Problem 48 - Master Resource Book in JEE Main Physics

Question

Assertion Mass and energy are not conserved separately, but are conserved as a single entity called 'mass-energy'. Reason This is because one can be obtained at the cost of the other as per Einstein equation. $$ E=m c^{2} $$

Step-by-Step Solution

Answer

Both the assertion and the reason are correct; the reason explains the assertion.

Step 1: Understanding the Assertion

The statement says that mass and energy are not conserved independently, but as a combined entity known as 'mass-energy'. This implies that in physical processes, the creation or destruction of mass corresponds to the creation or destruction of energy respectively, when both are considered together, they are conserved.

Step 2: Understanding the Reason

The reason given is rooted in Einstein's equation, \( E=mc^2 \), which shows a direct relationship between mass and energy. Mass can be converted into energy and vice versa, and this means their combined amount (mass-energy) is constant in any closed system.

Step 3: Analyzing the Connection

The reason supports the assertion because Einstein's equation shows that mass and energy can interconvert, directly implying their 'combined' conservation as one entity—mass-energy. The reason provides the explanation for why mass and energy are considered together instead of separately.

Key Concepts

Conservation of Energy

The conservation of energy is a fundamental concept in physics. It states that energy cannot be created or destroyed, only transformed from one form to another. For instance, when a ball is thrown into the air, its kinetic energy is converted into potential energy as it rises. When it falls back down, that potential energy is converted back into kinetic energy.
To better understand this concept:

• Think of energy like water in a closed bottle; it changes its form or location but its total amount stays constant.
• In closed systems, the total energy remains the same, despite transformations.

Einstein's equation adds a fascinating twist to this, showing that energy is not just tied to movement or position, but also to mass itself, befitting the idea of mass-energy conservation.

Einstein's Equation

Einstein's equation, famously expressed as \( E=mc^2 \), revolutionized our understanding of how mass and energy are related. Here, \( E \) stands for energy, \( m \) for mass, and \( c \) represents the speed of light in a vacuum, approximately \( 3 \times 10^8 \) meters per second. This equation demonstrates that a small amount of mass can be converted into a tremendous amount of energy.
Einstein's equation implies:

• Mass and energy are two sides of the same coin, easily transformable under the right conditions.
• It allows for the conversion of mass into energy, such as in nuclear reactions where a small amount of matter is converted into vast energy.

This equation fundamentally alters how we view the conservation laws that previously considered mass and energy separately, showcasing their interconnection and mutual transformability.

Mass-Energy Conservation

Mass-energy conservation broadens the traditional concept of energy conservation to include mass, in the framework provided by \( E=mc^2 \). According to this principle, the sum total of mass and energy in a closed system remains constant. If energy appears to be lost, it has merely transformed into mass or another form of energy and vice versa.
Key takeaways about mass-energy conservation include:

• This principle integrates mass as an interchangeable form of energy, suggesting a holistic conservation law applicable to the universe.
• Practical examples include nuclear fusion and fission, where measurable mass defects lead to significant energy changes, thoroughly explained by mass-energy conservation.

This principle has profound implications in technological applications, from everyday electronics to the cosmos's most energetic phenomena, underscoring the fundamental links between mass and energy.