Calculate Work from Energy - Physics

1. Fundamental Concepts

Definition: Work is the energy transferred to or from an object via a force acting on the object in the direction of its displacement.
Formula: The work \(W\) done by a constant force \(F\) along a displacement \(d\) is given by \(W = F \cdot d\).
Units: The unit of work is the joule (J), where \(1 \text{ J} = 1 \text{ N} \cdot \text{m}\).

2. Key Concepts

Work-Energy Principle: The net work done on an object equals the change in its kinetic energy: \(W_{\text{net}} = \Delta K\).

Gravitational Potential Energy: Potential energy due to gravity is given by \(U_g = mgh\), where \(m\) is mass, \(g\) is gravitational acceleration, and \(h\) is height.

Conservation of Mechanical Energy: In the absence of non-conservative forces, the total mechanical energy (kinetic plus potential) remains constant: \(E_{\text{initial}} = E_{\text{final}}\).

3. Examples

Example 1 (Basic)

Problem: A force of \(5 \text{ N}\) acts on an object moving in a straight line for a distance of \(10 \text{ m}\). Calculate the work done.

Step-by-Step Solution:

Use the formula for work: \(W = F \cdot d\).
Substitute the values: \(W = 5 \cdot 10 = 50 \text{ J}\).

Validation: Given \(F = 5 \text{ N}\) and \(d = 10 \text{ m}\), the work done is \(50 \text{ J}\).

Example 2 (Intermediate)

Problem: An object with a mass of \(2 \text{ kg}\) is lifted vertically through a height of \(5 \text{ m}\). Calculate the work done against gravity.

Step-by-Step Solution:

Calculate the gravitational force: \(F = mg = 2 \cdot 9.8 = 19.6 \text{ N}\).
Use the formula for work: \(W = F \cdot d = 19.6 \cdot 5 = 98 \text{ J}\).

Validation: Given \(m = 2 \text{ kg}\), \(g = 9.8 \text{ m/s}^2\), and \(h = 5 \text{ m}\), the work done is \(98 \text{ J}\).

4. Problem-Solving Techniques

Identify Forces: List all forces acting on the object and determine their directions.
Choose Reference Points: Establish a reference point for calculating potential energy changes.
Apply Conservation Laws: Use the conservation of energy principle when appropriate.