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Dynamics

- Astrophysics
- Absolute Magnitude
- Astronomical Objects
- Astronomical Telescopes
- Black Body Radiation
- Classification by Luminosity
- Classification of Stars
- Cosmology
- Doppler Effect
- Exoplanet Detection
- Hertzsprung-Russell Diagrams
- Hubble's Law
- Large Diameter Telescopes
- Quasars
- Radio Telescopes
- Reflecting Telescopes
- Stellar Spectral Classes
- Telescopes
- Atoms and Radioactivity
- Fission and Fusion
- Medical Tracers
- Nuclear Reactors
- Radiotherapy
- Random Nature of Radioactive Decay
- Thickness Monitoring
- Circular Motion and Gravitation
- Applications of Circular Motion
- Centripetal and Centrifugal Force
- Circular Motion and Free-Body Diagrams
- Fundamental Forces
- Gravitational and Electric Forces
- Gravity on Different Planets
- Inertial and Gravitational Mass
- Vector Fields
- Conservation of Energy and Momentum
- Dynamics
- Application of Newton's Second Law
- Buoyancy
- Drag Force
- Dynamic Systems
- Free Body Diagrams
- Friction Force
- Normal Force
- Springs Physics
- Superposition of Forces
- Tension
- Electric Charge Field and Potential
- Charge Distribution
- Charged Particle in Uniform Electric Field
- Conservation of Charge
- Electric Field Between Two Parallel Plates
- Electric Field Lines
- Electric Field of Multiple Point Charges
- Electric Force
- Electric Potential Due to Dipole
- Electric Potential due to a Point Charge
- Electrical Systems
- Equipotential Lines
- Electricity
- Ammeter
- Attraction and Repulsion
- Basics of Electricity
- Batteries
- Capacitors in Series and Parallel
- Circuit Schematic
- Circuit Symbols
- Circuits
- Current Density
- Current-Voltage Characteristics
- DC Circuit
- Electric Current
- Electric Motor
- Electrical Power
- Electricity Generation
- Emf and Internal Resistance
- Kirchhoff's Junction Rule
- Kirchhoff's Loop Rule
- National Grid Physics
- Ohm's Law
- Potential Difference
- Power Rating
- RC Circuit
- Resistance
- Resistance and Resistivity
- Resistivity
- Resistors in Series and Parallel
- Series and Parallel Circuits
- Simple Circuit
- Static Electricity
- Superconductivity
- Time Constant of RC Circuit
- Transformer
- Voltage Divider
- Voltmeter
- Electricity and Magnetism
- Benjamin Franklin's Kite Experiment
- Changing Magnetic Field
- Circuit Analysis
- Diamagnetic Levitation
- Electric Dipole
- Electric Field Energy
- Magnets
- Oersted's Experiment
- Voltage
- Electromagnetism
- Electrostatics
- Energy Physics
- Big Energy Issues
- Conservative and Non Conservative Forces
- Elastic Potential Energy
- Electrical Energy
- Energy and the Environment
- Forms of Energy
- Geothermal Energy
- Gravitational Potential Energy
- Heat Engines
- Heat Transfer Efficiency
- Kinetic Energy
- Mechanical Power
- Potential Energy
- Potential Energy and Energy Conservation
- Pulling Force
- Renewable Energy Sources
- Wind Energy
- Work Energy Principle
- Engineering Physics
- Angular Momentum
- Angular Work and Power
- Engine Cycles
- First Law of Thermodynamics
- Moment of Inertia
- Non-Flow Processes
- PV Diagrams
- Reversed Heat Engines
- Rotational Kinetic Energy
- Second Law and Engines
- Thermodynamics and Engines
- Torque and Angular Acceleration
- Fields in Physics
- Alternating Currents
- Capacitance
- Capacitor Charge
- Capacitor Discharge
- Coulomb's Law
- Electric Field Strength
- Electric Fields
- Electric Potential
- Electromagnetic Induction
- Energy Stored by a Capacitor
- Escape Velocity
- Gravitational Field Strength
- Gravitational Fields
- Gravitational Potential
- Magnetic Fields
- Magnetic Flux Density
- Magnetic Flux and Magnetic Flux Linkage
- Moving Charges in a Magnetic Field
- Newton’s Laws
- Operation of a Transformer
- Parallel Plate Capacitor
- Planetary Orbits
- Synchronous Orbits
- Fluids
- Absolute Pressure and Gauge Pressure
- Application of Bernoulli's Equation
- Archimedes' Principle
- Conservation of Energy in Fluids
- Fluid Flow
- Fluid Systems
- Force and Pressure
- Force
- Air resistance and friction
- Conservation of Momentum
- Contact Forces
- Elastic Forces
- Force and Motion
- Gravity
- Impact Forces
- Moment Physics
- Moments Levers and Gears
- Moments and Equilibrium
- Pressure
- Resultant Force
- Safety First
- Time Speed and Distance
- Velocity and Acceleration
- Work Done
- Fundamentals of Physics
- Further Mechanics and Thermal Physics
- Bottle Rocket
- Charles law
- Circular Motion
- Diesel Cycle
- Gas Laws
- Heat Transfer
- Heat Transfer Experiments
- Ideal Gas Model
- Ideal Gases
- Kinetic Theory of Gases
- Models of Gas Behaviour
- Newton's Law of Cooling
- Periodic Motion
- Rankine Cycle
- Resonance
- Simple Harmonic Motion
- Simple Harmonic Motion Energy
- Temperature
- Thermal Equilibrium
- Thermal Physics
- Volume
- Work in Thermodynamics
- Geometrical and Physical Optics
- Kinematics Physics
- Air Resistance
- Angular Kinematic Equations
- Average Velocity and Acceleration
- Displacement, Time and Average Velocity
- Frame of Reference
- Free Falling Object
- Kinematic Equations
- Motion in One Dimension
- Motion in Two Dimensions
- Rotational Motion
- Uniformly Accelerated Motion
- Linear Momentum
- Magnetism
- Ampere force
- Earth's Magnetic Field
- Fleming's Left Hand Rule
- Induced Potential
- Magnetic Forces and Fields
- Motor Effect
- Particles in Magnetic Fields
- Permanent and Induced Magnetism
- Magnetism and Electromagnetic Induction
- Faraday's Law
- Induced Currents
- LC Circuit
- Lenz's Law
- Magnetic Field of a Current-Carrying Wire
- Magnetic Flux
- Magnetic Materials
- Monopole vs Dipole
- RL Circuit
- Measurements
- Mechanics and Materials
- Acceleration Due to Gravity
- Bouncing Ball Example
- Bulk Properties of Solids
- Centre of Mass
- Collisions and Momentum Conservation
- Conservation of Energy
- Density
- Elastic Collisions
- Force Energy
- Friction
- Graphs of Motion
- Linear Motion
- Materials
- Materials Energy
- Moments
- Momentum
- Power and Efficiency
- Projectile Motion
- Scalar and Vector
- Terminal Velocity
- Vector Problems
- Work and Energy
- Young's Modulus
- Medical Physics
- Absorption of X-Rays
- CT Scanners
- Defects of Vision
- Defects of Vision and Their Correction
- Diagnostic X-Rays
- Effective Half Life
- Electrocardiography
- Fibre Optics and Endoscopy
- Gamma Camera
- Hearing Defects
- High Energy X-Rays
- Lenses
- Magnetic Resonance Imaging
- Noise Sensitivity
- Non Ionising Imaging
- Physics of Vision
- Physics of the Ear
- Physics of the Eye
- Radioactive Implants
- Radionuclide Imaging Techniques
- Radionuclide Imaging and Therapy
- Structure of the Ear
- Ultrasound Imaging
- X-Ray Image Processing
- X-Ray Imaging
- Modern Physics
- Bohr Model of the Atom
- Disintegration Energy
- Franck Hertz Experiment
- Mass Energy Equivalence
- Nucleus Structure
- Quantization of Energy
- Spectral Lines
- The Discovery of the Atom
- Wave Function
- Nuclear Physics
- Alpha Beta and Gamma Radiation
- Binding Energy
- Half Life
- Induced Fission
- Mass and Energy
- Nuclear Instability
- Nuclear Radius
- Radioactive Decay
- Radioactivity
- Rutherford Scattering
- Safety of Nuclear Reactors
- Oscillations
- Energy Time Graph
- Energy in Simple Harmonic Motion
- Kinetic Energy in Simple Harmonic Motion
- Mechanical Energy in Simple Harmonic Motion
- Pendulum
- Period of Pendulum
- Period, Frequency and Amplitude
- Phase Angle
- Physical Pendulum
- Restoring Force
- Simple Pendulum
- Spring-Block Oscillator
- Torsional Pendulum
- Velocity
- Particle Model of Matter
- Physical Quantities and Units
- Converting Units
- Physical Quantities
- SI Prefixes
- Standard Form Physics
- Units Physics
- Use of SI Units
- Physics of Motion
- Acceleration
- Angular Acceleration
- Angular Displacement
- Angular Velocity
- Centrifugal Force
- Centripetal Force
- Displacement
- Equilibrium
- Forces of Nature Physics
- Galileo's Leaning Tower of Pisa Experiment
- Inclined Plane
- Inertia
- Mass in Physics
- Speed Physics
- Static Equilibrium
- Radiation
- Antiparticles
- Antiquark
- Atomic Model
- Classification of Particles
- Collisions of Electrons with Atoms
- Conservation Laws
- Electromagnetic Radiation and Quantum Phenomena
- Isotopes
- Neutron Number
- Particles
- Photons
- Protons
- Quark Physics
- Specific Charge
- The Photoelectric Effect
- Wave-Particle Duality
- Rotational Dynamics
- Angular Impulse
- Angular Kinematics
- Angular Motion and Linear Motion
- Connecting Linear and Rotational Motion
- Orbital Trajectory
- Rotational Equilibrium
- Rotational Inertia
- Satellite Orbits
- Third Law of Kepler
- Scientific Method Physics
- Data Collection
- Data Representation
- Drawing Conclusions
- Equations in Physics
- Uncertainties and Evaluations
- Space Physics
- Thermodynamics
- Heat Radiation
- Thermal Conductivity
- Thermal Efficiency
- Thermodynamic Diagram
- Thermodynamic Force
- Thermodynamic and Kinetic Control
- Torque and Rotational Motion
- Centripetal Acceleration and Centripetal Force
- Conservation of Angular Momentum
- Force and Torque
- Muscle Torque
- Newton's Second Law in Angular Form
- Simple Machines
- Unbalanced Torque
- Translational Dynamics
- Centripetal Force and Velocity
- Critical Speed
- Free Fall and Terminal Velocity
- Gravitational Acceleration
- Gravitational Force
- Kinetic Friction
- Object in Equilibrium
- Orbital Period
- Resistive Force
- Spring Force
- Static Friction
- Turning Points in Physics
- Cathode Rays
- Discovery of the Electron
- Einstein's Theory of Special Relativity
- Electromagnetic Waves
- Electron Microscopes
- Electron Specific Charge
- Length Contraction
- Michelson-Morley Experiment
- Millikan's Experiment
- Newton's and Huygens' Theories of Light
- Photoelectricity
- Relativistic Mass and Energy
- Special Relativity
- Thermionic Electron Emission
- Time Dilation
- Wave Particle Duality of Light
- Waves Physics
- Acoustics
- Applications of Ultrasound
- Applications of Waves
- Diffraction
- Diffraction Gratings
- Doppler Effect in Light
- Earthquake Shock Waves
- Echolocation
- Image Formation by Lenses
- Interference
- Light
- Longitudinal Wave
- Longitudinal and Transverse Waves
- Mirror
- Oscilloscope
- Phase Difference
- Polarisation
- Progressive Waves
- Properties of Waves
- Ray Diagrams
- Ray Tracing Mirrors
- Reflection
- Refraction
- Refraction at a Plane Surface
- Resonance in Sound Waves
- Seismic Waves
- Snell's law
- Standing Waves
- Stationary Waves
- Total Internal Reflection in Optical Fibre
- Transverse Wave
- Ultrasound
- Wave Characteristics
- Wave Speed
- Waves in Communication
- X-rays
- Work Energy and Power
- Conservative Forces and Potential Energy
- Dissipative Force
- Energy Dissipation
- Energy in Pendulum
- Force and Potential Energy
- Force vs. Position Graph
- Orbiting Objects
- Potential Energy Graphs and Motion
- Spring Potential Energy
- Total Mechanical Energy
- Translational Kinetic Energy
- Work Energy Theorem
- Work and Kinetic Energy

How do the objects we see in everyday life move the way they do? For instance, have you looked up in the sky and wondered how planes and birds fly across effortlessly? Or perhaps considered why a ball thrown from one end of a football pitch to another, follows a curved trajectory. The field of **dynamics **in physics will explore these situations and explain the answers to these questions for us in terms of vectors and mathematical equations. Forces are responsible for every change in motion, and the principles of dynamics help us understand precisely how this happens.

Fig. 1 - An airplane in the sky is kept afloat due to its dynamics and the dynamics of the surrounding atmosphere.

To understand dynamics, we need to understand what a force is and the rules of motion. We'll discuss these below, as well as free-body diagrams and how to use them with Newton's Laws to solve dynamics problems.

Before we dive into Newton's laws of motion and free-body diagrams, let's cover some essential knowledge about dynamics and forces to build a strong foundation.

As we mentioned earlier, dynamics is the study of the motion of objects we see around us. We can specifically define it as the following.

**Dynamics **is the study of the relationship between forces and the motion of bodies.

A **force **is a push or a pull due to an interaction between two or more objects which can cause the interacting objects' motion to change.

An example of a force being exerted on you is when you're being pushed by your friend while sitting on a swing. This pushing force in combination with the tension in the string between the seat and the bar creates a rotational force that allows you to swing backward and forwards. On the other hand, an example of a pulling force could be when you are walking up a snowy hill and pulling your sled behind you. By pulling the string attached to the sled, you are exerting a force on the sled to work against gravity.

An object can't exert a force on itself; a force requires at least two objects (including things like surfaces and fluids) to occur.

There are many different types of forces. The examples above are **applied forces**, which result from someone or something applying an external force to an object. **Contact forces** result from objects touching: such as the friction force, spring force, buoyant force, drag force, and normal force. There are also **long-range forces—**where the interacting objects don't have to touch to exert forces—such as gravitational, electric, and magnetic forces. The image below depicts these forces.

Any time an object starts moving, stops, slows down, speeds up, or changes direction, a force causes the change. These changes are examples of changes in **acceleration. **Therefore, whenever an object changes acceleration, we know that something has applied a force to it.

Forces are vectors, meaning they have magnitude and direction**. **Therefore, they determine how strongly an object is pulled and how it moves. The magnitude and direction of the force are directly related to the magnitude and direction of the resulting change in acceleration. Forces can also act in opposition to each other and effectively cancel each other out, so even if an object isn't moving, forces are still acting on it.

Since force causes acceleration, a force is measured by how much acceleration it creates. The SI unit of force is a newton \(\text N\), where one newton is equivalent to \(1 \, \mathrm{N} = 1 \, \mathrm{\frac{\text{kg}\,\text{m}}{\text s^2}}\). We can visualize this unit by thinking of a \(1\,\mathrm{kg}\) weight. If we push it across a table with an acceleration of \(1 \, \mathrm{\frac{\text{m}}{\text s^2}}\), with no other forces acting on it, we would be applying \(1\,\mathrm{N}\) of force on the weight in the direction of our push.

The term dynamic implies movement. Therefore, dynamic equilibrium refers to something that is moving, but yet still in equilibrium. How does that work? How can something be moving, and yet be in equilibrium? The answer lies in forces and acceleration.

When an object is in **dynamic equilibrium**, that object is not accelerating and has a net force of zero acting on it.

For example, in the absence of any outside forces, kicking a ball would cause it to accelerate. However, once that ball leaves your foot, no net forces will be acting on it. Since the acceleration of an object requires a nonzero net force, that ball will be in dynamic equilibrium; it will keep moving but have a constant velocity: no acceleration.

Another example is objects falling at terminal velocity. At first, when skydivers jump out of a plane, they accelerate rapidly. However, as the drag force of the air builds and builds, their acceleration decreases until the force of gravity equalizes with the drag force of the air acting upward on them. This causes the net forces acting on them to be balanced. Therefore, they stop accelerating, have a constant velocity, and are at dynamic equilibrium.

The friction force, spring force, and gravitational force each have equations we can use to calculate them. However, to solve for any of the other forces relevant to AP Physics 1 (tension and normal forces), we have to use the other forces acting on the object to solve for them. To do this, we use Newton's Laws of Motion and free-body diagrams, which will be discussed in later sections.

We calculate the friction force using the equation

\[ F_{\text{f}} = \mu F_{\text{N}} ,\]

where \( F_{\text{f}}\) is the frictional force measured in newtons \(\mathrm{N}\), \(\mu\) is the coefficient of friction, and \(F_{\text{N}}\) is the normal force measured in units of newtons \(\mathrm{N}\). It is important to note that the coefficient of friction \(\mu\) is dimensionless and is unique to the material of each surface. The normal force between an object and the surface is the component of the object's weight perpendicular to the surface.

Furthermore, we can also define the spring force equation as

\[ F_{\text{s}} = k x ,\]

where \(F_{\text{s}}\) is the spring force measured in newtons \(\mathrm{N}\), \(k\) is the spring constant of the spring measured in units of \(\mathrm{\frac{N}{m}}\), and \(x\) is the displacement of the spring from its resting position measured in meters \(\mathrm{m}\). This law is also referred to as Hooke's law and showcases that the spring force is proportional to the spring displacement. It is important to note that the spring can be both compressed and extended, thus the displacement vector \(x\) can take positive as well as negative values.

Finally, we have the gravitational force, also referred to as the weight force. We can define the gravitational force as

\[ F_{\text{g}} = mg ,\]

where \(F_{\text{g}}\) is the gravitational force measured in newtons \(\mathrm{N}\), \(m\) is the mass of the object measured in kilograms \(\mathrm{kg}\), and \(g\) is the gravitational acceleration. On the surface of Earth, the gravitational acceleration is given by a value of \( g = 9.81 \, \mathrm{\frac{m}{s^2}}\).

It's important to remember that mass is not the same thing as weight. Mass is measured in \(\mathrm{kilograms}\) and doesn't change based on location, whereas weight is a force (measured in \(\mathrm{newtons}\)) equal to mass times gravity, meaning it changes depending on the gravitational field it is in.

Newton's laws of motion explain the relationship between an object's motion and the forces acting on it. Newton's three Laws of motion are the following:

**Newton's First Law of motion—**Objects remain at rest or constant velocity unless acted on by a net external force.**Newton's Second Law**An object's acceleration depends on its amount of mass, and the amount of force applied. This law of motion results in the equation \(\sum\vec{F}=m\vec{a}\).**of motion—****Newton's Third Law**If an object exerts a force on a second object, the second object exerts a force with equal magnitude and opposite direction on the first one.**of motion—**

To represent external forces acting on an object, we draw **free-body diagrams**. Free-body diagrams allow us to visualize forces exerted on an object, which helps us write the equations representing the physical situation. We draw arrows representing forces in the direction of those forces, with lengths typically relating to the strengths of the forces. The image below is an example of a free-body diagram.

The object in the image has four forces acting on it—a normal force \(F_\mathrm{n}\) acting upwards, a gravitational force \(F_g\) acting downwards, a friction force \(F_\mathrm{f}\) acting to the left, and a tension force \(T\) acting to the right.

When drawing free-body diagrams, remember that the gravitational force acts straight down, and the normal force always acts perpendicularly away from the surface.

We can choose which object, or group of objects (called a **system**), to analyze based on which information we want to obtain. If we choose a system to analyze, we can group it into the same free-body diagram and act like the group is one object.

We have now covered the field of dynamics in a general sense, but a more specific study of dynamics is the field of fluid dynamics. In this field, we specifically look at the dynamics of liquids and gases. For instance, fluid dynamics would allow us to explain the lift an airplane experiences due to the flow of air underneath its wings. It also is an important foundation in the study of atmospheric physics and how different winds affect the weather across the globe. Fluid dynamics is a much more advanced field of study than what is covered in AP 1 , and will begin to be introduced in university physics.

Once we draw a free-body diagram, we can use Newton's laws to understand the motion of an object. Newton's Second Law of Motion is particularly useful because of the following equation:

$$\sum\vec{F}=m\vec{a}\mathrm{.}$$

Force \(F\) is measured in \(\text N\), mass \(m\) in \(\text{kg}\), and acceleration \(a\) in \(\frac{\text m}{\text s^2}\). This equation means that the vector sum of the forces (also known as a **net** or **resultant force**) acting on an object equals its mass multiplied by its acceleration.

The force and acceleration are both vectors, as indicated by the arrows above the variables. The direction of the net force determines the direction of the object's acceleration; this means we can only use the equation along individual directions (for example, the sum of the forces in the \(x\)-direction equals the object's mass times the acceleration in the \(x\)-direction alone). We can use the **principle of superposition of forces** to add the forces as vectors or to break a diagonal force into \(x\) and \(y\) components.

The direction of the force is the same as the acceleration, but that does not mean that the direction of the force is the same as the direction of the velocity. So, for example, if an object currently moving to the right is pushed to the left, the resulting acceleration acts to the left, but the object may continue moving to the right at a slower speed.

If a \(25\,\text{kg}\) box is slipping down an inclined plane where \(\theta=30^\circ\) and the coefficient of friction is \(0.20\), what is the acceleration of the box?

First, we want to draw a free-body diagram for the scenario, as shown below:

We included a normal force directed perpendicularly away from the surface, a friction force acting against the slipping motion, and a gravitational force acting directly down. Since most of the forces act on an axis corresponding to the surface, we chose a coordinate system with \(x\) parallel to the surface and y perpendicular to the surface, as shown. Since the gravitational force is acting diagonally in this coordinate system, we want to determine the \(x\) and \(y\) components of the force, shown in red. We will use trigonometry to find these force components (\(F_{gx}=F_g\sin\theta\) and \(F_{gy}=F_g\cos\theta\)).

To find the acceleration of the box, we can write Newton's Second Law equation in the \(x\)-direction:

$$-F_\mathrm{f}+F_{gx}=ma_x\mathrm{.}$$

To find the friction, we use the equation for friction. Since we know the box is slipping, we know the friction is equal to the friction coefficient times the normal force:

$$|F_\mathrm{f}|=\mu|F_\mathrm{n}|\mathrm{.}$$

To know the normal force, we need to look at the forces in the \(y\)-direction. Since the box is not accelerating in the \(y\)-direction, the sum of the forces equals zero

$$F_\mathrm{n}-F_{gy}=0\mathrm{.}$$

Now, rearrange to solve for the normal force and substitute \(mg\cos\theta\) for \(F_{gy}\)

$$F_\mathrm{n}=mg\cos\theta\mathrm{;}$$

substituting this into the friction equation yields

$$F_\mathrm{f}=\mu\,mg\cos\theta\mathrm{.}$$

We can substitute this into our first equation, and substitute \(mg\sin\theta\) for \(F_{gx}\)

$$-(\mu\,mg\cos\theta)+mg\sin\theta=ma$$

and rearrange our equation to solve for acceleration by dividing everything by the mass

$$a=g(\sin\theta-\mu\,\cos\theta)\mathrm{.}$$

Then, we can plug in our given numbers:

$$a=9.8 \frac{\text m}{\text s^2}\,(\sin{30^\circ}-0.2\cos{30^\circ})$$

$$a=3.2 \frac{\text m}{\text s^2\mathrm{.}}$$

The box will slide down the incline at an acceleration of \(3.2 \frac{\text m}{\text s^2}\).

And there you have it! You now know why things move the way they do and how objects exert forces on other objects. You have the proper foundation to become a physics wizard!

- Dynamics is the study of the relationship between force and motion.
- A force is a push or a pull due to an interaction between two or more objects.
- Forces are vectors, meaning they have magnitude and direction.
- Forces cause changes in acceleration. The direction of the net force determines the direction of the acceleration.
- Friction, spring, and gravitational forces have specific equations used to calculate them; tension and normal force don't.
- We use free-body diagrams to visualize forces acting on an object.
- We use free-body diagrams and Newton's Laws of Motion to solve force and acceleration problems.

- Fig. 1 - Airplane, Wikimedia Commons (https://commons.wikimedia.org/wiki/File:Finnair.a320-200.oh-lxf.arp.jpg) Licensed by Public Domain.
- Fig. 2- Hot air balloon, Wikimedia Commons (https://commons.wikimedia.org/wiki/File:Hot_air_balloon_and_moon.jpg) Licensed by CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/)
- Fig. 3 - Free body diagram, StudySmarter Originals.
- Fig. 4 - Dynamics example free-body diagram, StudySmarter Originals.

Dynamics is the study of the relationship between force and motion.

Dynamics can either be referring to linear or rotational motion.

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