c^{2} ?\) Rock has a density approximately 3 times that of water, translating to about \(3 \mathrm{mg}\) per cubic millimeter. So roughly how big would a chunk of rock material be to provide a year's worth of energy if converted to pure energy? Is it more like dust, a grain of sand, a pebble, a rock, a boulder, a hill, a mountain?

Short answer

A. Dust B. Grain of sand C. Pebble D. Rock E. Small Boulder

Step-by-step solution

Calculate the mass from the given energy

First, we need to find the mass by rearranging the equation \(E=mc^{2}\): \(m = \frac{E}{c^{2}}\) Given that the average American uses about \(3 \times 10^{11} \mathrm{~J}\) of energy per year, we can calculate the mass as follows: \(m = \frac{3 \times 10^{11} \mathrm{~J}}{(3 \times 10^{8} \mathrm{~m/s})^{2}}\) The mass is: \(m = 3.33 \times 10^{-3} \mathrm{~kg}\)

Calculate the volume of the rock

Given the density of rock, which is approximately \(3 \mathrm{mg/mm^{3}}\), we can convert it to \(\mathrm{kg/m^{3}}\) in SI units: Density of rock = \(3 \mathrm{mg/mm^{3}} \times \frac{10^{-6} \mathrm{~kg}}{1 \mathrm{~mg}} \times \frac{1 \mathrm{~mm^{3}}}{10^{-9} \mathrm{~m^{3}}}= 3000 \mathrm{~kg/m^{3}}\) Now, we can find the volume of the rock using the mass and density: \(V = \frac{m}{\rho}\) The volume is: \(V = \frac{3.33 \times 10^{-3} \mathrm{~kg}}{3000 \mathrm{~kg/m^{3}}}\) \(V = 1.11 \times 10^{-6} \mathrm{~m^{3}}\)

Determine the size of the rock

Finally, we will compare the volume of the rock obtained in the previous step to common objects to determine its size. The size of the rock with a volume of \(1.11 \times 10^{-6} \mathrm{~m^{3}}\) is equivalent to: \(1.11 \times 10^{-6} \mathrm{~m^{3}} \times \frac{1000 \mathrm{~mm^{3}}}{1 \mathrm{~m^{3}}} = 1.11 \times 10^{3} \mathrm{~mm^{3}}\) As a reference: - Dust: <\(1 \mathrm{~mm^{3}}\) - Grain of sand: 1 \(\mathrm{mm^{3}}\) - Pebble: 10-100 \(\mathrm{mm^{3}}\) - Rock: 100-1000 \(\mathrm{mm^{3}}\) - Boulder: >1000 \(\mathrm{~mm^{3}}\) Thus, the size of the rock with a volume of \(1.11 \times 10^{3} \mathrm{~mm^{3}}\) is similar to a small boulder.

Key concepts

E=mc^2 and Mass-Energy Equivalence

The equation E=mc² is one of the most famous in the world of physics, formulated by Albert Einstein as part of the theory of relativity. It explains that energy (E) and mass (m) are interchangeable; they are different forms of the same thing. In this relationship, c represents the speed of light in a vacuum, which is approximately 3 x 10⁸ meters per second. Because the speed of light squared (c²) is an astronomically large number, even a small amount of mass can be converted into a huge amount of energy.

Understanding this equation is critical in explaining numerous physical phenomena, including how the sun generates energy and the basics of nuclear power. If you know the amount of energy you need or have generated, rearranging the equation gives you the mass that would be equivalent to that energy.

Energy Usage in Everyday Life

When we talk about 'energy usage', we're referring to how much work or heat energy is consumed by actions or processes, such as heating a home, powering electronics, or traveling in a vehicle. Energy is measured in joules (J), and an average American's annual energy consumption is about 3 x 10¹¹ J.

To put that into perspective using E=mc², that amount of energy corresponds to a small mass when converted, signifying the massive potential energy stored within matter. This concept has practical applications in understanding the energy needs and conservation strategies, as well as the impacts of energy usage on the environment.

Density and Mass Relationships

Density is a property that describes how much mass is contained within a certain volume of a substance. The standard unit of density in the International System of Units (SI) is kilograms per cubic meter (kg/m³).

When solving problems with density, it's essential to use consistent units. In the given exercise, the rock's density is initially represented as 3 mg/mm³, and it needs to be converted into SI units to be able to calculate volume properly. The relationship between mass, density, and volume is given by the formula density (ρ) = mass (m)/volume (V). Once we know any two of these variables, we can easily solve for the third, which is crucial in many scientific and engineering applications.

Volume Calculation and Conceptual Understanding

Volume calculation is crucial when you're trying to understand the space that a particular amount of matter occupies. Volume is usually measured in cubic meters (m³) or liters, but sometimes smaller units like cubic millimeters (mm³) are used for more manageable, everyday comparisons.

The exercise's solution compared the calculated volume of the rock to familiar objects to help conceptualize the size. Understanding volume calculations is essential not just in science and engineering but also in daily life when dealing with quantities of materials or packing objects into spaces. It illustrates how mathematics can give us tangible insights into the physical world.