Question

Which of the following is a correct unit of energy? a) \(\mathrm{kg} \mathrm{m} / \mathrm{s}^{2}\) c) \(\mathrm{kg} \mathrm{m}^{2} / \mathrm{s}^{2}\) e) \(\mathrm{kg}^{2} \mathrm{~m}^{2} / \mathrm{s}^{2}\) b) \(\mathrm{kg} \mathrm{m}^{2} / \mathrm{s}\) d) \(\mathrm{kg}^{2} \mathrm{~m} / \mathrm{s}^{2}\)

Short answer

a) kg m / s² b) kg m² / s c) kg m² / s² d) kg² m / s² e) kg² m² / s² Answer: c) kg m² / s²

Step-by-step solution

Write the equation for work (energy) in terms of force and displacement

The work done \(W\) is given by the force \(F\) applied to the object multiplied by the displacement \(d\): \(W = F \cdot d\)

Replace force with its unit

The force \(F\) has the unit of newtons (\(\mathrm{N}\)), which is defined as: \(\mathrm{N} = \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}^{2}\) The work equation now becomes: \(W = \left(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}^{2}\right) \cdot d\)

Replace displacement with its unit

The displacement \(d\) has the unit of meters (\(\mathrm{m}\)), so the work equation becomes: \(W = \left(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}^{2}\right) \cdot \mathrm{m}\)

Multiply the units

Multiply the units together: \(W = \mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s}^{2}\)

Match the result with the given options

Now that we have the correct unit for energy, let's compare it with the given options: a) \(\mathrm{kg} \mathrm{m} / \mathrm{s}^{2}\) c) \(\mathrm{kg} \mathrm{m}^{2} / \mathrm{s}^{2}\) e) \(\mathrm{kg}^{2} \mathrm{~m}^{2} / \mathrm{s}^{2}\) b) \(\mathrm{kg} \mathrm{m}^{2} / \mathrm{s}\) d) \(\mathrm{kg}^{2} \mathrm{~m} / \mathrm{s}^{2}\) The correct unit of energy matches option c: \(\mathrm{kg} \mathrm{m}^{2} / \mathrm{s}^{2}\). So, the correct answer is (c).

Key concepts

Work-Energy Principle

The work-energy principle is fundamental in physics and ties together two crucial concepts: work and energy. When work is done on an object, it results in a change in the object's energy. The work performed by a force acting on an object is calculated as the product of the force applied and the displacement of the object in the direction of the force.

For instance, when you lift a book off the ground, you're doing work against gravity. The energy that you transfer to the book is equal to the work done. This energy is called potential energy because it can be converted back into work - in this case, when the book falls back to the ground. Understanding the work-energy principle helps us to analyze how objects store and transfer energy in physical processes.

Unit Conversion

Unit conversion is a critical skill in physics and engineering, which involves transforming a quantity expressed in one unit to another unit while maintaining the same quantity. Energy can be expressed in various units such as joules, calories, or kilowatt-hours, depending on the context.

Converting between these units requires an understanding of the relationship between them. For example, 1 joule is equivalent to 0.239 calories. Therefore, knowing how to convert between units is essential for interpreting and applying energy values correctly in different scientific and engineering contexts.

Dimensional Analysis

Dimensional analysis is the process of checking the correctness of an equation or conversion by ensuring that the dimensional units on both sides of an equation balance each other out. In physics, every measurement has dimensions such as length, mass, time, etc. Dimensional analysis helps to derive relationships between physical quantities and to convert from one system of units to another.

By using dimensional analysis, it is possible to confirm that the unit of energy, generally given in joules (or \(\mathrm{kg} \mathrm{m}^{2} / \mathrm{s}^{2}\)) in SI units, is correctly derived from the fundamental units for mass (kg), length (m), and time (s). This technique serves as a powerful tool to validate equations and conversions in physics and engineering.

Mechanics in Physics

Mechanics is a branch of physics concerned with the motion of objects and the forces that cause this motion. It's divided into two main branches: statics, which studies objects at rest, and dynamics, which studies objects in motion. The analysis of energy and work is a critical component of dynamics.

Mechanics employs physical quantities and principles like the work-energy principle to understand how and why objects move. A thorough comprehension of mechanics is essential for solving problems in physics and engineering, providing a foundation for further study in other areas of science and technology. Recognizing units of energy and how they relate to the forces and motions of objects is a vital element of practicing mechanics.