03 - Work and Energy

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Work and Energy

Why Energy Matters in Medicine and Biology

Energy, work, and power are often associated with engineering applications related to the functioning and performance of machines built by man. However, both from a historical point of view and in application and theory, these concepts have always been widely used in Medicine and Biology.

The principle of conservation of energy was introduced, in its first formulation, by Julius von Mayer, physician and physicist of the nineteenth century, who is counted among the founders of Thermodynamics.

The concept of power is used in Physiology to quantify cardiac and metabolic activity, and many diseases can lead to imbalances in the production and consumption of energy by organisms (e.g. the relationship between arterial hypertension and heart disease has a purely mechanical origin).

Energy as a Foundational Concept

• Energy is perhaps the most important concept in physics and science
• Its existence was definitively tested in 1850 (Joule)
• Every interaction (gravitational, electromagnetic, thermodynamic transformation) can be seen as an energy exchange
• Both mechanics and thermodynamics can be formulated in terms of energy

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Work

Definition

$L=\vec{F}\cdot \vec{s}=|\vec{F}|\cdot |\vec{s}|\cdot \cos (\alpha )$

Work is associated with the effort required to move an object. The larger the object or the displacement, the greater the effort. If an object is stationary and a force acts on it, the work performed is zero, even if we are making an effort.

Sign of Work

The sign of work $L$ depends on $\cos (\vartheta )$:

Condition	$\cos (\vartheta )$	$\vec{F}\cdot \vec{s}$	Type
$0\le \vartheta <\pi /2$	$>0$	$>0$	Active work
$\pi /2<\vartheta \le \pi$	$<0$	$<0$	Resisting work
$\vartheta =\pi /2$	$=0$	$=0$	Null work

Effective Force

Work is the product of the displacement magnitude and the projection of the force in the direction of displacement (effective force).

Infinitesimal Work

For a material point at position $\vec{r}$ within a force field $\vec{F}(\vec{r})$:

$dL=\vec{F}(\vec{r})\cdot d\vec{r}=|\vec{F}(\vec{r})|\cdot |d\vec{r}|\cdot \cos (\theta )$

Work Along a Trajectory

For a material point moving from initial position ${\vec{r}}_i$ to final position ${\vec{r}}_f$ along trajectory $TR$:

$L_{{\vec{r}}_i\stackrel{TR}{\to }{\vec{r}}_f}={\int }_{{\vec{r}}_i}^{{\vec{r}}_f}dL={\int }_{{\vec{r}}_i}^{{\vec{r}}_f}\vec{F}(\vec{r}),d\vec{r}$

Units of Work

Work is a scalar quantity with dimensionality $[F\cdot l]=[M{L}^2{T}^{-2}]$.

Unit	Conversion
Joule (SI)	$1\text{J}$
Electronvolt	$1\text{eV}=e\cdot V=1.6\times 10^{-19}\text{J}$
Calorie	$1\text{Cal}=4.184\text{J}$
Erg	$1\text{erg}=1\text{dyne}\times 1\text{cm}=10^{-5}\text{N}\times 10^{-2}\text{m}=10^{-7}\text{J}$

What is the formula for work done by a force $\vec{F}$ over a displacement $\vec{s}$?

$L=\vec{F}\cdot \vec{s}=|\vec{F}|\cdot |\vec{s}|\cdot \cos (\alpha )$ Work is zero when force is perpendicular to displacement ($\alpha =\pi /2$).

Anki cloze

Work is a {1:scalar} quantity. If the angle between force and displacement is between $\pi /2$ and $\pi$, the work is {2:negative (resisting work)}.

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Kinetic Energy

Definition

The kinetic energy of a point mass is:

$E_C=\frac{1}{2}m{v}^2$

where $m$ is the mass and $v$ its velocity.

The work–energy relation along a trajectory is:

$L_{{\vec{r}}_i\stackrel{TR}{\to }{\vec{r}}_f}=E_C({\vec{r}}_f)-E_C({\vec{r}}_i)=\frac{1}{2}m{v}_f^2-\frac{1}{2}m{v}_i^2$

A non-null work changes kinetic energy.

Fundamental Theorem of Kinetic Energy

Work–Energy Theorem

The change in kinetic energy of an object equals the total work done by all forces acting on it along its trajectory: $L_{tot}=\Delta E_C$

Demonstration (non-rigorous):

From $L=\vec{F}\cdot \vec{s}$ and Newton’s second law $\vec{F}=m\vec{a}$:

$L=m\vec{a}\cdot \vec{s}$

For uniformly accelerated motion: $v=at$ and $s=\frac{a{t}^2}{2}$, giving $s=\frac{v^2}{2a}$, thus:

$L=m\cdot a\cdot \frac{v^2}{2a}=\frac{m{v}^2}{2}$

Special Case: Centripetal Force

Centripetal Force Does No Work

• The centripetal force is always perpendicular to the displacement
• In uniform circular motion, the velocity magnitude is constant
• Therefore, centripetal force does not perform work ($\Delta E_C=0$)

State the Fundamental Theorem of Kinetic Energy.

The change in kinetic energy of a body equals the total work performed by all forces acting on it: $\Delta E_C=L_{tot}$

Anki cloze

The centripetal force does {1:zero} work because it is always {2:perpendicular} to the displacement, so the speed in uniform circular motion remains {3:constant}.

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Potential Energy

Force Fields

A force field is a region of space in which a force vector can be associated with each point. A material point moving in the field is subject at any point to a force $\vec{F}=\vec{F}(\vec{r})$.

Field Type	Description	Example
Uniform (constant) field	$\vec{F}$ does not depend on position; constant in magnitude, direction, and sense	Weight force $m\vec{g}$, capacitor
Non-uniform field	$\vec{F}$ varies with position	Gravitational field around Earth

Conservative Forces and Fields

In general, changing the trajectory from ${\vec{r}}_i$ to ${\vec{r}}_f$ changes the work $L$. However, conservative fieldsare those in which the work depends only on the endpoints, not on the path taken.

A uniform force ${\vec{F}}_K$ is conservative:

$L_{{\vec{r}}_i\stackrel{TR}{\to }{\vec{r}}_f}={\int }_{{\vec{r}}_i}^{{\vec{r}}_f}\vec{F}(\vec{r}),d\vec{r}={\vec{F}}_K\cdot {\int }_{{\vec{r}}_i}^{{\vec{r}}_f}d\vec{r}={\vec{F}}_K\cdot \Delta \vec{r}$

For gravitational forces:

$L_{{\vec{r}}_i\stackrel{TR}{\to }{\vec{r}}_f}=m\vec{g}\cdot ({\vec{r}}_f-{\vec{r}}_i)$

Every reference to the trajectory disappears.

Note on Fields and Forces

Associated with each force there is also a field (e.g., electric field $E=F/q$, gravitational field $\vec{g}$, magnetic field $\vec{B}$).

To define conservativity (and therefore the existence of a potential function), it is necessary to assess whether the work performed by the field (circulation) depends on the route or not.

In many cases, field and force are “the same thing” (same direction, same scalar product with displacement $\Delta s$) so no distinction is needed. In some cases there is a difference (e.g., the magnetic field).

Definition of Potential Energy

If a field is conservative, work can always be written as the difference between the value of a scalar function at ${\vec{r}}_i$ and ${\vec{r}}_f$. This function is called potential energy $U(\vec{r})$.

From the gravitational case:

$L({\vec{r}}_i\to {\vec{r}}_f)=m\vec{g}\cdot ({\vec{r}}_f-{\vec{r}}_i)=(-m\vec{g}\cdot {\vec{r}}_i)-(-m\vec{g}\cdot {\vec{r}}_f)$

$L({\vec{r}}_i\to {\vec{r}}_f)=mg\cdot (z_i-z_f)$

The function $U(z)=mgz$ is called the gravitational potential energy.

Properties of Potential Energy

• $U(\vec{r})$ is a scalar function of position
• Represents the ability of a body to perform work by virtue of its position in a conservative force field
• It is a state function (minima correspond to equilibrium positions)
• Defined up to an additive constant

For motion from $r_A$ to $r_B$: $L=-\Delta U=-[U(r_B)-U(r_A)]$

Gravitational Potential Energy

For the calculation of gravitational potential energy, we consider the projection along the $z$-axis:

$h=z_f-z_i$

$U(h)=mgh$

Elastic Potential Energy (Springs)

Elastic (Hooke’s) force:

$F=-k(x-l_0)$

If $l_0=0$:

$F=-kx$

The elastic potential energy:

$U=\frac{k{x}^2}{2}$

Derivation:

$\Delta U=F\cdot ds=kx\cdot dx⟹\int x,dx=\frac{x^2}{2}$

Summary of Potential Energies

Force	Expression	Potential Energy $U$
Weight force	$m\vec{g}$	$U(h)=mgh$
Gravitational (Newton)	$G\frac{m_1{m}_2}{r^2}$	$U(r)=-G\frac{m_1{m}_2}{r}$
Coulomb (electrostatic)	$\frac{1}{4\pi {ϵ}_0}\frac{q_1{q}_2}{r^2}$	$U(r)=\frac{1}{4\pi {ϵ}_0}\frac{Qq}{r}$
Elastic (spring)	$F=-kx$	$U(x)=\frac{1}{2}k{x}^2$

What is the elastic potential energy of a spring with constant $k$ displaced by $x$ from its rest position?

$U=\frac{1}{2}k{x}^2$

Anki cloze

Potential energy is defined only for {1:conservative} force fields and is specified only up to an {2:additive constant}. The work done by a conservative force equals $L=3:-\Delta U$.

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Mechanical Energy

Conservation of Mechanical Energy

If a material point moves in a conservative field, the sum of its kinetic energy and potential energy remains constant over time.

$E_M=E_C+U=\text{constant}$

This is the Theorem of Conservation of Mechanical Energy.

Example (gravitational field):

$E=\frac{1}{2}m{v}^2+mgz=\text{constant}$

Proof

From the work–energy theorem:

$L=\Delta E_C=[E_C(r_B)-E_C(r_A)]$

From potential energy:

$L=-\Delta U=-[U(r_B)-U(r_A)]$

Therefore:

$-\Delta U=\Delta E_C⟹E_C(r_A)+U(r_A)=E_C(r_B)+U(r_B)$

$\boxed{E_M(r_A)=E_M(r_B)}$

Example: Free Falling Body

At height $h$: all energy is potential $=mgh$

During fall: energy converts from potential to kinetic

At ground ($y=0$): all energy is kinetic

$mgh+0=mgy+\frac{1}{2}m{v}_f^2⟹v_f=\sqrt{2g(h-y)}$

Example: Compressed Spring

Position	Energy state
Maximum compression/elongation	All potential: $\frac{k{l}^2}{2}$, $E_C=0$ (motion reverses)
Rest (equilibrium) position	All kinetic: $\frac{m{v}^2}{2}$

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

The zero of potential energy is taken at the rest position.

Relation Between Force and Potential Energy

$\vec{F}=-\vec{\nabla }U=\left(-\frac{dU}{dx},-\frac{dU}{dy},-\frac{dU}{dz}\right)$

$U(r)=-\int F(r),dr$

Physical Interpretation

• Force pushes toward the minima of potential energy
• The force is zero where the slope of $U$ is zero (maximum or minimum of $U$)
• The system will settle in the position of minimum potential energy

Equilibrium and Potential Energy Minima

Stability of Equilibrium

• Minima of potential energy → positions of stable equilibrium
• Maxima of potential energy → positions of unstable equilibrium

Example: a spring’s equilibrium is at the minimum of its potential. Any perturbation moves the body away; the force then drives it back spontaneously.

Small Perturbations and Harmonic Motion

The region near the minimum of any potential can be approximated by an elastic potential. This is why small perturbations of an equilibrium state produce approximately harmonic (oscillatory) motion — a universally applicable result.

What does mechanical energy conservation state?

In a conservative field, the total mechanical energy $E_M=E_C+U$ remains constant. Work done by conservative forces converts kinetic energy to potential energy and vice versa, with no net loss.

Anki cloze

At the point of maximum compression of a spring, kinetic energy is {1:zero} and all mechanical energy is {2:potential}. At the equilibrium position, all energy is {3:kinetic}.

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Power

Definition

$P=\frac{\Delta L}{\Delta t}$

Time does not appear in the definition of work: there is only force and displacement. Work depends only on how a point moves from the initial to the final position, not on the time taken.

Power makes the temporal dependence explicit.

SI unit: Watt (W) $=\frac{\text{J}}{\text{s}}$

General Meaning of Power

Power measures the amount of energy exchanged per unit of time in anytransformation process: mechanical, electrical, thermal, or chemical.

Common Units of Power and Energy

$1\text{CV}=735\text{W};1\text{CV}=0.9863\text{HP}$

Energy Unit	Joule (J)	Calorie (Cal)	kWh
1 J	1	0.2389	$2.778\times 10^{-7}$
1 Cal	4.186	1	$1.163\times 10^{-6}$
1 kWh	$3.6\times 10^6$	$8.6\times 10^5$	1

Power and Velocity

For a constant force $\vec{F}$ causing displacement $\Delta \vec{r}$ in time $\Delta t$:

$P_{avg}=\frac{\Delta L}{\Delta t}=\vec{F}\cdot \frac{\Delta \vec{r}}{\Delta t}=\vec{F}\cdot {\vec{v}}_M$

Instantaneous power:

$P=\vec{F}\cdot \frac{d\vec{r}}{dt}=\vec{F}\cdot \vec{v}$

Basal Metabolic Rate

$P=\frac{10.0,m}{1\text{kg}}+\frac{6.25,h}{1\text{cm}}-\frac{5.0,a}{1\text{anno}}+s\left[\frac{\text{kcal}}{\text{day}}\right]$

Where:

• $m$ = body mass (kg)
• $h$ = height (cm)
• $a$ = age (years)
• $s$ = $+5$ for men, $-161$ for women

Worked Example

A 55-year-old woman weighing 59 kg and 168 cm tall: $P=\frac{10.0\times 59}{1}+\frac{6.25\times 168}{1}-\frac{5.0\times 55}{1}-161=1272\frac{\text{kcal}}{\text{day}}=53\frac{\text{kcal}}{\text{h}}\approx 61.3\text{W}$

Metabolic Scaling Laws

Metabolic rate is approximately proportional to body mass (volume) across animals.

What is the formula for instantaneous power in terms of force and velocity?

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

Anki cloze

The basal metabolic rate formula adds {1:+5} for men and {2:-161} for women as the sex-correction term $s$. Its SI unit of power is the {3:Watt (W)}, equivalent to {4:joule per second}.

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Exercises

Exercise 1 — Force on a Smooth Surface

A 60 kg block is placed on a perfectly smooth surface. A constant force $\vec{F}=100\text{N}$, parallel to the table, is applied for $t=3\text{s}$.

Find: Final speed and work done.

Solution:

$v=v_0+a\cdot t=\frac{F}{m}\cdot t=\frac{100\text{N}}{60\text{kg}}\cdot 3\text{s}=5\frac{\text{m}}{\text{s}}$

$v^2=2as⟹s=\frac{v^2}{2a}=7.5\text{m}$

$L=\Delta E_C=\frac{1}{2}m{v}^2-0=\frac{1}{2}\cdot 60\cdot 25=750\text{J}$

Verified: $L=F\cdot s=100\text{N}\cdot 7.5\text{m}=750\text{J}$

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Exercise 2 — Inclined Plane and Weight Force

A body of weight $\vec{P}$ slides along a frictionless inclined plane of length $l$.

Find: Work done by the weight force.

Solution:

$L_D=mg\cdot l\cdot \sin (\theta )$

$L_B=mg\cdot b\cdot \cos (\pi /2)=0$

$L_H=mg\cdot h\cdot \cos (0)=mgh$

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Exercise 3 — Climbing the Asinelli Tower

What work does a 70 kg man do to climb to the top of the Asinelli Tower ($h=100\text{m}$)?

$L=mgh=70\cdot 9.81\cdot 100\approx 68,670\text{J}$

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Exercise 4 — Rocket in Fireworks

Problem Statement

A 0.20 kg rocket is launched from rest and follows an erratic path to reach point P, which is 29 m above the starting point. During the flight, 425 J of work is done by the nonconservative force from the burning propellant. Ignoring air resistance and propellant mass loss, find the speed $v_f$ at point P.

Solution:

Applying the kinetic energy theorem — the change in kinetic energy equals the sum of all work done by all forces:

$\Delta E_K=L_{propellant}+L_{gravity}$

Since the rocket starts from rest: $\Delta E_K=E_{Kf}=\frac{1}{2}m{v}_f^2$

The total work is the positive work from combustion minus the negative work from gravity:

$\frac{1}{2}m{v}_f^2=425-mgh$

$v_f=\sqrt{\frac{2}{m}(425-mgh)}$

Substituting values ($m=0.20\text{kg}$, $g=9.81{\text{m/s}}^2$, $h=29\text{m}$):

$v_f=\sqrt{\frac{2}{0.20}(425-0.20\cdot 9.81\cdot 29)}=\sqrt{10\cdot (425-56.9)}\approx \sqrt{3681}\approx 60.7\frac{\text{m}}{\text{s}}$

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TLDR

Summary — Work and Energy

• Work $L=\vec{F}\cdot \vec{s}=|\vec{F}||\vec{s}|\cos \alpha$ is a scalar; positive when force and displacement are aligned, negative when opposed, zero when perpendicular. Units: Joule.
• Infinitesimal / path work is the integral ${\int }_{{\vec{r}}_i}^{{\vec{r}}_f}\vec{F}(\vec{r}),d\vec{r}$.
• Kinetic energy $E_C=\frac{1}{2}m{v}^2$; the work–energy theorem states $\Delta E_C=L_{tot}$.
• Conservative fields: work is path-independent; depends only on start and end points. Gravitational and elastic forces are conservative.
• Potential energy $U(\vec{r})$: scalar function of position for conservative fields; $L=-\Delta U$; defined up to an additive constant; minima = stable equilibrium.
• Key potential energies: gravitational $U=mgh$; gravitational (Newton) $U=-\frac{G{m}_1{m}_2}{r}$; elastic $U=\frac{1}{2}k{x}^2$; Coulomb $U=\frac{1}{4\pi {ϵ}_0}\frac{Qq}{r}$.
• Mechanical energy $E_M=E_C+U=\text{const}$ in a conservative field (conservation theorem).
• Force–potential relation: $\vec{F}=-\vec{\nabla }U$; force points toward potential minima; zero force at extrema of $U$.
• Near any potential minimum, the system behaves as a harmonic oscillator (elastic approximation for small perturbations).
• Power $P=\Delta L/\Delta t=\vec{F}\cdot \vec{v}$; SI unit is Watt. Quantifies energy exchanged per unit time in any process.
• Basal metabolic rate is calculated via the Mifflin–St Jeor formula; a typical adult woman produces ~60 W at rest.
• Centripetal force does zero work (always perpendicular to velocity).
• Metabolic rate scales approximately with body mass across animal species.