1. What dynamics studies

Dynamics is the study of motion and the forces that cause or respond to motion. It sits between statics and vibration-heavy advanced mechanics.

It is usually split into two parts:

Kinematics: describes motion without considering forces
Kinetics: relates motion to the forces and moments that produce it

The core variables are position, velocity, acceleration, force, mass, and moment of inertia.

Typical modeling assumptions

Bodies are often treated as particles when size and rotation do not matter.
Rigid bodies are treated as nondeforming.
Motion is usually measured in an inertial reference frame.
Gravity is often the only body force that needs explicit modeling unless stated otherwise.

Main laws

Newton's second law is the foundation:

\sum \mathbf{F} = m\mathbf{a}

For rotation:

\sum M_O = I_O \alpha

or, more generally,

\sum \mathbf{M}_G = \frac{d\mathbf{H}_G}{dt}

where \(G\) is the center of mass and \(\mathbf{H}\) is angular momentum.

2. Kinematics fundamentals

Kinematics describes how objects move.

Position, displacement, velocity, acceleration

For one-dimensional motion with position \(x(t)\):

v = \frac{dx}{dt}

a = \frac{dv}{dt} = \frac{d^2x}{dt^2}

Useful differential forms:

v\,dv = a\,dx

when acceleration is a function of position, and

a\,dt = dv

when acceleration is a function of time.

Vector form

For 2D or 3D motion:

\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}

\mathbf{v} = \frac{d\mathbf{r}}{dt}

\mathbf{a} = \frac{d\mathbf{v}}{dt}

The safest workflow is to write position as a vector first, then differentiate.

3. Rectilinear motion

Rectilinear motion is motion along a straight line. It is the simplest and most common starting point.

Constant acceleration

If acceleration \(a\) is constant:

$$ v = v_0 + at $$

x = x_0 + v_0 t + \frac{1}{2}at^2

$$ v^2 = v_0^2 + 2a(x-x_0) $$

These are valid only when \(a\) is constant.

Variable acceleration

If acceleration varies with time:

a(t) = \frac{dv}{dt}

Integrate to get velocity, then position.

If acceleration varies with position:

a(x) = v\frac{dv}{dx}

This is useful in spring, drag, and braking problems.

Example pattern

If a car starts from rest and accelerates at a constant \(2 \, \text{m/s}^2\) for \(5\) s:

v = 0 + 2(5) = 10 \, \text{m/s}

x = \frac{1}{2}(2)(5^2) = 25 \, \text{m}

4. Curvilinear motion

Curvilinear motion occurs along a curved path. Two common coordinate systems are Cartesian and normal-tangential coordinates.

Cartesian components

For a particle with coordinates \(x(t)\), \(y(t)\), and \(z(t)\):

\mathbf{v} = \dot{x}\mathbf{i} + \dot{y}\mathbf{j} + \dot{z}\mathbf{k}

\mathbf{a} = \ddot{x}\mathbf{i} + \ddot{y}\mathbf{j} + \ddot{z}\mathbf{k}

This is the best choice when the path is naturally described by coordinates.

Normal-tangential components

When motion follows a known curved path:

\mathbf{v} = v\mathbf{t}

\mathbf{a} = a_t\mathbf{t} + a_n\mathbf{n}

where

a_t = \frac{dv}{dt}

and

a_n = \frac{v^2}{\rho}

with \(\rho\) the radius of curvature.

Interpretation:

\(a_t\) changes speed
\(a_n\) changes direction

Polar coordinates

For planar motion:

\mathbf{v} = \dot{r}\mathbf{e}_r + r\dot{\theta}\mathbf{e}_\theta

\mathbf{a} = \left(\ddot{r} - r\dot{\theta}^2\right)\mathbf{e}_r + \left(r\ddot{\theta} + 2\dot{r}\dot{\theta}\right)\mathbf{e}_\theta

Polar coordinates are useful for motion involving rotation about a point.

5. Relative motion

Relative motion compares the motion of one body to another moving reference frame.

Translation of reference frames

For particles \(A\) and \(B\):

\mathbf{r}_B = \mathbf{r}_A + \mathbf{r}_{B/A}

Differentiating:

\mathbf{v}_B = \mathbf{v}_A + \mathbf{v}_{B/A}

\mathbf{a}_B = \mathbf{a}_A + \mathbf{a}_{B/A}

Common use cases

A person walking on a moving walkway
A boat moving in a current
A slider in a rotating slot
Two connected bodies with one body providing the frame for the other

Rotating frames

When the reference frame rotates, additional terms appear:

\mathbf{v}_B = \mathbf{v}_A + \mathbf{v}_{rel} + \boldsymbol{\omega} \times \mathbf{r}_{B/A}

\mathbf{a}_B = \mathbf{a}_A + \mathbf{a}_{rel} + 2\boldsymbol{\omega} \times \mathbf{v}_{rel} + \boldsymbol{\alpha} \times \mathbf{r}_{B/A} + \boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf{r}_{B/A})

The \(2\boldsymbol{\omega} \times \mathbf{v}_{rel}\) term is the Coriolis term.

6. Particle kinetics

Particle kinetics links forces to translational acceleration.

Newton's second law

For a particle:

\sum \mathbf{F} = m\mathbf{a}

Write the force balance in component form:

\sum F_x = ma_x,\quad \sum F_y = ma_y,\quad \sum F_z = ma_z

Free-body diagrams

A free-body diagram is the main tool for particle kinetics.

Include:

Weight \(W = mg\)
Normal reactions
Friction forces
Cable tensions
Applied forces

Friction

Static friction satisfies

f_s \le \mu_s N

Kinetic friction is usually modeled as

f_k = \mu_k N

The friction force opposes impending or relative motion along the contact surface.

Typical workflow

Draw the particle and isolate it.
Choose axes aligned with the motion if possible.
Write \(\sum \mathbf{F} = m\mathbf{a}\).
Add kinematic relations if needed.
Solve algebraically before substituting numbers.

7. Work-energy methods

Work-energy is often faster than force-balance when displacement is easier to track than time.

Work

The work of a force is

W = \int \mathbf{F}\cdot d\mathbf{r}

For a constant force in the direction of motion:

$$ W = Fs $$

Kinetic energy

For a particle:

T = \frac{1}{2}mv^2

Work-energy theorem:

T_1 + \sum W_{1\to2} = T_2

Conservative forces

Gravity and springs are conservative.

Potential energy forms:

$$ V_g = mgy $$

V_s = \frac{1}{2}kx^2

Mechanical energy form:

T_1 + V_1 + W_{nc} = T_2 + V_2

where \(W_{nc}\) is work by nonconservative forces such as friction.

Example pattern

For a block released on a rough incline, the work-energy method often gives speed after sliding a known distance without needing time explicitly.

8. Impulse-momentum methods

Impulse-momentum is useful when forces act over a short time, such as impacts or thrust events.

Linear impulse-momentum

For a particle:

\int_{t_1}^{t_2} \sum \mathbf{F}\,dt = m\mathbf{v}_2 - m\mathbf{v}_1

Component form is often easiest.

Angular impulse-momentum

About a point \(O\):

\int_{t_1}^{t_2} \sum M_O\,dt = H_{O2} - H_{O1}

When to use it

Impact problems
Short duration force histories
Average force from collision data
Problems where velocity change is known and time is short

Caution

Impulse-momentum is not ideal if the force is unknown and the time interval is long unless the impulse can be integrated or approximated.

9. Rigid-body kinematics

Rigid-body motion combines translation and rotation.

Plane motion

For planar motion of a rigid body:

\mathbf{v}_B = \mathbf{v}_A + \boldsymbol{\omega} \times \mathbf{r}_{B/A}

\mathbf{a}_B = \mathbf{a}_A + \boldsymbol{\alpha} \times \mathbf{r}_{B/A} + \boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf{r}_{B/A})

Pure translation

If every line in the body remains parallel to itself:

All points have the same velocity
All points have the same acceleration

Pure rotation about a fixed axis

For a point at radius \(r\):

v = \omega r

a_t = \alpha r

a_n = \omega^2 r

Direction matters:

\(a_t\) is tangent to the path
\(a_n\) points toward the center of rotation

Instantaneous center of zero velocity

In planar motion, a body can often be treated as instantaneously rotating about a point with zero velocity. This is a useful geometric shortcut for velocity analysis, but it does not replace full acceleration analysis.

10. Rigid-body kinetics

Rigid-body kinetics extends Newton's laws to bodies with rotation.

Translation of the mass center

For a rigid body:

\sum \mathbf{F} = m\mathbf{a}_G

where \(G\) is the center of mass.

Rotation about the mass center

For planar motion:

\sum M_G = I_G \alpha

where \(I_G\) is the mass moment of inertia about the center of mass.

Rotation about another point

About a fixed point \(O\):

\sum M_O = I_O \alpha

if the axis is fixed in an inertial frame.

Parallel-axis theorem:

$$ I_O = I_G + md^2 $$

where \(d\) is the distance between axes.

Common inertia facts

Thin rod about center: \(I = \frac{1}{12}mL^2\)
Thin rod about end: \(I = \frac{1}{3}mL^2\)
Solid disk about center: \(I = \frac{1}{2}mr^2\)
Solid sphere about center: \(I = \frac{2}{5}mr^2\)

Energy for rigid bodies

Total kinetic energy:

T = \frac{1}{2}mv_G^2 + \frac{1}{2}I_G\omega^2

This form is useful for rolling, pulleys, and rotating links.

11. Choosing the right method

The best method depends on what the problem gives and what it asks for.

Use kinematics when

You need position, velocity, or acceleration only
No forces are required
The motion is prescribed

Use Newton-Euler when

Forces or moments are explicitly requested
Unknown reactions matter
Acceleration is known or can be found

Use work-energy when

Displacement is known
Speed is needed rather than time
Conservative forces dominate

Use impulse-momentum when

The time interval is short
Impacts or average forces are involved
Velocity change is easier to track than acceleration

Practical decision rule

If the problem asks for a force, start with free-body diagrams and Newton's laws. If it asks for a speed after moving through a distance, work-energy is often shorter. If it asks about collision or impact, impulse-momentum is usually best.

12. Common pitfalls

Mixing up kinematics and kinetics
Using constant-acceleration formulas when acceleration is not constant
Forgetting to define a positive sign convention
Leaving out the normal or tangential acceleration component
Treating the center of mass as the same thing as a geometric center without checking symmetry
Using \(I_G\) when the problem needs \(I_O\), or vice versa
Forgetting that friction direction is opposite impending or relative motion
Applying rigid-body formulas to deformable systems
Skipping the free-body diagram and trying to solve from memory

Sanity checks

Units must match at every step
Acceleration direction should agree with the chosen coordinates
A force balance should reduce correctly in special cases
Energy methods should conserve mechanical energy only when the model allows it

13. Formula sheet

Particle motion

\mathbf{v} = \frac{d\mathbf{r}}{dt},\quad \mathbf{a} = \frac{d\mathbf{v}}{dt}

\sum \mathbf{F} = m\mathbf{a}

T = \frac{1}{2}mv^2

\int \sum \mathbf{F}\,dt = m\mathbf{v}_2 - m\mathbf{v}_1

Curvilinear motion

a_t = \frac{dv}{dt}

a_n = \frac{v^2}{\rho}

\mathbf{a} = \left(\ddot{r} - r\dot{\theta}^2\right)\mathbf{e}_r + \left(r\ddot{\theta} + 2\dot{r}\dot{\theta}\right)\mathbf{e}_\theta

Rigid-body motion

\mathbf{v}_B = \mathbf{v}_A + \boldsymbol{\omega} \times \mathbf{r}_{B/A}

\mathbf{a}_B = \mathbf{a}_A + \boldsymbol{\alpha} \times \mathbf{r}_{B/A} + \boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf{r}_{B/A})

\sum M_G = I_G \alpha

T = \frac{1}{2}mv_G^2 + \frac{1}{2}I_G\omega^2

Energy

T_1 + \sum W_{1\to2} = T_2

T_1 + V_1 + W_{nc} = T_2 + V_2

Friction

f_s \le \mu_s N,\quad f_k = \mu_k N

Problem-solving checklist

Draw the system and the free-body diagram.
Identify whether the body is a particle or rigid body.
Choose coordinates that simplify the equations.
Decide whether force, energy, or impulse methods are best.
Write the governing equations before plugging in numbers.
Check units and limiting cases.

Sources

Engineering LibreTexts
Hibbeler, Engineering Mechanics
Nilsson and Riedel, Electric Circuits
Sedra and Smith, Microelectronic Circuits
Oppenheim and Willsky, Signals and Systems
Nise, Control Systems Engineering
Incropera et al., Fundamentals of Heat and Mass Transfer
Fox, McDonald, and Pritchard, Introduction to Fluid Mechanics
Groover, Fundamentals of Modern Manufacturing
Callister and Rethwisch, Materials Science and Engineering
Montgomery, Introduction to Statistical Quality Control
Kerzner, Project Management: A Systems Approach to Planning, Scheduling, and Controlling
Law, Simulation Modeling and Analysis
Fraden, Handbook of Modern Sensors
Leake and Borger, Engineering Design Graphics
Parell GitHub repository