Work, Energy & Power – Notes

Work, energy and power form a single conceptual thread that explains why objects move, stop, or remain at rest and how living beings and machines perform useful tasks.

What fundamental concepts are interconnected to explain motion and task performance?

Everyday vs. Scientific Notion

• Everyday speech labels any strenuous mental or physical effort (studying, singing, carrying a load, etc.) as “work”.

How does the everyday definition of "work" differ from its scientific definition?

• Scientifically, work requires TWO simultaneous conditions:1. A force must act on an object.

  2. The object must undergo displacement in the direction of (or with a component along) that force.

What are the two essential scientific conditions that must be met for work to be done?

• If either condition is missing, no physical work is done even though energy may be expended physiologically (e.g.- pushing a wall that does not move,

  • standing still with a heavy suitcase).

Provide an example of a situation where energy is expended, but no physical work is performed?

Quantitative Definition (Constant Force)

If a constant force $F$ moves an object through a displacement $s$ along the same line,

$W = F\,s$

Unit: $1\;\text{J} = 1\;\text{N}\cdot 1\;\text{m}$.

What is the formula for work done by a constant force, and what is the SI unit of work?

Sign Convention for Work

• Positive work – force and displacement in same direction (e.g., baby pulling toy car, one lifts a book upward.)

When is work considered positive?

• Negative work – force opposite displacement (e.g., friction on a sliding body, gravity acting while you lift an object.)

When is work considered negative?

• Zero work – either $F=0$ (no force) or $s=0$ (no displacement) or force ? displacement.

What are the conditions that result in zero work being done?

Illustrative Numerical Examples

1. $F=5\;\text{N},\;s=2\;\text{m} \;\Rightarrow\; W = 10\;\text{J}.$

If a force of $5\;\text{N}$ moves an object $2\;\text{m}$, how much work is done?

2. Porter lifting $15\,\text{kg}$ suitcase to $1.5\,\text{m}:$

   $W = m g h = 15\times10\times1.5 = 225\;\text{J}.$

Calculate the work done by a porter lifting a $15\,\text{kg}$ suitcase to a height of $1.5\,\text{m}$ (Assume $g=10\,\text{m/s}^2$)?

Concept-check Activities

• Push pebble, pull trolley, lift book – all satisfy both conditions.

Give an example of an everyday action that satisfies the scientific conditions for work.

• Think of cases with force but no motion (zero work) versus motion without force (e.g., coasting ice puck – in ideal frictionless state, work done by you after release is zero).

Explain the difference between situations where force is applied but no work is done, and situations where there is motion but no work done by a specific force (after initial push).

Energy

Energy is the capacity of a system to do work. Same SI unit as work → Joule.

Define energy and state its SI unit.

• Biological Context: Food supplies chemical energy; strenuous tasks require more energy.

How is energy utilized in a biological context?

• Mechanical Context: Fast cricket ball dislodging bails, stretched bow, compressed spring show stored ability to work.

Provide examples of energy in a mechanical context.

Forms

1. Mechanical = Kinetic + Potential

What are the two main components of mechanical energy?

2. Heat (thermal)

Is heat considered a form of energy?

3. Chemical

How is chemical energy stored, typically?

4. Electrical

What is electrical energy?

5. Light (radiant)

Give an example of light energy.

6. Nuclear, geothermal, tidal, etc.

Can you name other less common forms of energy?

Criterion for Calling Something a “Form of Energy”

If it can be quantified in Joules and can be converted into or from mechanical work, it qualifies.

What is the criterion for something to be classified as a "form of energy"?

Kinetic Energy

Energy due to motion.

What type of energy is associated with motion?

Operational Definition

Amount of work needed to accelerate a body of mass $m$ from rest to velocity $v$.

How is kinetic energy operationally defined?

Derivation

Starting from equations of motion:

$v^{2}-u^{2}=2 a s$, $F=ma$.

Substitute into work formula:

$W = F s = m a s = m a \left(\frac{v^{2}-u^{2}}{2a}\right)=\frac12 m (v^{2}-u^{2}).$

If initial speed $u=0$, kinetic energy $E<em>k$ becomes $E</em>k = \frac12 m v^{2}.$

Derive the formula for kinetic energy $(E_k = \frac12 m v^{2})$ starting from the work-energy principle and equations of motion.

Properties

• Proportional to mass and to square of speed.

What is the relationship between kinetic energy, mass, and speed?

• Doubling velocity quadruples $E_k$.

What happens to kinetic energy if the velocity of an object is doubled?

Examples

• $m=15\,\text{kg},\;v=4\,\text{m/s}\;\Rightarrow\;E_k=120\,\text{J}.$

Calculate the kinetic energy of a $15\,\text{kg}$ object moving at $4\,\text{m/s}$.

• Car $m=1500\,\text{kg}$ from $30\,\text{km/h}\to 60\,\text{km/h}:$

  $\Delta E_k = 156250\,\text{J}$ of work required.

If a $1500\,\text{kg}$ car accelerates from $30\,\text{km/h}$ to $60\,\text{km/h}$, how much additional kinetic energy does it gain?

Potential Energy

Stored energy of position or configuration.

Define potential energy.

Elastic Potential Energy

• Stretching rubber band or slinky, compressing spring, bending bow stores energy in molecular bindings.

How is elastic potential energy stored in objects like a rubber band or a spring?

• Demonstrated by release causing motion (arrow shot, slinky recoil).

How can stored elastic potential energy be demonstrated?

Gravitational Potential Energy

When an object of mass $m$ is raised through height $h$ (against gravity $g$):

$E_p = m g h.$

What is the formula for gravitational potential energy?

Path Independence

Work done by gravity depends only on vertical displacement, not on path (straight lift vs. ramp → same $mgh$).

Is gravitational potential energy path-dependent or path-independent? Explain.

Examples

1. $m=10\,\text{kg},\;h=6\,\text{m}\Rightarrow E_p=588\,\text{J}$ (using $g=9.8\,\text{m/s}^{2}$).

Calculate the gravitational potential energy of a $10\,\text{kg}$ mass raised $6\,\text{m}$. (Use $g=9.8\,\text{m/s}^{2}$).

2. Given $E_p=480\,\text{J},\;m=12\,\text{kg},\;g=10\,\text{m/s}^{2}\Rightarrow h=4\,\text{m}.$

If an object has a gravitational potential energy of $480\,\text{J}$ and a mass of $12\,\text{kg}$, what is its height? (Use $g=10\,\text{m/s}^{2}$).

Interconversion of Energy

Nature continuously converts energy:

What fundamental principle does nature demonstrate through continuous energy conversion?

• Photosynthesis: Radiant → chemical.

Describe the energy conversion in photosynthesis.

• Water cycle: Solar → heat (evaporation) → potential (clouds) → kinetic (rainfall) → hydroelectric.

Trace the energy transformations involved in the water cycle.

• Wind: Solar heating → pressure gradients → kinetic of air.

How does solar energy lead to wind energy?

• Fuels (coal, petroleum): Ancient solar energy → chemical bonds.

Where does the energy in fossil fuels originate from?

• Human devices: Batteries (chemical) → electrical → light/heat; Engines: chemical → thermal → mechanical.

Give two examples of energy conversions in human-made devices.

Law of Conservation of Energy

“Energy can neither be created nor destroyed; it only changes form.”

State the Law of Conservation of Energy.

Consider free fall from height $h$:

• At top: $E<em>p = mgh,\;E</em>k=0.$

In free fall, what are the kinetic and potential energies at the highest point?

• Mid-way: $E<em>p = mg(h/2),\;E</em>k = \frac12 m v^{2} = m g (h/2).$

  Total remains $mgh$.

During free fall, what happens to the sum of potential and kinetic energy midway down?

• Just before ground: $E<em>p \to 0,\;E</em>k \to mgh.$

What are the kinetic and potential energies of an object just before it hits the ground in free fall?

Energy losses in real life (sound, heat, air drag) are conversions, not destruction.

Why are "energy losses" in real-world scenarios not considered destruction of energy?

Power

Rate at which work is done or energy transferred:

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

Unit: watt (W) where $1\,\text{W} = 1\,\text{J/s}$.

Larger unit: $1\,\text{kW} = 1000\,\text{W}.$

Define power and list its SI unit and a larger common unit.

Average vs. Instantaneous

If rate varies, average power $= \frac{\text{total}\;E}{\text{total}\;t}.$

How is average power calculated when the rate of work or energy transfer varies?

Practical Examples

• Girl A climbs 8 m in 20 s: $P = \frac{400\times8}{20}=160\,\text{W}.$

If a girl with a weight of $400\,\text{N}$ climbs $8\,\text{m}$ in $20\,\text{s}$, what is her power output?

• Girl B same height in 50 s: $64\,\text{W}.$

If another girl with the same weight climbs the same height in $50\,\text{s}$, how does her power output compare?

• Boy (50 kg) up staircase 6.75 m in 9 s: $P = 375\,\text{W}.$

Calculate the power of a $50\,\text{kg}$ boy climbing a staircase of $6.75\,\text{m}$ in $9\,\text{s}$. (Assume $g=10\,\text{m/s}^2$).

• Household energy billing: 1 electrical “unit” = $1\,\text{kWh} = 3.6\times10^{6}\,\text{J}.$

What is an electrical "unit" in household energy billing, and how many Joules does it represent?

Synthesising Concepts

1. Mechanical Energy $= E<em>k + E</em>p$. In many mechanical systems (pendulum, roller coaster, projectile ignoring friction) this sum is constant.

What is mechanical energy, and under what conditions is it conserved?

2. Work–Energy Theorem: Net work done on body equals change in kinetic energy.

State the Work-Energy Theorem.

3. Real-world Relevance: From muscle physiology (chemical → mechanical) to renewable tech (solar, wind, hydro) the conversion principles govern design and efficiency.

How do energy conversion principles apply to real-world applications like muscle physiology and renewable technology?

4. Ethical & Practical Implication: The inexorability of conservation means we cannot “make” energy, only transform; motivates sustainable harnessing and minimal dissipation.

What ethical and practical implications arise from the law of conservation of energy?

Recap of Key Formulae

• Work: $W = F s \cos\theta$ (general form – here $\theta$ is angle between $\vec F, \vec s$; for parallel $\theta=0$ → $W=Fs$.)

What is the general formula for work, including the angle between force and displacement?

• Kinetic energy: $E_k = \tfrac12 m v^{2}.$

What is the formula for kinetic energy?

• Gravitational potential energy: $E_p = mgh.$

What is the formula for gravitational potential energy?

• Power: $P = \dfrac{W}{t}=\dfrac{E}{t}.$

What are the two main formulas for power?

Worked Problem Set (Condensed)

• Change of velocity from $5\,\text{m/s}$ to $2\,\text{m/s}$ for $20\,\text{kg}$:

  $\Delta E<em>k = \tfrac12 m (v</em>f^{2}-v_i^{2}) = \tfrac12\times20(4-25)= -210\,\text{J}$ (negative indicates energy removed; same magnitude of work done by retarding force).

A $20\,\text{kg}$ object changes its velocity from $5\,\text{m/s}$ to $2\,\text{m/s}$. Calculate the change in its kinetic energy and explain the meaning of the negative result.

• Electric heater 1500 W for 10 h: $E = Pt = 1500\times10\,\text{h} = 15\,\text{kWh} = 5.4\times10^{7}\,\text{J}.$

An electric heater uses $1500\,\text{W}$ for $10\,\text{h}$. Calculate the total energy consumed in both kilowatt-hours and Joules.

Philosophical Note

If energy transformations were impossible, complex processes from star formation to metabolism in cells would freeze—no gradients, no motion, no life. Conservation coupled with transformability is thus central to existence.

Why is the transformability of energy, alongside its conservation, crucial for the existence of complex processes and life?

Here's an explanation of some words from your notes that an 8th grader might find a bit tricky:

• Conceptual thread: Think of this like a string that connects different ideas together. In your notes, Work, Energy, and Power are all linked by one 'conceptual thread' because they are all really about how things move and do stuff.

• Strenuous: This means something that takes a lot of effort, either physically or mentally. Like a strenuous workout is one that makes you super tired.

• Simultaneously: This means at the exact same time. If two things happen simultaneously, they happen together.

• Displacement: In science, displacement isn't just moving anywhere, it's the specific distance and direction an object moves from its starting point. Imagine drawing a straight arrow from where something began to where it ended up – that's its displacement.

• Physiologically: This relates to how living things (like our bodies) work. So, when you push a wall, you're using energy 'physiologically' in your body, even if the wall doesn't move.

• Quantitative: This means something that can be measured using numbers or amounts. A 'quantitative definition' is one that uses formulas and measurements.

• Constant force: A force that stays the same in strength and direction. It doesn't get stronger, weaker, or change which way it's pushing or pulling.

• Component: Think of this as part of something bigger. If a force has a 'component' along a direction, it means part of that force is pushing or pulling in that specific direction.

• Illustrative: This means something that helps to explain or show something clearly, usually through examples. 'Illustrative numerical examples' are numbers that help make the ideas clear.

• Porter: This is a person whose job is to carry luggage or other loads, usually at a train station or airport.

• Kinetic: This means related to motion. 'Kinetic energy' is the energy an object has because it's moving.

• Potential: This means something that is stored and has the possibility to do something in the future. 'Potential energy' is stored energy, like a ball held high up has the potential to fall and do work.

• Configuration: This refers to how parts are arranged or put together. For things like springs, their 'configuration' (whether they're stretched or squeezed) affects their stored energy.

• Molecular bindings: This refers to the forces or connections between the tiny, tiny particles (molecules) that make up materials. When you stretch a rubber band, you're changing these 'molecular bindings'.

• Recoil: This is when something springs back quickly after being compressed or stretched, like a slinky snapping back into shape.

• Gravitational: This is related to gravity, the force that pulls things towards each other (like pulling you down to Earth).

• Path Independence: This means something doesn't depend on the route or way you get there, only the start and end points. For gravitational potential energy, it being 'path independent' means it doesn't matter if you lift a book straight up or carry it up a spiral ramp; the energy change is the same if the height change is the same.

• Interconversion: This means changing from one form to another. 'Energy interconversion' is when energy changes from light to chemical, or chemical to electrical, etc.

• Photosynthesis: This is the process plants use to make their own food using sunlight, water, and carbon dioxide.

• Radiant: This describes energy that travels in waves, like light or heat from the sun. 'Radiant energy' is often another name for light energy.

• Evaporation: This is when a liquid turns into a gas, like water turning into vapor when it gets hot.

• Hydroelectric: This describes electricity generated using the power of moving water, usually from dams.

• Pressure gradients: Think of this as a difference in pressure from one place to another. Wind blows because there are 'pressure gradients' in the air.

• Fossil fuels: These are fuels like coal, oil, and natural gas formed over millions of years from the remains of ancient plants and animals.

• Metabolism: This is all the chemical processes that happen inside a living body to keep it alive, like converting food into energy.

• Conservation coupled with transformability: This means two ideas working together: energy is 'conserved' (it's never lost or made new) AND it can be 'transformed' (changed from one type to another). This combined idea is super important for how the universe works.

• Inexorability: This means something cannot be stopped or changed. The 'inexorability of conservation' means the law of conservation of energy is absolute and always true.

• Dissipation: When energy 'dissipates', it spreads out or is wasted, usually turning into heat or sound that isn't useful for the main purpose. We want minimal dissipation to be efficient.

• General form: This refers to a formula or rule that works in all situations, not just specific ones. The 'general form' of the work formula includes an angle, making it useful no matter the direction of force.

• Retarding force: This is a force that slows something down or opposes its motion, like friction or air resistance.