Work and Kinetic Energy (Physics 211, Module 10)

Work and Kinetic Energy

Introduction

This module explores the concepts of work and kinetic energy, foundational ideas in classical and modern physics. It covers the equivalence of mass and energy, the principle of energy conservation, and the mathematical definitions and applications of work and kinetic energy.

Mass-Energy Equivalence

Einstein's Famous Equation

• Mass-energy equivalence is described by Einstein's equation:

$E = mc^2$

• This equation states that mass (m) and energy (E) are interchangeable; they are different forms of the same physical quantity.

• The speed of light (c) is a very large number, so even a small amount of mass can be converted into a large amount of energy.

• In practical terms, $c^2 = 10^{17} (m/s)^2$.

• Historically, Newtonian physics treated mass and energy as separate, but modern physics recognizes their deep connection.

Creation of Matter and Antimatter

Particle Creation

• In high-energy processes, such as those involving quantum light (photons), matter and antimatter can be created from energy.

• Example: A photon interacts with a proton (hydrogen nucleus) to produce an electron and an anti-electron (positron).

• In these processes, mass is not conserved, but energy is always conserved.

Diagram (described): A photon (quantum of light) interacts with a proton, resulting in the creation of an electron and an anti-electron.

Conservation of Energy

Principle of Energy Conservation

• Energy is always conserved in physical processes, even if mass is not.

• Energy can be transformed between different forms (e.g., mechanical, thermal, chemical), but the total energy remains constant.

• Analogy: Like money in a bank account, energy can be transferred between objects or forms, but the total amount is conserved.

• Mechanical energy (kinetic + potential) is not always conserved if non-conservative forces (like friction) are present, but total energy is.

Work

Definition of Work

• In physics, work is the change in energy of an object due to an external force moving it along a path.

• Mathematically, work is defined as:

$W = \Delta E = \int_a^b \vec{F} \cdot d\vec{s}$

• For motion along a straight line (x-direction):

$W = \int_{x_i}^{x_f} F_x dx = \int_{x_i}^{x_f} F \cos \theta dx = F \cos \theta \int_{x_i}^{x_f} dx = F \cos \theta (x_f - x_i)$

• Where $\theta$ is the angle between the force and the direction of displacement.

Key Points about Work

• Work is only done by the component of force in the direction of displacement.

• If the force is perpendicular to the displacement ($\theta = 90^\circ$), then $\cos 90^\circ = 0$ and no work is done.

• Work can be positive, negative, or zero depending on the direction of force relative to displacement.

Example: Carrying a Stone

• If a person carries a stone horizontally across a field, the gravitational force acts vertically, while the displacement is horizontal.

• Since the angle between force and displacement is $90^\circ$, the work done by gravity is zero.

Kinetic Energy

Definition and Formula

• Kinetic energy is the energy of motion, given by:

$K = \frac{1}{2}mv^2$

• Where m is mass and v is velocity.

• Kinetic energy can be associated with any object in motion.

Work-Energy Theorem

• The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy:

$W = \Delta K = K_f - K_i$

• This relationship holds even if the force is not constant, as long as the net work is calculated.

Example: Pulling a Box

• Pulling a box with a force $F$ over a distance $\Delta x$ at an angle $\theta$:

$K_f = K_i + F \cos \theta \Delta x$

• If friction is present, the net work is reduced by the work done against friction.

Work Done by Gravity

Work-Energy Analysis

• When an object is thrown upward, gravity does negative work, reducing the object's kinetic energy and increasing its potential energy.

• At the highest point, all kinetic energy is converted to potential energy.

$W_{gravity} = -mgh$

• Where $h$ is the height gained.

Work Done by a Spring

Hooke's Law and Spring Force

• The force exerted by a spring is given by Hooke's Law:

$F = -kx$

• Where $k$ is the spring constant and $x$ is the displacement from equilibrium.

• The work done by a spring as it moves from $x_i$ to $x_f$:

$W_{spring} = \int_{x_i}^{x_f} F dx = \int_{x_i}^{x_f} (-kx) dx = -\frac{1}{2}k(x_f^2 - x_i^2)$

• The work done on the spring by an external force is equal in magnitude and opposite in sign.

Spring as a Launcher

• When a spring launches a projectile, the potential energy stored in the spring is converted into the kinetic energy of the projectile.

$\frac{1}{2} k x^2 = \frac{1}{2} m v^2$

• Where $x$ is the compression/stretch of the spring, $m$ is the mass of the projectile, and $v$ is its velocity upon release.

Summary Table: Types of Work and Energy

Type	Formula	Key Points
Work (general)	$W = \int \vec{F} \cdot d\vec{s}$	Depends on force direction and displacement
Kinetic Energy	$K = \frac{1}{2}mv^2$	Energy of motion
Work by Gravity	$W = -mgh$	Negative when moving upward
Work by Spring	$W = -\frac{1}{2}k(x_f^2 - x_i^2)$	Depends on displacement from equilibrium

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Examples and analogies have been added to reinforce conceptual understanding.