Work Done by a Force

Introduction

Key Concepts

Definition of Work

In physics, work is defined as the product of the component of a force acting in the direction of displacement and the magnitude of that displacement. Mathematically, it is expressed as:

$$ W = \vec{F} \cdot \vec{d} = Fd \cos(\theta) $$

where:

• W is the work done by the force.
• F is the magnitude of the force.
• d is the displacement of the object.
• θ is the angle between the force and the direction of displacement.

Work is measured in joules (J) in the International System of Units (SI), where 1 joule equals 1 newton-meter (1 J = 1 N.m).

Types of Forces and Their Work

Different types of forces can do work, and the nature of the work depends on the force's direction relative to the displacement:

• Positive Work: When the force component is in the same direction as displacement ($0 \leq \theta < 90^\circ$), work is positive, indicating energy transfer to the object.
• Negative Work: When the force component is opposite to the direction of displacement ($90^\circ < \theta \leq 180^\circ$), work is negative, indicating energy removal from the object.
• Zero Work: When the force is perpendicular to the displacement ($\theta = 90^\circ$), no work is done by the force.

Energy and Work

Work is directly related to energy transfer. When work is done on an object, energy is transferred to or from that object. This relationship is foundational in understanding kinetic and potential energy:

• Kinetic Energy: The work-energy theorem states that the work done by all forces acting on an object equals the change in its kinetic energy: $$ W_{\text{total}} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 $$ where $m$ is mass, $v_f$ is final velocity, and $v_i$ is initial velocity.
• Potential Energy: Work done against conservative forces like gravity or springs is stored as potential energy:
  • Gravitational Potential Energy: $$ U = mgh $$ where $h$ is the height above a reference point.
  • Elastic Potential Energy: $$ U = \frac{1}{2}kx^2 $$ where $k$ is the spring constant and $x$ is the displacement from equilibrium.

Power and Work

Power is the rate at which work is done. It quantifies how quickly energy is transferred or transformed:

$$ P = \frac{W}{t} $$

where:

• P is power.
• W is work done.
• t is the time taken.

Power is measured in watts (W), where 1 watt equals 1 joule per second (1 W = 1 J/s).

Work Done by Variable Forces

When the force varies with displacement, work is calculated using integral calculus:

$$ W = \int_{x_i}^{x_f} F(x) \, dx $$

For example, with a spring force $F(x) = -kx$, the work done in stretching the spring from $x_i$ to $x_f$ is:

$$ W = \int_{x_i}^{x_f} (-kx) \, dx = \frac{1}{2}k(x_i^2 - x_f^2) $$

This negative work signifies that energy is stored in the spring as potential energy.

Work in Multiple Dimensions

In scenarios involving forces in multiple dimensions, work is calculated using the dot product of vectors. For a force $\vec{F} = (F_x, F_y, F_z)$ and displacement $\vec{d} = (d_x, d_y, d_z)$:

$$ W = \vec{F} \cdot \vec{d} = F_x d_x + F_y d_y + F_z d_z $$

This formulation allows for precise calculation of work when forces act at various angles and directions.

Applications of Work

Understanding work is crucial in various physical applications:

• Mechanical Systems: Calculating work done by engines, motors, and other machinery.
• Ergonomics: Assessing the effort required in human activities.
• Energy Transfer: Analyzing how energy moves within systems, such as in electricity generation.
• Engineering: Designing systems that optimize work and energy efficiency.

Work and Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. Work plays a central role in this principle, as it represents the transfer of energy between objects or systems. For instance, when lifting a mass against gravity, work is done against the gravitational force, increasing the system's potential energy. Conversely, when an object falls, gravitational work converts potential energy back into kinetic energy.

Dimensional Analysis of Work

Dimensional analysis ensures that the equations for work are dimensionally consistent. The dimensions of work (W) are derived from force and displacement:

$$ [W] = [F][d] = \text{MLT}^{-2} \times \text{L} = \text{ML}^2\text{T}^{-2} $$

This dimensional consistency is crucial for validating physical equations and ensuring correct unit usage.

Examples and Problem-Solving

Applying the concept of work often involves solving problems that require identifying the force components, displacement, and the angle between them. For example:

• Example 1: Calculate the work done by a force of 50 N acting at an angle of $30^\circ$ to move an object 10 meters horizontally.

  Solution:

  Using the work formula:

  $$ W = Fd \cos(\theta) = 50 \times 10 \times \cos(30^\circ) = 500 \times 0.866 = 433 \text{ J} $$
• Example 2: Determine the work done by friction if a box is pushed 5 meters with a constant force of 20 N opposite to the direction of motion.

  Solution:

  Here, $\theta = 180^\circ$, so $\cos(180^\circ) = -1$:

  $$ W = 20 \times 5 \times \cos(180^\circ) = 100 \times (-1) = -100 \text{ J} $$

Advanced Concepts

Work-Energy Theorem

The work-energy theorem is a pivotal principle in mechanics, articulating that the net work done on an object equals its change in kinetic energy. Formally, it is expressed as:

$$ W_{\text{net}} = \Delta KE $$

where $\Delta KE = KE_f - KE_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$. This theorem bridges the concepts of force, displacement, and energy, allowing for the analysis of motion without explicitly solving differential equations of motion.

Derivation:

Starting from Newton's second law, $F = ma$, and integrating with respect to displacement:

$$ W = \int F \, dx = \int ma \, dx = m \int a \, dx $$

Since acceleration $a = \frac{dv}{dt}$ and $v = \frac{dx}{dt}$, we can rewrite the integral in terms of velocity:

$$ W = m \int \frac{dv}{dt} \frac{dx}{dt} \, dt = m \int v \, dv = \frac{1}{2}mv^2 \bigg|_{v_i}^{v_f} = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 $$

Thus, $W_{\text{net}} = \Delta KE$.

Non-Conservative Forces and Work

Non-conservative forces, such as friction and air resistance, do work that dissipates mechanical energy, typically converting it into thermal energy. The work done by non-conservative forces affects the total mechanical energy of a system:

$$ W_{\text{non-cons}} = \Delta KE + \Delta PE $$

For example, when an object slides on a rough surface, friction does negative work, reducing the object's kinetic energy and increasing thermal energy.

Path Dependence of Work

The work done by a force can depend on the path taken between two points, especially for non-conservative forces. For conservative forces, work is path-independent and depends only on the initial and final positions. This distinction is crucial in energy conservation and potential energy calculations.

Work in Rotational Motion

In rotational dynamics, work is analogous to its linear counterpart but involves torque and angular displacement:

$$ W = \tau \theta \cos(\phi) $$

where:

• W is work.
• τ is torque.
• θ is the angular displacement in radians.
• φ is the angle between torque and angular displacement vectors.

This formulation is essential in analyzing systems like engines and turbines, where rotational work is prevalent.

Interdisciplinary Connections

The concept of work extends beyond classical mechanics, finding applications in various fields:

• Engineering: Designing efficient machines and systems by optimizing work cycles.
• Biology: Understanding muscle work and energy expenditure in organisms.
• Economics: Modeling work as a function of labor input and productivity.
• Environmental Science: Calculating the work involved in processes like pumping water or wind energy generation.

These interdisciplinary connections highlight the versatility and fundamental nature of work in both natural and applied sciences.

Complex Problem-Solving

Advanced problems involving work often require multi-step reasoning and integration of various concepts:

• Example 3: A slingshot launches a projectile with an initial speed of 20 m/s. Calculate the work done by the slingshot on the projectile.

  Solution:

  Assuming all work is converted into kinetic energy:

  $$ W = \Delta KE = \frac{1}{2}mv^2 - 0 = \frac{1}{2}m(20)^2 = 200m \text{ J} $$

  where $m$ is the mass of the projectile.

• Example 4: A force varies with displacement as $F(x) = 3x^2$ N over a distance from $x = 1$ m to $x = 3$ m. Calculate the work done.

  Solution:

  Using the integral of force over displacement:

  $$ W = \int_{1}^{3} 3x^2 \, dx = 3 \left[ \frac{x^3}{3} \right]_1^3 = [x^3]_1^3 = 27 - 1 = 26 \text{ J} $$

Work in Energy Conservation Equations

In systems where multiple forces act, including both conservative and non-conservative, the conservation of energy equation incorporates work done by all forces:

$$ \Delta KE + \Delta PE = W_{\text{non-cons}} $$

This equation allows for the analysis of systems where energy is not conserved due to external factors like friction or applied forces.

Work in Thermodynamics

In thermodynamics, work represents the energy transfer between a system and its surroundings due to macroscopic forces acting through displacements. For example, the work done by an expanding gas against external pressure is given by:

$$ W = P \Delta V $$

where:

• P is the external pressure.
• ΔV is the change in volume.

This formulation is critical in understanding heat engines, refrigerators, and various thermodynamic cycles.

Relativistic Considerations of Work

At velocities approaching the speed of light, classical definitions of work require modification. Relativistic mechanics introduces concepts where mass and energy are interdependent, and work done by forces must account for time dilation and length contraction effects. While beyond the scope of IB Physics HL, it is essential for advanced studies in modern physics.

Mathematical Derivations

Deriving work-related equations often involves calculus and vector analysis. For instance, deriving the work done by a variable force requires integrating the force function over the displacement:

$$ W = \int_{x_i}^{x_f} F(x) \, dx $$

In three-dimensional space, using vector calculus, work is expressed as the line integral of the force vector along the path of displacement:

$$ W = \int_{\mathcal{C}} \vec{F} \cdot d\vec{r} $$

where $\mathcal{C}$ represents the path from the initial to the final position.

Comparison Table

Aspect	Conservative Forces	Non-Conservative Forces
Definition	Forces where work done is path-independent and can be associated with potential energy.	Forces where work done depends on the path and cannot be associated with potential energy.
Examples	Gravity, spring force.	Friction, air resistance, applied force.
Energy Conservation	Mechanical energy is conserved when only conservative forces do work.	Mechanical energy is not conserved due to energy dissipation.
Work Done	Path-independent; work can be stored as potential energy.	Path-dependent; work results in energy dissipation.
Mathematical Representation	Can be expressed as the gradient of a potential energy function.	No potential energy function can represent the work done.

Summary and Key Takeaways