Work, Kinetic Energy, and the Energy Principle

Energy: Concepts and Forms

Introduction to Energy

Energy is a fundamental property of physical systems, essential for understanding how systems change and interact. In physics, energy has a precise definition, distinct from its colloquial use. We classify energy into several types, each with unique characteristics and roles in physical processes.

• Kinetic Energy (K): The energy of motion. All moving objects possess kinetic energy, which increases with mass and speed.

• Potential Energy (U): Stored energy associated with an object's position. Gravitational potential energy, for example, depends on an object's height above the ground.

• Thermal Energy (Eth): The collective kinetic and potential energies of all atoms and bonds within an object. Thermal energy increases as an object becomes hotter.

Energy in the System vs. the Environment

Energy is a property of a system, so it is crucial to define the system's boundaries. Energy can be transformed within the system or transferred between the system and its environment:

• Transformation: Energy changes from one form to another within the system, but the total energy remains constant.

• Transfer: Energy moves into or out of the system via two main processes:

  • Work (W): Energy transferred by mechanical means.

  • Heat (Q): Energy transferred by thermal means.

These processes can change the total energy of the system.

Energy Transformation vs. Energy Transfer

• Energy Transfer Example: A shot putter does work on the shot, transferring kinetic energy. A boy stretches a slingshot, transferring energy as potential energy.

• Energy Transformation Example: A diver transforms gravitational potential energy into kinetic energy during a dive. A meteor transforms kinetic energy into thermal energy as it heats the air.

The Energy Principle

Statement of the Energy Principle

The energy principle states that the change in a system's energy equals the work done on the system by the environment:

$\Delta E = W_{\text{ext}}$

• Sign Convention: Energy entering the system (from the environment) is positive; energy leaving is negative.

• Work: $W_{\text{ext}} > 0$ increases the system's energy.

Work and Kinetic Energy for a Single Particle

Work-Energy Theorem

Consider a particle of mass $m$ acted on by a constant force $F_s$ parallel to its displacement $\Delta s$:

$F_s = m a_s = m \frac{dv_s}{dt}$

Using calculus and integrating both sides, we find:

$\int_{v_i}^{v_f} m v_s dv_s = \int_{s_i}^{s_f} F_s ds$

$\frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2 = F_s (s_f - s_i)$

Defining kinetic energy as $K = \frac{1}{2} m v^2$, the change in kinetic energy is:

$\Delta K = K_f - K_i = W$

Units and Properties of Kinetic Energy

• Units: Joules (J), where $1\,\text{J} = 1\,\text{N} \cdot \text{m} = 1\,\text{kg} \cdot \text{m}^2/\text{s}^2$

• Scalar Quantity: Kinetic energy depends only on speed and mass, not direction.

Work Done by a Constant Force

General Expression for Work

For a constant force $\vec{F}$ acting at an angle $\theta$ to the displacement $\Delta \vec{r}$:

$W = F \cos \theta \Delta r$

Or, using the dot product:

$W = \vec{F} \cdot \Delta \vec{r}$

Determining the Sign of Work

• Positive Work: Force and displacement in the same direction ($0 < \theta < 90^\circ$), increases kinetic energy.

• Negative Work: Force and displacement in opposite directions ($90^\circ < \theta < 180^\circ$), decreases kinetic energy.

• Zero Work: Force perpendicular to displacement ($\theta = 90^\circ$) or no displacement.

Calculating Work: Tactics and Examples

Force and Displacement	Angle $\theta$	Work $W$	Sign of $W$	Energy Transfer
Same direction	$0^\circ$	$F \Delta r$	+	Energy into system, $K$ increases
Angle $< 90^\circ$	$< 90^\circ$	$F \Delta r \cos \theta$	+	Energy into system, $K$ increases
Opposite direction	$180^\circ$	$-F \Delta r$	-	Energy out of system, $K$ decreases

Work with Multiple Forces and Particles

• Multiple Forces: Total work is the sum of work done by each force: $W_{\text{tot}} = W_1 + W_2 + \dots + W_N$

• Multiple Particles: Total kinetic energy is the sum for all particles: $K_{\text{tot}} = K_1 + K_2 + \dots + K_N$

• Energy Principle for Systems: $\Delta K_{\text{tot}} = W_{\text{tot}}$

Mathematical Tools: The Dot Product

Definition and Properties

The dot product (scalar product) of two vectors $\vec{A}$ and $\vec{B}$ with angle $\alpha$ between them is:

$\vec{A} \cdot \vec{B} = AB \cos \alpha$

• Result: A scalar (not a vector).

• Sign: Positive if $0 < \alpha < 90^\circ$, negative if $90^\circ < \alpha < 180^\circ$, zero if $\alpha = 90^\circ$.

Component Form

If $\vec{A} = A_x \hat{i} + A_y \hat{j}$ and $\vec{B} = B_x \hat{i} + B_y \hat{j}$, then:

$\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y$

Work Done by a Variable Force

If the force varies in magnitude or direction, work is calculated as:

$W = \int_{s_i}^{s_f} \vec{F} \cdot d\vec{s}$

This is the area under the $F_s$ vs. $s$ graph.

Elastic Forces and Hooke's Law

Restoring Forces and Hooke's Law

Elastic objects like springs exert a restoring force proportional to their displacement from equilibrium:

$F_{\text{Sp}} = -k \Delta s = -k (s - s_{\text{eq}})$

• k: Spring constant (stiffness of the spring).

• Direction: Always opposite to displacement.

Work Done by a Spring

The work done by a spring as it moves from $s_i$ to $s_f$:

$W = \int_{s_i}^{s_f} F_{\text{Sp},s} ds = \frac{1}{2} k (s_i - s_{\text{eq}})^2 - \frac{1}{2} k (s_f - s_{\text{eq}})^2$

Thermal Energy and Dissipative Forces

Thermal Energy

Thermal energy ($E_{th}$) is the sum of the microscopic kinetic and potential energies of all atoms and bonds in an object. Changes in $E_{th}$ correspond to temperature changes.

Dissipative Forces

Dissipative forces, such as kinetic friction and drag, convert macroscopic kinetic energy into thermal energy, increasing the temperature of objects in contact.

• Example: Dragging a box across a carpet at constant speed increases the thermal energy of both the box and the carpet.

• Energy Principle: $\Delta E_{th} = f_k \Delta s$ (where $f_k$ is the kinetic friction force).

Work and Power

Definition of Power

Power is the rate at which energy is transferred or transformed:

$P = \frac{dE_{sys}}{dt}$

When energy changes due to work by a force:

$P = \frac{dW}{dt}$

• SI Unit: Watt (W), where $1\,\text{W} = 1\,\text{J}/\text{s}$

• English Unit: Horsepower (hp), where $1\,\text{hp} = 746\,\text{W}$

Power Delivered by a Force

For a constant force $\vec{F}$ moving a particle with velocity $\vec{v}$:

$P = \vec{F} \cdot \vec{v} = F v \cos \theta$