Suppose that an object with the mass of Earth \(\left(M_{\text {Earth }}=5.97 \times 10^{24} \mathrm{kg}\right)\) fell into a supermassive black hole with a 10 percent energy conversion. a. How much energy (in joules) would be radiated by the black hole? b. Compare your answer with the energy radiated by the Sun each second: \(3.85 \times 10^{26} \mathrm{J}\)

Energy conversion is the process of changing one form of energy into another. In the context of our exercise, the mass of Earth is being converted into energy as it falls into a supermassive black hole. This is explained by Einstein's famous mass-energy equivalence formula, which reveals that mass can be transformed into vast amounts of energy.
Imagine a scenario where something as massive as Earth falls into a black hole with a 10 percent energy conversion efficiency. Ten percent of that mass is converted into energy. That's a huge number! It underscores how powerful the process of energy conversion can be in astronomical events.

A supermassive black hole is an extremely massive black hole, typically found at the center of galaxies. They have masses ranging from millions to billions of times the mass of our Sun.
When objects fall into these cosmic giants, intense gravitational forces lead to extraordinary energy emissions.
In our exercise, Earth’s mass falling into a supermassive black hole results in significant energy radiation, demonstrating its vast energy conversion capabilities. These events are critical in shaping the dynamics of galaxies and influencing their evolution.

The mass-energy equivalence formula is expressed as \[ E = mc^2 \] This equation, proposed by Albert Einstein, indicates that energy (E) is equal to mass (m) times the speed of light (c) squared.
This formula shows a direct relationship between mass and energy, revealing that a small amount of mass can be converted into a tremendous amount of energy. To put this into practice, if we take Earth’s mass \(5.97 \times 10^{24} \text{ kg}\), and convert 10 percent of it to energy, we use the equation \[ E = 0.10 \times 5.97 \times 10^{24} \text{ kg} \times (3 \times 10^8 \text{ m/s})^2 \] It highlights how incredibly large numbers come into play when dealing with astronomical masses.

Joules (J) are the unit of measurement for energy in the International System of Units (SI). One joule is the energy transferred when one newton of force is applied over a distance of one meter.
When we calculated the energy radiated by converting Earth’s mass into energy, we found the result to be \[5.373 \times 10^{40} \text{ J} \]. To give you a sense of scale, the energy radiated by the Sun each second is \[ 3.85 \times 10^{26} \text{ J} \]. Our result is significantly larger, demonstrating the immense energy associated with supermassive black holes and the process of mass-energy conversion.