Theoretical Mechanics PDF

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©ZamirMohyedin Theoretical Mechanics Muhammad Zamir Mohyedin Universiti Teknologi MARA Ilustrasi oleh: Syafy Zahin 1 ©ZamirMohyedin 1. Vectors 1.1 Fundamental Dimension 1.2 Vectors 1.3 Scalar Product 1.4 Vector Product 1.5 Transformation of Coordinate System 1.6 Vector Derivative 1.7 Velocity and Ac...

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Fowles ClassicalMechanics Izuma Sarut obi NAT IONAL OPEN UNIVERSIT Y OF NIGERIA SCHOOL OF SCIENCE AND T ECHNOLOGY COURSE CODE: PHY… Fet t y Shamy Lin Yahaya UNIT -I Review of t he t hree laws of mot ion and vect or algebra VJ kum R

©ZamirMohyedin

Theoretical Mechanics

Muhammad Zamir Mohyedin Universiti Teknologi MARA

Ilustrasi oleh: Syafy Zahin

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©ZamirMohyedin

1. Vectors 1.1 Fundamental Dimension 1.2 Vectors 1.3 Scalar Product 1.4 Vector Product 1.5 Transformation of Coordinate System 1.6 Vector Derivative 1.7 Velocity and Acceleration in Cartesian Coordinates 1.8 Velocity and Acceleration in Plane Polar Coordinates 1.9 Velocity and Acceleration in Cylindrical Coordinates 1.10 Velocity and Acceleration in Spherical Coordinates 1.11 Velocity and Acceleration in Plane Polar Coordinates 1.12 Velocity and Acceleration in Cylindrical and Spherical Coordinates

2. Newtonian Mechanics 2.1 Newton’s Law of Motion 2.2 Straight-Line Motion 2.3 The Concepts of Kinetic and Potential Energy 2.4 Fluid Resistance and Terminal Velocity

3. Oscillations 3.1 Introduction 3.2 Harmonic Motion 3.3 Rotating Vector Projection 3.4 Damped Harmonic Motion 3.5 Resonance

4. Non-Inertial Reference Systems 4.1 Accelerated Coordinate Systems 4.2 Rotating Coordinate Systems 4.3 Dynamics of a Particle in a Rotating Coordinate System

5. Gravitation and Central Forces 5.1 Introduction 2

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5.2 Kepler’s Laws of Planetary Motion 5.3 The Law of Ellipse 5.4 The Law of Equal Areas 5.5 The Law of Harmonic

6. Dynamics of Particles Systems 6.1 Relationship between the Centre of Mass and Linear Momentum of the System 6.2 Angular Momentum of a System 6.3 Kinetic Energy of a System 6.4 Two Interacting Particles 6.5 Collisions

7. Mechanics of Rigid Bodies 7.1 Centre of Mass of Rigid Body 7.2 Moment of Inertia 7.3 Calculation of the Moment of Inertia 7.4 The Physical Pendulum

8. Motion of Rigid Bodies in Three Dimensions 8.1 Rotation of a Rigid Body about an Arbitrary Axis 8.2 Principal Axes of a Rigid Body 8.3 Euler’s Equations of Motion of a Rigid Body

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Preface Material of this book is based on the Analytical Mechanics by Fowles and Cassiday with only certain topic based on the Mara University of Technology syllabus. The mathematics knowledge is based on my notes during previous semesters when I took Calculus and Differential Equation, with helping on textbooks of Differential Equations with Boundary-Value Problems by Dennis G. Zill and Warren S. Wright, and Calculus by James Stewart. Due to the lack of mathematical approach on Analytical Mechanics textbook, I presented more on derivation and mathematical approach on this book. The structure of the content is a bit same as the textbook mention. I adjusted a bit on structure in accordance with the student flow of understanding. Explanation on theory in this book is simple and short compare to the textbooks, and I mention only necessary. If you want further explanation on theory, better you read on the textbooks. This book is more on mathematical explanation. On matter of reference, maybe some people thought this book require many references since this book is made for reference. My goal for this book is not necessarily to made it as primary reference for student, it just secondary reference or handbook for student on helping them in mathematical approach in classical mechanics. Since the topics are based on the university syllabus and the Analytical Mechanics as the primary reference, so, it was not so difficult for me to construct the content. I hope this book could help the student to understand nicely on classical mechanics subject. Good luck!

Muhammad Zamir Mohyedin

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1.

Vectors

1.1 Fundamental Dimension Three-dimensional space is Euclidian, and positions of points in that space are specified by a set of three numbers (x, y, z) relative to the origin (0, 0, 0) of a Cartesian coordinate system. A length is the spatial separation of two points relative to some standard length. The Unit of Length (m)

The meter is the distance light travels in 1/299,792,458 s in a vacuum.

The Unit of Time (s)

The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the cesium-133 atom.

The Unit of Mass (kg)

The kilogram, its primary standard is stored in a vault in Sevres, Frances.

Dimensional Analysis Describe relationships between different physical quantities that can be used to determine the truth of a result of equation.

Example:

The dimension is equally same.

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1.2 Vectors The motion of dynamical systems is described in terms of scalars and vectors. i.

Scalar has magnitude only.

ii.

Vector has both magnitude and direction.

Example of scalar quantity

Example of vector quantity

-

Length

-

Acceleration

-

Mass

-

Displacement

-

Area

-

Velocity

-

Volume

-

Force

-

Density

-

Momentum

A vector is identifying as the set of components. Vector P has set of components,

as its

components. Thus, the equation,

The operation of vector is as follows, Equality

Addition

Subtraction

Scalar Multiply

Null

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The law of vector is as follows, Addition Commutative Law

Distributive Law

Associative Law

Magnitude

A unit vector is a unity magnitude of vector. Unit vector is identified by the parameter e.

Equivalently,

Notation i, j, and k is used in Cartesian unit vectors. The equation is as follows,

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1.3 Scalar Product For vectors P and Q, the scalar product also known as ‘dot’ product can be expressed as follows,

Could be described the relationship as follows,

.

Then, detailed it,

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Projection Vector

Figure (1.3.1)

The equation for projection vector is based on Figure (1.3.1). The derivation of equation is as follows,

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Cosines Law

Figure (1.3.2)

Cosines Law equation could be obtain from the Figure (1.3.2). The derivation is as follows,

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1.4 Vector Product Vector product also known as cross product is defined as follows,

Then, to prove

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Then, to prove

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Based on the definitions of the unit coordinate vectors,

The cross product expressed in ijk form is,

The magnitude of cross product,

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ϴ = angle between P and Q

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1.5 Transformation of Coordinate System A vector P in term of ijk,

Then, in term of

,

Then, it can be expressed as follows,

The regular components can be expressed as follows,

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1.6 Vector Derivative A vector can represent any physical quantity such as position, velocity, acceleration, displacement, etc.

The parameter ‘u’ can be representing any quantity that determines the component .

The derivative,

of a function of a real variable measures the sensitivity to change of a quantity

This means, we approaches a smaller time intervals in

to a limit

.

.

The sum of two vectors derivative,

Rules of differentiating vector products which is Product Rule,

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1.7 Velocity and Acceleration in Cartesian Coordinates Cartesian Coordinates Cartesian Coordinates bore a name from Natural Philosopher and Mathematician Rene Descartes. He was the one create the idea of coordinates system. His idea was developed in 1637 and revolutionized mathematics by providing systematic relationship between algebra and Euclidean geometry. Thus, this idea is a fundamental on establishment of calculus by Isaac Newton and Gottfried Wilhelm Leibniz.

The position vector of Cartesian Coordinates is expressed as follows,

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As mention earlier in topic, the derivative of a function of a real variable measures the sensitivity to change of a quantity. The dots indicate differentiation with respect to t. In this case, these dots also refer to velocity.

Speed is the magnitude of velocity. In Cartesian components, the speed , is,

The change in velocity in a function of time, or in other word, the second derivative of a position and differentiation of

would give acceleration,

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1.8 Velocity and Acceleration in Plane Polar Coordinates Polar Coordinates Polar Coordinates are based on the concept of radius and angle. It first discovered millennium years ago by Greek astronomer, Hipparchus. However, it does not extend to a full coordinate system. Since 8th century AD onward, astronomers developed methods to calculate and approximating the direction of Mecca for qibla. Eventually, they found polar coordinate for the method. Then, introduction of polar coordinates is wide among the mathematician.

The position vector of Polar Coordinates is expressed as follows,

The velocity vector,

The acceleration vector,

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Radial component Transverse component

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1.9 Velocity and Acceleration in Cylindrical Coordinates The position vector of Cylindrical Coordinates is expressed as follows,

Unit radial vector in the

plane

Unit vector in z-direction

The velocity vector of particle,

The acceleration of particle,

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1.10 Velocity and Acceleration in Spherical Coordinates The position vector is written as the product of the radial distance

and the unit radial vector

, as

with plane polar coordinates. Thus,

Velocity vector of the particle,

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Acceleration of the particle,

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Problems

1.1.Given the two vectors (a)

and

, find the following:

and

(b) (c) (d)

and

1.2. Given the three vectors (a)

and

(b)

and

(c)

and

, find the following:

and

1.3. Given the time-varying vector

Where

and

and

.

are constants, find the first and second time derivatives

1.4. Prove the vector identity

1.5. Show that

1.6. Express the vector rotated about the

in the primed triad through an angle

in which the

are

.

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1.7. The ball moves in an elliptical path given by the equation,

where

and

at

and

are constants. Find the speed of the ball as a function of . In particular, find , at which times the ball is, respectively, at its minimum and

maximum distances from the origin.

1.8. Helical path given by the equation

Show that the magnitude of the acceleration is constant, provided

and

are constant.

1.9. Given the position of particle,

Find the velocity and the acceleration of the particle

1.10. Find the velocity and acceleration of the particle in plane polar coordinates. Given the position of the particle,

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1.11. Find the velocity and acceleration of a particle in cylindrical coordinates. Given the position of the particle,

Given also the relationship between fixed unit triad and the rotated triad

1.12. Prove the relationship between fixed unit triad and rotated triad of cylindrical coordinates,

Sketch a diagram to prove your answer.

1.13. Find the velocity and acceleration of a particle in spherical coordinates. Given the position of the particle,

Given also the relationship between fixed unit triad and rotated triad of spherical coordinates,

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2. Newtonian Mechanics 2.1 Newton’s Laws of Motion i.

Every object continues in its state of rest or in constant motion, until there is a force impressed upon it.

ii.

The vector sum of forces is directly proportional to the product of mass and acceleration, follow the formula of F = ma.

iii.

When there is force upon an object, there is an equal opposite force.

Newton’s First Law. Known as inertia. Inertia is the resistance of any physical object to undergoing its state changed. If an object remained at rest state, it resists being moved, which means, there is a force needed to move it. If an object is in motion, it resists being lead to rest, which means, there is a force needed to make it rest. Frame of reference Framework of mathematical description which is coordinates in configuration time-space that is used to identify the position, velocity, and acceleration of an object at any instantaneous time.

An inertial reference frame is reference of Newton’s First law of motion. This law present an ideal concept of inertial accelerated frames of reference, because an object at rest or in motion at constant velocity ‘forever’.

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Newton’s Second and Third Laws Mass is the inertia quantitative measure. The mass of object is directly proportional to the resistive of it to the acceleration.

The definition of the ratio of the two masses, Equivalent to As we learn in chapter 1, the derivative,

of a function of a real variable measures the sensitivity to

change of a quantity, . So,

This equation give Newton’s Second Law, which F = ma. Then, from Newton’s Second Law, would give you Newton’s Third Law,

Newton’s Third Law states that, when a body exert a force upon another body, the opposite force is equal to the exerting force.

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Linear Momentum

From Newton’s Second Law,

From Newton’s Third Law,

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2.2 Straight-Line Motion Straight-line motion Also know as rectilinear motion. This type of motion explains the movement of a particle or an object in a straight line. An object experience rectilinear motion if any of two particles of the object move at the same distance along two parallel straight lines.

The straight-line motion equation is as follows,

Distance Velocity Time

To find the value of C, lets

.

, because at initial time, t which is t = 0, the

velocity should be at initial also, which in notation of

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To find the value of C, lets

because at initial time, t which is t = 0, the

position should be at initial also, which in notation of

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Then, from equation 1 and equation 2,

And

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2.3 The Concepts of Kinetic and Potential Energy Kinetic Energy Energy possesses by the object during a motion Potential Energy Energy possesses by the object at rest

Equation of straight-line motion,

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Then, from an equation (2.3.3),

Then, from an equation (2.3.6),

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The velocity of the particle, from an equation (2.3.7),

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Free Fall Free Fall Downward movement of an object due to the force of gravity only.

Consider an object is throwing positive upward direction; opposite to the gravitational force. The derivation is as follows,

The energy equation,

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Gravity variation due to the altitude (height) The gravitational force is varying with varying height. Different height would have different force of gravity. This is because Earth is not really sphere in shape. The sphere is not perfect. So, object at the pole and equator would have different distance from the centre of the Earth.

Newton’s constant gravitation Mass of the Earth Distance from the centre of the Earth to the body.

By definition,

Radius of the Earth

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Derivation of relationship between the energy and variation of gravity

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Escape Velocity Escape Velocity Minimum velocity needed to break free from the attraction of gravity of any matter.

The derivation of escape velocity is as follows,

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2.4 Fluid Resistance and Terminal Velocity Fluid Resistance In fluid dynamics, fluid is a type of friction which refers to opposite force relative to the motion of any object travel with respect to a surrounding fluid.

Terminal velocity Highest velocity attainable by an object as it falls through air. It occurs once the sum of the drag force and buoyancy equals the downward force of gravity acting on the object. Since the net force on the object is zero, the object has zero acceleration - Wikipedia

Force in term of function v can be described in two forms,

Any independent constant force

Approximate measurement for fluid resistance is as follows,

Constant values of shape and size of the object The value of the constant that depend on shape of sphere on air is as follows,

Diameter

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Horizontal Motion with Linear Resistance Lets an object is launch with initial velocity . Then, there is fluid resistance which is air which the linear term dominates,

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Horizontal Motion with Quadratic Resistance If the parameters are the quadra...