Problem 88 - General Chemistry

Question

The potential energy of an object in the gravitational field of the earth is \(E_{p}=m g h .\) What must be the SI unit of \(g\) if this equation is to be consistent with the SI unit of energy for \(E_{p} ?\)

Step-by-Step Solution

Answer

The SI unit of \(g\) must be \(\text{m} \cdot \text{s}^{-2}\).

Step 1: Write Down the Given Formula

We start with the given formula for potential energy of an object in a gravitational field: \(E_p = mgh\).

Step 2: Identify the SI Units for Known Quantities

The potential energy \(E_p\) is measured in Joules (J) in the SI unit system. The mass \(m\) is measured in kilograms (kg), and the height \(h\) is measured in meters (m).

Step 3: Express the Joule in Base SI Units

A Joule, which is the unit for energy, can be expressed in base SI units as \(\text{kg} \, \text{m}^2 \, \text{s}^{-2}\).

Step 4: Set Up the Equation with Units

To ensure the equation is dimensionally consistent, set up the equation with the units: \([E_p] = [m][g][h]\) which translates to: \(\text{J} = \text{kg} \cdot [g] \cdot \text{m}\).

Step 5: Solve for the Unit of g

From \(\text{J} = \text{kg} \cdot [g] \cdot \text{m}\), we substitute \(\text{J} = \text{kg} \, \text{m}^2 \, \text{s}^{-2}\) and solve for \([g]\):\[[g] = \frac{\text{J}}{\text{kg} \cdot \text{m}} = \frac{\text{kg} \, \text{m}^2 \, \text{s}^{-2}}{\text{kg} \cdot \text{m}} = \text{m} \, \text{s}^{-2}.\]So, the SI unit of \(g\) must be \(\text{m} \, \text{s}^{-2}\).

Key Concepts

SI units

The International System of Units, or SI Units, is the standardized system used globally to ensure consistency in measurements. Each physical quantity has a specific SI unit associated with it. For instance:

• Energy is measured in Joules (J).
• Mass is measured in kilograms (kg).
• Height or length is measured in meters (m).

The SI units provide a common language for scientists and engineers everywhere, ensuring that calculations and measurements are reliable and universally understood. In any physical equation, it's crucial to use these units to maintain dimensional consistency, meaning all terms must be expressed in equivalent SI units to accurately compare or calculate results.

dimensional analysis

Dimensional analysis is a technique used to check the consistency of equations involving physical quantities. It relies on the fundamental idea that you can only compare or calculate quantities when they are expressed in the same dimensional terms. This technique helps verify that equations are correctly set up by ensuring all units align correctly. For instance, consider the potential energy formula: \[E_p = mgh\]Here, the potential energy \(E_p\) in Joules (J) is expected to equate to the product of mass \(m\), gravity \(g\), and height \(h\). By expressing each quantity in terms of their SI units and ensuring they match the expressed units for energy, we verify the formula's validity. Understanding the balance in equations through dimensional analysis is key to accurate scientific work.

acceleration due to gravity

Acceleration due to gravity, denoted as \(g\), is a measure of how quickly an object accelerates when it is in free fall near a massive body like Earth. On Earth's surface, this acceleration is approximately \(9.81 \, \text{m} \, \text{s}^{-2}\).
This specific value is used in various physics equations, including those for gravitational potential energy. Notably, the SI unit of \(g\) is \(\text{m} \, \text{s}^{-2}\), aligning with both the dimensional requirements and the nature of acceleration as a change in velocity per time unit. Grasping this concept helps in understanding how gravitational force plays a crucial role in motions observed on Earth and in calculations beyond our planet.