Work, Energy, and Power
Summary
This chapter introduces one of physics' most powerful concepts: energy and its conservation. You'll learn how forces do work to transfer energy, and how energy transforms between kinetic energy (energy of motion) and potential energy (stored energy). The work-energy theorem provides an alternative approach to solving mechanics problems, often simpler than using Newton's laws directly. You'll distinguish between conservative and non-conservative forces, analyze energy transformations using diagrams, and explore power as the rate of energy transfer. Simple machines demonstrate how mechanical advantage can amplify forces, though energy conservation always applies. These energy principles will reappear throughout the rest of physics.
Concepts Covered
This chapter covers the following 15 concepts from the learning graph:
- Work
- Work by Constant Force
- Work by Variable Force
- Work-Energy Theorem
- Kinetic Energy
- Potential Energy
- Gravitational Potential Energy
- Elastic Potential Energy
- Conservative Forces
- Non-conservative Forces
- Conservation of Energy
- Mechanical Energy
- Energy Diagrams
- Power
- Efficiency
Prerequisites
This chapter builds on concepts from:
- Chapter 1: Scientific Foundations and Mathematical Tools
- Chapter 2: Motion in One Dimension
- Chapter 4: Forces and Newton's Laws
Introduction to Energy Concepts
Energy is everywhere. It powers your phone, moves cars down the highway, and even keeps your heart beating. In this chapter, you'll explore one of the most fundamental ideas in all of physics: energy and how it transforms from one form to another. Unlike previous chapters where we focused on forces and motion directly, energy provides an alternative perspective that often makes solving problems much simpler.
The beauty of energy analysis lies in its universality. Whether you're analyzing a roller coaster, a bow and arrow, or a hydroelectric dam, the same energy principles apply. Energy is conserved—it never disappears, but constantly transforms between different forms. This powerful idea will help you understand not just physics problems, but real-world phenomena from renewable energy to how your muscles work.
What is Work?
In everyday language, "work" means many things—homework, a job, effort. In physics, work has a precise definition that connects force and motion. Understanding this definition is your first step toward mastering energy concepts.
The Physics Definition of Work
Work is the process of transferring energy by applying a force over a distance. For work to occur, three conditions must be met:
- A force must be applied to an object
- The object must move in some direction
- At least some component of the force must be in the direction of motion
If any of these conditions is missing, no work is done in the physics sense. You can push against a brick wall all day until you're exhausted, but if the wall doesn't move, you've done zero work in physics terms (though your muscles certainly expended energy!).
Work by a Constant Force
When a constant force acts on an object that moves in a straight line, calculating work is straightforward. The work done by a constant force is given by:
$$W = F \cdot d \cdot \cos(\theta)$$
Where:
- $W$ = work (measured in joules, J)
- $F$ = magnitude of the applied force (newtons, N)
- $d$ = displacement of the object (meters, m)
- $\theta$ = angle between the force vector and displacement vector
This equation reveals several important insights:
- When force and displacement are in the same direction ($\theta = 0°$), work is maximum: $W = Fd$
- When force is perpendicular to displacement ($\theta = 90°$), no work is done: $W = 0$
- When force opposes displacement ($\theta = 180°$), work is negative: $W = -Fd$
Example: If you push a 20 kg box with a force of 50 N across a floor for 3 meters in the direction you're pushing, the work done is:
$$W = (50 \text{ N})(3 \text{ m})\cos(0°) = 150 \text{ J}$$
Work in Different Scenarios
Let's examine how work behaves in different situations:
| Scenario | Force Direction | Angle | Work Done |
|---|---|---|---|
| Pushing a cart forward | Same as motion | 0° | Positive (maximum) |
| Lifting a box upward | Upward (against gravity) | 0° | Positive |
| Friction on sliding object | Opposite to motion | 180° | Negative |
| Carrying a box horizontally | Perpendicular to motion | 90° | Zero |
| Pulling a sled at an angle | Between parallel and perpendicular | 30°-60° | Positive (reduced) |
Diagram: Work Scenario Interactive Diagram
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Work by a Variable Force
Many real-world forces aren't constant—they change as an object moves. Springs are a perfect example: the more you compress or stretch a spring, the harder it pushes back. Calculating work by a variable force requires a different approach.
For a force that varies with position, work is calculated using integration:
$$W = \int_{x_1}^{x_2} F(x) \, dx$$
Graphically, this represents the area under a force-versus-position graph between the initial and final positions.
Spring Example: For an ideal spring following Hooke's Law ($F = kx$), the work done in stretching or compressing the spring from its natural length by a distance $x$ is:
$$W = \frac{1}{2}kx^2$$
This quadratic relationship means doubling the stretch requires four times the work.
Diagram: Variable Force Work Calculation MicroSim
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Kinetic Energy: Energy of Motion
Now that we understand work as energy transfer, let's explore the first major form of energy: kinetic energy.
Kinetic energy (KE) is the energy an object possesses due to its motion. Any moving object—from a rolling ball to a speeding bullet—has kinetic energy. The faster it moves or the more massive it is, the more kinetic energy it has.
The equation for kinetic energy is:
$$KE = \frac{1}{2}mv^2$$
Where:
- $KE$ = kinetic energy (joules, J)
- $m$ = mass (kilograms, kg)
- $v$ = speed (meters per second, m/s)
Notice that kinetic energy depends on the square of velocity. This means:
- Doubling the speed quadruples the kinetic energy
- Tripling the speed increases kinetic energy by a factor of nine
- Halving the speed reduces kinetic energy to one-quarter
This quadratic relationship has important real-world implications. A car traveling at 60 mph has four times the kinetic energy of the same car at 30 mph, which is why high-speed collisions are so much more destructive.
Example: A 1200 kg car traveling at 25 m/s (about 56 mph) has kinetic energy:
$$KE = \frac{1}{2}(1200 \text{ kg})(25 \text{ m/s})^2 = 375,000 \text{ J}$$
That's enough energy to lift the car 32 meters straight up!
The Work-Energy Theorem
One of the most powerful tools in physics connects work and kinetic energy through a simple, elegant relationship.
The Fundamental Connection
The work-energy theorem states that the net work done on an object equals its change in kinetic energy:
$$W_{net} = \Delta KE = KE_f - KE_i$$
Or expanded:
$$W_{net} = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$$
This theorem provides an alternative to using Newton's second law for solving motion problems. Instead of analyzing forces and accelerations, you can often solve problems more easily by tracking energy changes.
Why This Theorem Matters
The work-energy theorem is powerful because:
- It's independent of path: Only the initial and final states matter, not the details of motion in between
- It handles variable forces naturally: No need to track changing accelerations
- It simplifies complex problems: Many multi-step force problems become single-step energy problems
- It's universally applicable: Works for any object, any force, any path
Example Problem: A 5 kg block sliding at 8 m/s encounters a rough surface that exerts a friction force of 20 N. How far does the block slide before stopping?
Using forces (Newton's second law): 1. Find acceleration: $a = F/m = -20/5 = -4$ m/s² 2. Use kinematic equation: $v_f^2 = v_i^2 + 2ad$ 3. Solve for $d$: $0 = 64 + 2(-4)d$, so $d = 8$ m
Using work-energy theorem: 1. Initial KE: $\frac{1}{2}(5)(8)^2 = 160$ J 2. Final KE: 0 J (block stops) 3. Work by friction: $W = -Fd = -20d$ 4. Apply theorem: $-20d = 0 - 160$, so $d = 8$ m
Same answer, but the energy approach is often more intuitive and requires fewer steps.
Diagram: Work-Energy Theorem Interactive Demonstration
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Potential Energy: Stored Energy
While kinetic energy is energy of motion, potential energy (PE) is stored energy based on an object's position or configuration. It represents the potential to do work in the future.
Think of potential energy like money in a bank account—it's available to be spent (converted to kinetic energy) when needed. There are several types of potential energy, but we'll focus on the two most common: gravitational and elastic.
Gravitational Potential Energy
Gravitational potential energy is the energy stored in an object due to its height above a reference point. Lift a book above your desk, and you've given it gravitational potential energy. Release it, and that stored energy converts to kinetic energy as it falls.
The equation for gravitational potential energy near Earth's surface is:
$$PE_g = mgh$$
Where:
- $PE_g$ = gravitational potential energy (joules, J)
- $m$ = mass (kg)
- $g$ = gravitational acceleration (9.8 m/s²)
- $h$ = height above reference point (m)
Key points about gravitational potential energy:
- The reference point (where $h = 0$) is arbitrary—choose what's convenient
- Only changes in height matter, not the absolute height
- Gravitational PE increases linearly with height
- The same amount of work is required to lift an object regardless of the path taken
Example: Lifting a 10 kg backpack from the floor to a shelf 2 meters high increases its gravitational potential energy by:
$$PE_g = (10 \text{ kg})(9.8 \text{ m/s}^2)(2 \text{ m}) = 196 \text{ J}$$
Elastic Potential Energy
Elastic potential energy is energy stored in objects that can be stretched, compressed, or deformed and then return to their original shape. Springs, rubber bands, and drawn bows all store elastic potential energy.
For an ideal spring obeying Hooke's Law, the elastic potential energy is:
$$PE_s = \frac{1}{2}kx^2$$
Where:
- $PE_s$ = elastic potential energy (J)
- $k$ = spring constant (N/m), measuring the spring's stiffness
- $x$ = displacement from natural length (m)
Notice this is the same expression we found earlier for work done on a spring. That makes sense: the work you do compressing the spring is stored as elastic potential energy.
Important characteristics:
- Elastic PE depends on the square of displacement (quadratic relationship)
- Doubling the compression/extension quadruples the stored energy
- Energy is the same whether spring is compressed or stretched by distance $x$
- Stiffer springs (larger $k$) store more energy for the same displacement
Example: Compressing a spring with $k = 200$ N/m by 0.3 meters stores:
$$PE_s = \frac{1}{2}(200 \text{ N/m})(0.3 \text{ m})^2 = 9 \text{ J}$$
Diagram: Potential Energy Comparison Chart
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Conservative vs. Non-Conservative Forces
Not all forces are created equal when it comes to energy. Some forces conserve mechanical energy, while others dissipate it. This distinction is crucial for understanding energy conservation.
Conservative Forces
Conservative forces are forces for which the work done is independent of the path taken—only the starting and ending points matter. These forces have an associated potential energy.
Common examples of conservative forces:
- Gravitational force
- Elastic spring force
- Electrostatic force (covered in later chapters)
Key characteristics:
- Work done is path-independent
- Work done around a closed loop is zero
- Associated with a potential energy function
- Mechanical energy is conserved
Path Independence Example: Whether you lift a book straight up 2 meters or carry it up a winding staircase that rises 2 meters, the work done against gravity is the same: $mgh = mg(2)$.
Non-Conservative Forces
Non-conservative forces are forces for which the work done depends on the path taken. These forces convert mechanical energy to other forms (typically thermal energy) and have no associated potential energy function.
Common examples of non-conservative forces:
- Friction (kinetic and static)
- Air resistance
- Tension in a rope (when energy is dissipated)
- Normal force (when used to dissipate energy)
- Applied forces from motors or muscles
Key characteristics:
- Work done depends on the path
- Work done around a closed loop is not zero
- No potential energy function exists
- Mechanical energy is not conserved (converted to heat, sound, etc.)
Path Dependence Example: Sliding a box 10 meters in a straight line requires less work against friction than sliding it along a zigzag path totaling 15 meters, even if both paths start and end at the same points.
Why This Distinction Matters
Understanding conservative versus non-conservative forces tells you whether you can use energy conservation equations or need to account for energy dissipation:
- Conservative forces only → Total mechanical energy is conserved
- Non-conservative forces present → Mechanical energy decreases (usually converted to thermal energy)
This guides your problem-solving approach and determines which equations apply.
Conservation of Energy
We've arrived at one of the most profound principles in all of physics: energy conservation. This principle underlies everything from roller coasters to rocket propulsion to the universe itself.
The Law of Conservation of Energy
The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. The total energy of an isolated system remains constant.
In equation form:
$$E_{total} = E_{kinetic} + E_{potential} + E_{thermal} + E_{chemical} + ... = \text{constant}$$
For mechanical systems where we focus on kinetic and potential energy, we often refer to conservation of mechanical energy:
$$E_{mechanical} = KE + PE = \text{constant}$$
Or more explicitly:
$$KE_i + PE_i = KE_f + PE_f$$
This equation is valid only when no non-conservative forces do work (or when we account for energy converted to other forms).
When Mechanical Energy is Conserved
Mechanical energy is conserved when:
- Only conservative forces do work
- The system is isolated (no external forces)
- No energy is converted to non-mechanical forms (heat, sound, light, etc.)
Examples where mechanical energy is approximately conserved:
- A pendulum swinging in a vacuum (no air resistance)
- A satellite orbiting Earth (neglecting atmospheric drag)
- A ball on a frictionless track
- An ideal spring-mass system
When Mechanical Energy is Not Conserved
When non-conservative forces like friction are present, mechanical energy decreases (it's converted to thermal energy, not destroyed):
$$KE_i + PE_i = KE_f + PE_f + E_{thermal}$$
Or equivalently:
$$W_{non-conservative} = \Delta KE + \Delta PE$$
The work done by non-conservative forces equals the change in mechanical energy.
Example: A 2 kg ball is dropped from 5 meters. Without air resistance, it would hit the ground at:
Using energy conservation: $mgh = \frac{1}{2}mv^2$
$$v = \sqrt{2gh} = \sqrt{2(9.8)(5)} = 9.9 \text{ m/s}$$
With air resistance doing -20 J of work, the final speed is reduced:
$$mgh - 20 = \frac{1}{2}mv^2$$ $$(2)(9.8)(5) - 20 = \frac{1}{2}(2)v^2$$ $$v = 8.1 \text{ m/s}$$
The "missing" 20 J was converted to thermal energy heating the air and ball.
Diagram: Energy Conservation Roller Coaster Simulation
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Energy Diagrams
Energy diagrams provide a visual way to understand potential energy, stability, and motion without doing detailed calculations. They're particularly useful for analyzing complex systems.
Energy diagrams plot potential energy as a function of position. By adding a horizontal line representing total mechanical energy, you can quickly determine where an object can move and how fast it will be going.
Key features of energy diagrams:
- PE curve: Shows potential energy at each position
- Total energy line: Horizontal line representing $E_{total} = KE + PE$
- Allowed regions: Where total energy line is above PE curve (KE would be positive)
- Forbidden regions: Where total energy line is below PE curve (KE would be negative—impossible!)
- Turning points: Where total energy equals PE (KE = 0, object stops and reverses)
- Equilibrium positions: Where PE curve has zero slope (no net force)
Stable vs. Unstable Equilibrium
Energy diagrams reveal the stability of equilibrium positions:
- Stable equilibrium: PE curve has a local minimum (like a ball at the bottom of a bowl)
- Small displacement results in restoring force back toward equilibrium
-
System oscillates around this point
-
Unstable equilibrium: PE curve has a local maximum (like a ball balanced on top of a hill)
- Small displacement results in force away from equilibrium
-
System accelerates away from this point
-
Neutral equilibrium: PE curve is flat (like a ball on a flat surface)
- No restoring or destabilizing force
- System remains in new position after displacement
Reading Energy Diagrams
From an energy diagram, you can determine:
- Speed at any position: $KE = E_{total} - PE(x)$, so $v = \sqrt{2(E_{total} - PE)/m}$
- Maximum speed: Occurs where PE is minimum
- Range of motion: Between turning points where $E_{total} = PE$
- Force direction: Force points from higher PE toward lower PE (down the PE slope)
Diagram: Energy Diagram Interactive Explorer
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Power: The Rate of Energy Transfer
Imagine two construction workers lifting identical bricks to the top of a building. One takes an hour; the other takes all day. They do the same amount of work, but one does it much faster. This rate of energy transfer is called power.
Power is the rate at which work is done or energy is transferred. It measures how quickly energy changes form or location.
The definition of power is:
$$P = \frac{W}{t} = \frac{\Delta E}{t}$$
Where:
- $P$ = power (watts, W)
- $W$ = work done (joules, J)
- $\Delta E$ = energy transferred (J)
- $t$ = time interval (seconds, s)
One watt equals one joule per second: $1 \text{ W} = 1 \text{ J/s}$
For constant power, the work done over time $t$ is:
$$W = Pt$$
Power in Different Contexts
Power appears in many forms:
- Mechanical power: Rate of doing mechanical work (lifting, pushing, pulling)
- Electrical power: Rate of electrical energy transfer (covered in later chapters)
- Metabolic power: Rate of chemical energy consumption in living organisms
- Solar power: Rate of energy reception from the Sun
Example: A motor lifts a 500 kg elevator 20 meters in 10 seconds. The power output is:
Work done: $W = mgh = (500)(9.8)(20) = 98,000$ J
Power: $P = W/t = 98,000/10 = 9,800$ W = 9.8 kW
Instantaneous Power and Force
When a force acts on a moving object, the instantaneous power delivered is:
$$P = F \cdot v = Fv\cos(\theta)$$
Where:
- $F$ = force magnitude (N)
- $v$ = velocity magnitude (m/s)
- $\theta$ = angle between force and velocity vectors
When force and velocity are in the same direction ($\theta = 0°$):
$$P = Fv$$
This relationship shows that for a given power output:
- Larger forces require slower speeds
- Higher speeds are possible with smaller forces
This is why cars have different gears: lower gears provide more force but less speed, while higher gears provide less force but more speed, all for approximately the same engine power output.
Example: A car engine producing 75 kW (about 100 horsepower) can exert different forces at different speeds:
| Speed | Force Available |
|---|---|
| 10 m/s (22 mph) | 7,500 N |
| 20 m/s (45 mph) | 3,750 N |
| 30 m/s (67 mph) | 2,500 N |
At higher speeds, less force is available for acceleration—which is why cars accelerate more slowly at highway speeds than from rest.
Efficiency: Real vs. Ideal Energy Transfer
No real machine or process transfers energy perfectly. Some energy is always "lost" to non-useful forms, typically heat and sound. Efficiency quantifies how well a device converts input energy to useful output energy.
Efficiency is defined as:
$$\text{Efficiency} = \frac{E_{output}}{E_{input}} = \frac{P_{output}}{P_{input}}$$
Often expressed as a percentage:
$$\text{Efficiency} = \frac{E_{output}}{E_{input}} \times 100\%$$
Key points about efficiency:
- Efficiency is always less than 100% for real devices (second law of thermodynamics)
- Efficiency equals 100% only for idealized, frictionless systems
- Higher efficiency means less wasted energy
- Improving efficiency saves energy and money
Efficiency Examples
Different devices and processes have vastly different efficiencies:
| Device/Process | Typical Efficiency |
|---|---|
| LED light bulb | 80-90% |
| Electric motor | 70-90% |
| Human muscles | 20-25% |
| Gasoline engine | 20-30% |
| Coal power plant | 33-40% |
| Incandescent bulb | 5-10% |
Example: An electric motor draws 2000 W of electrical power and delivers 1600 W of mechanical power. Its efficiency is:
$$\text{Efficiency} = \frac{1600}{2000} = 0.80 = 80\%$$
The remaining 400 W (20%) is converted to heat warming the motor and surroundings.
Why Efficiency Matters
Efficiency has profound real-world implications:
- Energy conservation: Higher efficiency reduces energy consumption
- Cost savings: Less wasted energy means lower operating costs
- Environmental impact: Reduced energy use decreases pollution and resource depletion
- Performance: More efficient devices often perform better (less heat, longer lifespan)
- Sustainability: Efficient use of resources is essential for long-term sustainability
Diagram: Energy Efficiency Comparison Infographic
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Simple Machines and Mechanical Advantage
Throughout history, humans have used simple machines to make work easier by trading force for distance or vice versa. While simple machines can't reduce the amount of work required (energy is conserved!), they can make tasks more achievable by changing how force is applied.
The Six Classical Simple Machines
- Lever - Bar that pivots on a fulcrum
- Pulley - Wheel with a grooved rim for a rope or cable
- Inclined plane - Flat surface tilted at an angle
- Wedge - Two inclined planes back-to-back
- Screw - Inclined plane wrapped around a cylinder
- Wheel and axle - Large wheel attached to smaller axle
All complex machines are combinations of these basic types.
Mechanical Advantage
Mechanical advantage (MA) is the factor by which a simple machine multiplies force:
$$MA = \frac{F_{output}}{F_{input}}$$
A mechanical advantage greater than 1 means the machine multiplies your input force—you can lift a heavy load with less force. However, there's always a tradeoff:
Work-Energy Principle for Machines:
$$W_{input} = W_{output}$$ (ideal, no friction)
$$F_{input} \cdot d_{input} = F_{output} \cdot d_{output}$$
Therefore:
$$MA = \frac{F_{output}}{F_{input}} = \frac{d_{input}}{d_{output}}$$
The tradeoff: If you gain force (MA > 1), you must move through a greater distance. If you gain distance (MA < 1), you must apply greater force.
Examples of Mechanical Advantage
Lever: - Fulcrum near load, far from effort → Large MA (force multiplier) - Example: Crowbar, wheelbarrow - $MA = \frac{d_{effort}}{d_{load}}$ (distances from fulcrum)
Pulley System: - Single fixed pulley: MA = 1 (changes direction only) - Single movable pulley: MA = 2 (force divided between two rope segments) - Multiple pulleys: MA = number of rope segments supporting load
Inclined Plane: - $MA = \frac{\text{length of ramp}}{\text{height of ramp}} = \frac{L}{h}$ - Longer, more gradual ramps have higher MA (less force needed but greater distance) - Example: Wheelchair ramps, roads up mountains
Real vs. Ideal Mechanical Advantage
Ideal Mechanical Advantage (IMA): Mechanical advantage calculated assuming no friction or energy loss
Actual Mechanical Advantage (AMA): Mechanical advantage measured in real conditions with friction and inefficiencies
Always: $AMA < IMA$
The efficiency of a simple machine relates the two:
$$\text{Efficiency} = \frac{AMA}{IMA} \times 100\%$$
Diagram: Simple Machines Comparison Table
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| Simple Machine | How It Works | Ideal MA Formula | Common Examples | Typical Uses |
|---|---|---|---|---|
| Lever | Rigid bar pivots on fulcrum; effort and load at different distances from pivot | $MA = \frac{d_{effort}}{d_{load}}$ | Crowbar, scissors, seesaw, wheelbarrow | Lifting heavy objects, cutting, balancing |
| Pulley | Rope runs over grooved wheel; can be fixed or movable | Fixed: MA = 1 Movable: MA = 2 System: MA = # of ropes supporting load |
Flagpole, cranes, blinds, elevators | Lifting loads vertically, changing direction of force |
| Inclined Plane | Flat surface at angle to horizontal; trade distance for reduced force | $MA = \frac{L}{h}$ (length ÷ height) | Ramps, roads, stairs | Moving heavy objects to higher elevation |
| Wedge | Two inclined planes back-to-back; converts downward force to sideways splitting force | $MA = \frac{L}{W}$ (length ÷ width) | Axe, knife, chisel, doorstop | Splitting, cutting, holding objects in place |
| Screw | Inclined plane wrapped around cylinder; rotational motion becomes linear | $MA = \frac{2\pi r}{p}$ (circumference ÷ pitch) | Bolts, jar lids, vise, cork screw | Fastening, lifting, pressing |
| Wheel & Axle | Larger wheel attached to smaller axle; both rotate together | $MA = \frac{r_{wheel}}{r_{axle}}$ | Doorknob, steering wheel, winch, screwdriver | Turning with less force, winding rope |
Diagram: Pulley System Mechanical Advantage Simulator
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Connecting Work, Energy, and Power
Let's bring together the key concepts from this chapter and see how they interconnect:
The Energy Flow Chain
- Work transfers energy from one object or system to another
- Kinetic energy represents energy of motion
- Potential energy represents stored energy due to position or configuration
- Conservation of energy ensures total energy remains constant (though it transforms)
- Power measures how quickly energy transfers or transforms
- Efficiency quantifies how much input energy becomes useful output energy
Problem-Solving Strategies
When approaching work-energy problems, consider these strategies:
Use the Work-Energy Theorem when: - You know initial and final speeds (or can find them) - Forces may vary with position - You don't need to know time or acceleration details
Use Conservation of Energy when: - Only conservative forces act (or you can account for non-conservative work) - You know energy at one point and want to find it at another - The problem involves height changes or spring compression/extension
Use Power equations when: - The problem involves time rates - You need to find how long energy transfer takes - The question asks about "how fast" energy is used or delivered
Use Force and kinematics (Newton's laws) when: - You need detailed information about acceleration, time, or force - Energy methods aren't applicable - The problem specifically asks about forces
Key Equations Summary
Work: - Constant force: $W = Fd\cos(\theta)$ - Variable force: $W = \int F(x) \, dx$
Energy: - Kinetic: $KE = \frac{1}{2}mv^2$ - Gravitational potential: $PE_g = mgh$ - Elastic potential: $PE_s = \frac{1}{2}kx^2$
Conservation: - Mechanical energy (conservative forces only): $KE_i + PE_i = KE_f + PE_f$ - With non-conservative forces: $KE_i + PE_i + W_{nc} = KE_f + PE_f$ - Work-energy theorem: $W_{net} = \Delta KE$
Power: - Average power: $P = W/t = \Delta E/t$ - Instantaneous power: $P = Fv\cos(\theta)$
Efficiency: - $\text{Efficiency} = \frac{E_{output}}{E_{input}} = \frac{P_{output}}{P_{input}}$
Mechanical Advantage: - $MA = \frac{F_{output}}{F_{input}} = \frac{d_{input}}{d_{output}}$ (ideal case)
Real-World Applications
Energy principles appear everywhere in technology and nature. Understanding work, energy, and power helps explain:
Transportation: - Why cars have better fuel efficiency at constant highway speeds than in stop-and-go traffic - How regenerative braking in electric vehicles recovers kinetic energy - Why aerodynamic design matters more at higher speeds (air resistance increases with v²)
Sports: - How pole vaulters convert running kinetic energy to gravitational potential energy - Why trampolines help gymnasts jump higher (elastic potential energy storage and release) - How downhill skiers maximize speed by minimizing friction and choosing optimal paths
Construction and Engineering: - How pile drivers use gravitational potential energy to drive posts into ground - Why cranes use pulley systems to lift heavy loads - How dam spillways convert gravitational PE to kinetic energy safely
Energy Production: - Hydroelectric dams convert gravitational PE of water to electrical energy - Wind turbines extract kinetic energy from moving air - Solar panels convert light energy to electrical energy
Biology: - Muscles convert chemical energy to mechanical work - Heart does work pumping blood throughout the body - ATP molecules store and release chemical potential energy
Key Takeaways
As you complete this chapter, keep these essential ideas in mind:
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Work is energy transfer by applying force through a distance. Only the component of force parallel to displacement does work.
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Energy comes in many forms but can be categorized as kinetic (motion) or potential (position/configuration).
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Energy is conserved: It transforms between types but the total remains constant in an isolated system.
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The work-energy theorem provides a powerful alternative to force analysis: net work equals change in kinetic energy.
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Conservative forces (like gravity and springs) conserve mechanical energy and have associated potential energies.
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Non-conservative forces (like friction) convert mechanical energy to other forms, usually thermal energy.
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Power measures rate: How quickly work is done or energy is transferred. Same work done faster requires more power.
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Efficiency quantifies losses: Real devices always have efficiency less than 100% due to energy conversion to non-useful forms.
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Simple machines trade force for distance: They can multiply force (MA > 1) but you must move through greater distance. Work is still conserved.
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Energy analysis often simplifies problems: Many situations difficult to solve with forces become straightforward using energy methods.
Energy concepts will reappear throughout the rest of this course—in momentum, rotation, oscillations, waves, and even electricity. Mastering these principles now will pay dividends as you continue your physics journey.