Problem 18 - College Physics

Question

Einstein's famous mass-energy equivalence is expressed by the equation \(E=m c^{2},\) where \(E\) is energy, \(m\) is mass, and \(c\) is the speed of light. (a) What are the SI base units of energy? (b) Another equation for energy is \(E=m g h,\) where \(m\) is mass, \(g\) is the acceleration due to gravity, and \(h\) is height. Does this equation give the same units as in part (a)?

Step-by-Step Solution

Answer

(a) SI units for energy are \( kg \cdot m^2/s^2 \). (b) Yes, both equations have the same units.

Step 1: Understanding SI Units

In the SI system, each physical quantity is expressed using a combination of base units. The base units include meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), and candela (cd). Our task is to express energy in terms of these base units.

Step 2: Analyze Mass-Energy Equivalence

The mass-energy equivalence equation is given by \( E = mc^2 \). Here, \( m \) is mass, measured in kilograms (kg), and \( c \) is the speed of light, measured in meters per second (m/s). This means \( c^2 \) has units of \((m/s)^2\). Consequently, the units for \( E \) are: \[ \text{Units of } E = \text{kg} \times (\text{m/s})^2 = \text{kg} \cdot \text{m}^2/\text{s}^2 \]

Step 3: Define SI Base Units of Energy

The derived unit for energy, joule (J), is composed of base units as follows: \( 1 \text{ J} = 1 \text{ kg} \cdot \text{m}^2/\text{s}^2 \). This is the standard SI base unit representation for energy.

Step 4: Analyze Gravitational Potential Energy

The gravitational potential energy equation is \( E = mgh \). In this equation, \( m \) is mass (kg), \( g \) is the acceleration due to gravity \((\text{m}/\text{s}^2)\), and \( h \) is height (m). Multiplying these units gives: \[ \text{Units of } E = \text{kg} \times (\text{m}/\text{s}^2) \times \text{m} = \text{kg} \cdot \text{m}^2/\text{s}^2 \]

Step 5: Compare Units of Both Equations

Both the mass-energy equivalence equation \( E = mc^2 \) and the gravitational potential energy equation \( E = mgh \) yield the same SI base units for energy: \( \text{kg} \cdot \text{m}^2/\text{s}^2 \). This confirms consistency in the units used to express energy.

Key Concepts

SI Base Units

The International System of Units, known as SI, provides a standardized set of measurement units for scientific and everyday use. There are seven base units, from which all other units are derived:

• Meter (m): Used to measure length or distance.
• Kilogram (kg): The unit of mass.
• Second (s): The unit of time.
• Ampere (A): Represents electric current.
• Kelvin (K): The unit of thermodynamic temperature.
• Mole (mol): Measures the amount of substance.
• Candela (cd): The unit of luminous intensity.

To express any physical quantity in the SI system, these base units are combined. For instance, energy is derived from these base units by combining mass, distance, and time. Understanding these foundation concepts is key to grasping more complex scientific equations and principles.

Energy

Energy is a fundamental concept in physics, representing the capacity to do work or produce change. It is present in various forms, such as kinetic, potential, thermal, and chemical energy. In the realm of physics, energy is usually measured in joules (J).

The formula for energy can vary depending on the situation. For mass-energy equivalence, Albert Einstein gave us the iconic equation:\[ E = mc^2 \]Here:

• E is the energy.
• m denotes mass measured in kilograms.
• c represents the speed of light, approximately \(3.00 \times 10^8\) m/s.

Another energy expression is through gravitational potential energy, which is given by:\[ E = mgh \]Where:

• m is mass.
• g is the acceleration due to gravity (around \(9.81\) m/s² on Earth).
• h is the height above ground.

In both cases, the resulting unit for energy is \( \text{kg} \cdot \text{m}^2/\text{s}^2 \), confirming the consistency of the energy unit in equations.

Gravitational Potential Energy

Gravitational potential energy (GPE) is a form of potential energy related to the position of an object within a gravitational field. It highlights the potential energy stored due to its height above a reference point.

The formula to calculate gravitational potential energy is:\[ E = mgh \]Let's dive deeper:

• Mass (m): The object’s mass in kilograms.
• Gravity (g): The gravitational force acting on the object. On Earth, it is approximately \(9.81\) m/s².
• Height (h): Vertical height from the reference point, measured in meters.

When you multiply these elements, you get units of \( \text{kg} \cdot \text{m}^2/\text{s}^2 \), which are the SI base units for energy. This calculation confirms that energy, regardless of how it's obtained or described, aligns well with the same unit framework. Understanding gravitational potential energy helps in illustrating how energy can be stored due to position and how it can later transform into other forms.