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Statics and Dynamics

The concepts of statics and dynamics are basically a categorisation of rigid body mechanics. Dynamics is the branch of mechanics that deals with the analysis of physical bodies in motion, and statics deals with objects at rest or moving with constant velocity. This means that dynamics implies change and statics implies changelessness, where change in both cases is associated with acceleration. …

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Statics and Dynamics

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Area Between Two Curves
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits at Infinity and Asymptotes
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Radius of Convergence
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
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- Second Derivative Test
- Separable Equations
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- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
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- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
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- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
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- Washer Method
- Decision Maths
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- Constructing Cayley Tables
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- The Travelling Salesman Problem
- Using a Dummy
- Utility
- Geometry
- 2 Dimensional Figures
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- Altitude
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- Area of Parallelograms
- Area of Plane Figures
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- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
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- Convexity in Polygons
- Coordinate Systems
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- Distance and Midpoints
- Equation of Circles
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- Figures
- Fundamentals of Geometry
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- Glide Reflections
- HL ASA and AAS
- Identity Map
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- Law of Cosines
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- Linear Measure and Precision
- Median
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- Plane Geometry
- Polygons
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- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
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- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
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- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
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- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Conservation of Mechanical Energy
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Damped harmonic oscillator
- Direct Impact and Newton's Law of Restitution
- Elastic Energy
- Elastic Strings and Springs
- Force as a Vector
- Kinematics
- Newton's First Law
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- Newton's Third Law
- Power
- Problems involving Relative Velocity
- Projectiles
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- Relative Motion
- Resolving Forces
- Rigid Bodies in Equilibrium
- Stability
- Statics and Dynamics
- Tension in Strings
- The Trajectory of a Projectile
- Variable Acceleration
- Vertical Oscillation
- Work Done by a Constant Force
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
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- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
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- Algebra
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- Argand Diagram
- Arithmetic Sequences
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- Bijective Functions
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- Chain Rule
- Circle Theorems
- Circles
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- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
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- Congruence Equations
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Coupled First-order Differential Equations
- Cubic Function Graph
- Data transformations
- De Moivre's Theorem
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- Definite Integrals
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- Determinant of Inverse Matrix
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- Differential Equations
- Differentiation
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- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Eigenvalues and Eigenvectors
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Fermat's Little Theorem
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- First-order Differential Equations
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Group Mathematics
- Growth and Decay
- Growth of Functions
- Harmonic Motion
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trigonometric Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Invariant Points
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- L'Hopital's Rule
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Leibnitz's Theorem
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
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- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Calculations
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modelling with First-order Differential Equations
- Modular Arithmetic
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
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- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
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- Partial Fractions
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- Percentage
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- Perimeter of a Triangle
- Permutations and Combinations
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- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
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- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Reducible Differential Equations
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Products
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Second-order Differential Equations
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
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- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
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- Sum and Difference of Angles Formulas
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- The Quadratic Formula and the Discriminant
- Transformations
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- Confidence Interval for Population Mean
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- Type I Error
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- Types of Data in Statistics
- Variance for Binomial Distribution
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Jetzt kostenlos anmeldenThe concepts of statics and dynamics are basically a categorisation of rigid body mechanics. Dynamics is the branch of mechanics that deals with the analysis of physical bodies in motion, and statics deals with objects at rest or moving with constant velocity. This means that dynamics implies change and statics implies changelessness, where change in both cases is associated with acceleration.

Statics is concerned with the forces that act on bodies at rest under equilibrium conditions. This is expressed in the first part of Newton's first law of motion, where equilibrium conditions are met:

A body will remain at rest (zero displacement).

A body will remain in uniform motion.

Acceleration is always zero in statics, so the right-hand side of the equation of Newton's second law of motion will always amount to zero as well. This means that most statics problems are going to be associated with the analysis of force – on the left-hand side of Newton's second law of motion.${F}_{net}=ma$

Forces possess both magnitude and direction so they are considered vectors. The magnitude of vectors describes the size and strength of the force. When objects interact with each other, force is exerted on them. The force ceases to exist when the interaction stops. Force is what makes it possible for the conditions regarding objects in equilibrium to be possible. It takes force for objects to stay at rest, and it takes force for objects to be in uniform motion.

Now, let's look at resultant force. This is one force that provides the same effect as all the other forces have on the particle. In this section, we are only concerned with ones that affect particles in equilibrium.

Find the magnitude of ${F}_{1}$ and ${F}_{2}$ acting on the particle in equilibrium in the diagram below.

Answer:

Since our particle is in equilibrium

$\Sigma F=0$

We will need to write equations for both x and y components equating them to zero.

By resolving the x component we get,

$\Sigma {F}_{x}=0$$8+4\mathrm{cos}60\xb0-{F}_{2}\mathrm{cos}30\xb0=0$

${F}_{2}=\frac{20}{\sqrt{3}}N$

By resolving the y component we get

$\Sigma {F}_{y}=0$

${F}_{1}+4\mathrm{sin}60\xb0-{F}_{2}\mathrm{sin}30\xb0=0$

${F}_{1}+\frac{4\sqrt{3}}{2}-\frac{{F}_{2}}{2}=0$

${F}_{1}=\frac{{F}_{2}}{2}-2\sqrt{3}$

${F}_{1}=\frac{10}{\sqrt{3}}-2\sqrt[]{3}$

${F}_{1}=\frac{4}{\sqrt{3}}N$

Dynamics in mechanics studies the forces that cause or modify the movement of an object. It deals with the analysis of physical bodies in motion. Therefore, acceleration is a factor in these problems.

Dynamics can be subdivided into Kinematics and Kinetics. Kinematics is an area of study that focuses on the movement of objects, disregarding the forces that cause the movements. It studies motion that relates to displacement, velocity, acceleration, and time. Kinetics on the other hand studies motion that relates to the forces that affect these motions.

Kinematics focuses on the movement of objects, disregarding the forces that cause the movements. It deals with forces and the geometric aspects of motion, which is related to velocity and acceleration.

In kinematics, we can have problems associated with either acceleration being constant or acceleration changing over time (variable acceleration). Kinematic equations associated with constant acceleration are only valid when acceleration is constant, and motion is constrained to a straight line. Variable acceleration problems deal with kinematics where acceleration changes over time.

Differentiation is used to convert displacement to velocity, and velocity to acceleration. Integration is used to cover acceleration to velocity and velocity back to displacement. This makes velocity the first derivative, and acceleration the second derivative with respect to time.

- $s=f\left(t\right)$ [Location in reference to an origin]
- $v=\frac{ds}{dt}=f\text{'}\left(t\right)$ [Derivatives of displacement]
- $a=\frac{dv}{dt}=\frac{{d}^{2}s}{d{t}^{2}}=f\text{'}\text{'}\left(t\right)$ [Derivative of velocity]

The four equations of motion used in solving kinematics problems are:

- $v=u+at$
- $s=\frac{(u+v)}{2}t$
- ${v}^{2}={u}^{2}+2as$
- $s=ut+\frac{1}{2}a{t}^{2}$

Let's look at an example:

If a particle is travelling in a straight line with a constant acceleration of $5m{s}^{-2}$and at t = 0 s the particle has a speed of $3m{s}^{-1}$, what is the speed of the particle when t = 4 s?

Answer:

This is a constant acceleration problem so we can use the equation of motion that involves the variable we are going to be working with.

$u=3m{s}^{-1}$

$v=?$

$a=5m{s}^{-2}$

$t=4s$

From the data, we can see that the equation is best suited for this problem is:

$v=u+at$

We can now substitute in what we know.

$v=3+5\left(4\right)$

$v=23m{s}^{-1}$

Speed is only the scalar quantity for velocity, therefore:$v=23m{s}^{-1}$

A significant concept in kinematics is projectiles. Projectile motion occurs when objects are projected through the air and gravity acts on them. A good example is a ball being thrown. The path of a projectile is called a trajectory. Projectile problems are mostly tackled with trigonometric functions, by resolving the components of the path into x and y components.

- Statics is concerned with the forces that act on bodies at rest under equilibrium conditions.
- Dynamics in mechanics studies the forces that cause or modify the movement of an object.
- Dynamics can be subdivided into Kinematics and Kinetics.
- A resultant force is one force that provides the same effect as all those forces have on the particle.
- With respect to Newton, most statics problems are going to be associated with the analysis of forces.
- Projectile problems are mostly tackled with trigonometric functions by resolving the components of the path into x and y components.

The sum of the forces must be zero for the system to be in equilibrium.

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