A.3.2 Work and energy transfer

Work and Energy

1. Imagine you’re pushing a box across the floor.
2. You apply a force, and the box moves.
3. But what exactly are you doing? You’re doing work.

Work is a way to quantify how much energy is transferred when a force causes an object to move.

Definition of Work: The Relationship Between Force and Displacement

Mathematically, this is expressed as:

$$W=Fs\cos \theta$$

where:

• $W$ is the work done (measured in joules, J).
• $F$ is the magnitude of the force applied (in newtons, N).
• $s$ is the displacement of the object (in meters, m).
• $\theta$ is the angle between the force and the displacement.

Positive, Negative, and Zero Work

1. Positive Work: When the force and displacement are in the same direction ($0^{\circ }\le \theta <{90}^{\circ }$), work is positive.
2. Negative Work: When the force opposes the displacement (${90}^{\circ }<\theta \le {180}^{\circ }$), work is negative.
3. Zero Work: When the force is perpendicular to the displacement ($\theta ={90}^{\circ }$), no work is done.

The Work-Energy Theorem: Connecting Work Done to Changes in Kinetic Energy

In other words, when you do work on an object, you are transferring energy to or from it, changing its motion.

Kinetic Energy

It is given by the formula:

$$E_K=\frac{1}{2}m{v}^2$$

where:

• $E_K$ is the kinetic energy (in joules, J).
• $m$ is the mass of the object (in kilograms, kg).
• $v$ is the velocity of the object (in meters per second, m/s).

Work-Energy Theorem

The work-energy theorem can be expressed as:

$$W_{\text{net}}=\mathrm{\Delta }{E}_K=E_{K,\text{final}}-E_{K,\text{initial}}$$

This means that the net work done on an object is equal to the change in its kinetic energy.

Energy Transformations: Analyzing Energy Conversion in Systems

Energy can change forms but is never lost. This principle is the foundation of the law of conservation of energy.

Pendulum Example (discussed in A.3.2)

1. Consider a pendulum swinging back and forth.
2. At its highest point, the pendulum has maximum gravitational potential energy and zero kinetic energy.
3. As it swings down, potential energy is converted into kinetic energy.
4. At the lowest point, kinetic energy is maximized, and potential energy is minimized.

Graphical Representation: Using Force-Displacement Graphs to Calculate Work Done

When a force varies with displacement, you can use a force-displacement graph to calculate the work done.

The work done is equal to the area under the curve on the graph.

Calculating Work from a Graph

• Constant Force:
  • The graph is a horizontal line, and the area is a rectangle.
  • Work is simply $F\times s$.
• Varying Force:
  • The graph may be a curve or a series of straight lines.
  • Divide the area into simple shapes (rectangles, triangles, trapezoids) and sum their areas to find the total work done.

Sankey Diagrams: Visualizing Energy Transfers

1. Energy transfers in physical systems can be efficiently represented using Sankey diagrams, which provide a clear visual representation of how energy is distributed and lost in a process.
2. In a Sankey diagram, the width of each arrow is proportional to the amount of energy being transferred or lost.

Interpreting a Sankey Diagram

• Input Energy:
  • Represented by a thick arrow entering the system, showing the total energy supplied.
• Useful Energy Output:
  • A thinner arrow pointing forward, representing energy that is successfully converted into a useful form (e.g., kinetic energy in a motor).
• Wasted Energy:
  • Arrows branching off, usually pointing downward or sideways, representing energy lost due to inefficiencies (e.g., heat due to friction or air resistance).

Reflection and Broader Implications

Work and energy are foundational concepts in physics, underpinning everything from simple machines to complex ecosystems.