Question

Energy equivalent of \(0.001 \mathrm{mg}\) is (a) \(9 \times 10^{7} \mathrm{ergs}\) (b) \(9 \times 10^{9}\) ergs (c) \(9 \times 10^{7} \mathrm{~J}\) (d) \(9 \times 10^{5} \mathrm{~J}\)

Short answer

The correct answer is (c) \( 9 \times 10^7 ext{ J} \).

Step-by-step solution

Understand Mass-Energy Equivalence

The mass-energy equivalence principle is given by Einstein's famous equation: \[ E = mc^2 \] where \( E \) is energy, \( m \) is mass, and \( c \) is the speed of light in a vacuum (approximately \( 3 \times 10^8 \) meters per second). Our task is to compute the energy equivalent of a given mass.

Convert Mass from Milligrams to Kilograms

The given mass is \( 0.001 \) mg. First, we need to convert this mass into kilograms, as SI units require mass in kilograms.1 mg = \( 10^{-6} \) kg, therefore:\[ 0.001 ext{ mg} = 0.001 imes 10^{-6} ext{ kg} = 10^{-9} ext{ kg} \]

Calculate the Energy Equivalent

Using the mass-energy equivalence formula, substitute the values for mass and the speed of light to find the energy.\[ E = (10^{-9} ext{ kg}) imes (3 imes 10^8 ext{ m/s})^2 \]\[ E = 10^{-9} imes 9 imes 10^{16} ext{ J} \]\[ E = 9 imes 10^7 ext{ J} \]

Match with Provided Options

Compare the calculated energy value: \( 9 \times 10^7 ext{ J} \), with the options provided.- Option (c): \( 9 imes 10^7 ext{ J} \) matches the calculated result.Therefore, the correct answer is option (c).

Key concepts

Einstein's equation

Einstein's equation is one of the most famous formulas in physics: \( E = mc^2 \). This elegant equation links mass \( m \) with energy \( E \). Here, \( c \) represents the speed of light in a vacuum, which is approximately \( 3 \times 10^8 \) meters per second. This equation tells us that mass can be converted into energy.

The principle behind this equation suggests that even a small amount of mass holds an enormous amount of energy. This is because the speed of light \( c \) is such a large number. When squared, it amplifies the mass, demonstrating a substantial energy equivalent.

This relationship is crucial to understanding processes in the universe, such as nuclear reactions in stars or particles in accelerators.

- **Mass \( m \):** Measured in kilograms (kg).- **Speed of light \( c \):** \( 3 \times 10^8 \) m/s.- **Energy \( E \):** Calculated in Joules (J).

energy conversion

Energy conversion, in this context, refers to the process of determining how much energy corresponds to a specific mass using Einstein's equation. When we talk about converting mass to energy, we're using the formula \( E = mc^2 \) to convert a mass in kilograms into energy in joules.

This conversion requires understanding of how units work in this equation:

• Mass in kilograms (kg) must be used for all calculations.
• The speed of light \( c \), due to its squared factor, has a profound impact on the total energy calculated.
• The outcome is given as energy in Joules (J), indicating how much energy theoretically can be derived from the mass in question.

This process demonstrates the potential energy stored within mass. In practical terms, this conversion concept is used in fields such as nuclear physics and astrophysics.

During steps in calculations, decimal adjustments might be seen, emphasizing the precision necessary in physics.

speed of light

The speed of light \( c \) is a fundamental constant in nature used extensively in physics equations. In the context of Einstein's mass-energy equivalence equation, it is approximately \( 3 \times 10^8 \) meters per second. Its importance cannot be overstated:

• **It is a constant value:** Never changes regardless of where it's applied in physical contexts.
• **Creates large energy results:** When squared in the formula \( E = mc^2 \), it massively increases the result, evidencing just how much energy can be derived from even a minuscule mass.

Thus, the speed of light not only sets a universal speed limit for how fast information or matter can travel but also dominates the calculation of energy from mass. This makes understanding \( c \)'s role essential in energy conversion.

The constancy and magnitude of the speed of light make measurements and predictions across cosmic and quantum scales possible.

kilogram conversion

Conversion of units is often the first step in physics problems, as seen when converting mass from milligrams to kilograms. For calculations in SI units, which is the standard in science, mass is required to be in kilograms:

• **Milligram to kilogram:** 1 milligram (mg) is \( 10^{-6} \) kilograms (kg).
• **Example:** If you start with a mass of \( 0.001 \) mg, convert it by multiplying by \( 10^{-6} \). The conversion becomes \( 0.001 \times 10^{-6} = 10^{-9} \) kg.

This conversion is a vital step since precision in units ensures correct energy calculations. Missteps in unit conversions can lead to significant errors in results, especially when calculating large energy values derived from tiny masses.

By attending to these unit transformations, we uphold the integrity of equations like \( E = mc^2 \), as it relies heavily on accurate inputs of mass and other constants.