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majed majeed lecture notes of chaper 12 first lecture with problems and solutions...

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Chapter 12 Introduction and Rectilinear Motion Section 12.1-2

Dynamics ENG-203

Spring 2020

AN OVERVIEW OF MECHANICS Mechanics: the study of how bodies react to forces acting on them

Statics: the study of bodies in equilibrium

Dynamics: the study of bodies in motion

Newton’s second law of motion forms the basis for most of the analysis in dynamics. If the force F is applied to a particle

• • •

A force (F) acting on an object is the interaction between that object and its environment. Forces have magnitude and direction—we use vectors to mathematically represent forces. The mass (m) of an object is a measure of the amount of matter in it. The acceleration (𝑎) is the time rate of change of momentum of a particle.

Particle: A particle has a mass but negligible size and shape. Rotation is ignored and is considered as motion. The motion of rockets and airplanes can often be analyzed as if they were particles, provided motion of the body is characterized by motion of its mass center and any rotation of the body is neglected. Rigid Body: A rigid body is a body whose mass is (1) distributed over a region of space (2) such that the distance between any two points on it never changes or negligible compared with the overall dimensions of the body. Kinematics: Kinematics concerns geometry of motion with no mention of force Kinetics: Kinetics concerns the relation of force to motion.

SPACE AND TIME: • •

•

Space: the environment in which physical phenomena happen. Often use a right-handed Cartesian coordinate system to locate points in space (point P). 𝒓󰇍𝑷 = 𝑥𝑃 𝑖 + 𝑦𝑃 𝑗 + 𝑧𝑃 𝑘 Time: a scalar variable that allows us to order sequences of events.

VECTORS AND THEIR CARTESIAN REPRESENTATION : • • • •

Vectors are denoted by an arrow over a letter. A symbol with no arrow generally denotes a scalar. Unit vectors are denoted by a “hat” over a letter. In figures, double-headed arrows identify “rotational” quantities, e.g., moments, angular velocities, and angular accelerations.

-1D:\Dropbox\Teaching\ENG203 Dynamics Summer 2019\Dynamics ClassNotes Presentations\Chapter 12\D12.1-2 Introduction

Dr. M. Majeed

Chapter 12 Introduction and Rectilinear Motion Section 12.1-2 • •

Dynamics ENG-203

Spring 2020

The expression 𝑟𝑃/𝑂 is the position of P relative to the origin O measured from point O. When there is no ambiguity, we denote position by 𝑟 and write rx and ry are the (scalar) Cartesian components:



VECTOR O PERATIONS:

( )

𝑟 = 𝑟𝑥 𝑖 + 𝑟𝑦 𝑗 & 𝑤 = 𝑤𝑥 𝑖 + 𝑤𝑦 𝑗 •

Multiplication by scalar:

•

Vector addition:

•

Dot or scalar product: (gives scalar) 𝑟∙ 𝑤 󰇍󰇍 = 𝑟𝑥 𝑤𝑥 + 𝑟𝑦 𝑤𝑦 = |𝑟||𝑤󰇍󰇍 | cos 𝜃

•

Cross Product: (gives vector)

UNITS:

ACCURACY AND APPROXIMATIONS:

-2D:\Dropbox\Teaching\ENG203 Dynamics Summer 2019\Dynamics ClassNotes Presentations\Chapter 12\D12.1-2 Introduction

Dr. M. Majeed

Chapter 12 Introduction and Rectilinear Motion Section 12.1-2

Dynamics ENG-203

Spring 2020

SOME BASIC CONSTANTS: Gravitational Acceleration (g) Universal Gravitational Constant (G) Mass of Earth (me) Radius of Earth (Re)

= 9.81 m/s2 = 32.2 ft/s 2 = 6.673×10 -11 = 5.976×10 24 kg = 6371× 103 m

USEFUL APPROXIMATIONS:

TIME D ERIVATIVE: We use a shorthand notation for time derivatives. Let f (t) be a function of time. Then the time derivative of this function will be denoted by a dot over the function, i.e.,

-3D:\Dropbox\Teaching\ENG203 Dynamics Summer 2019\Dynamics ClassNotes Presentations\Chapter 12\D12.1-2 Introduction

Dr. M. Majeed

Chapter 12 Introduction and Rectilinear Motion Section 12.1-2

Spring 2020

Dynamics ENG-203

12.2 RECTILINEAR KINEMATICS (CONTINUOUS MOTION) Objective: To find the kinematic quantities (position, displacement, velocity, and acceleration) of a particle traveling along a rectilinear or straight-line path. Note that these quantities are vectors. These quantities need a reference to be measured from and a coordinate system (e.g., 𝑥-𝑦-𝑧). Position. The instantaneous position of particle P can be described by 𝑠 = 𝑠(𝑡) or 𝑟 = 𝑟(𝑡). The position of the particle at locations 𝑷 and 𝑷’ are defined as, 𝑟𝑃′ = 𝑠 ′ 𝑖 𝑟𝑃 = 𝑠 𝑖 The displacement of the particle (𝛥𝑠 = 𝑠2 𝑖 − 𝑠1 𝑖) or (Δ𝑠 = 𝑠 ′ 𝑖 − 𝑠 𝑖) is defined as the change in its position measured from a given reference during time interval 𝛥𝑡. Since all motions are in the same direction, the unit vector is usually deleted and the position and its corresponding displacements are written as 𝑟𝑃 = 𝑠 and Δ𝑠 = 𝑠 ′ − 𝑠 . The distance 𝑆𝑇 is defined as the total distance traveled by the particle during time Δ𝑡. Velocity. The average velocity 𝑣𝑎𝑣𝑔 of the particle during the time interval 𝛥𝑡 is defined as follows: ∆𝑠 ∆𝑟 = 𝑣𝑎𝑣𝑔 = Δ𝑡 Δ𝑡 If we take smaller and smaller values of 𝛥𝑡, the magnitude of 𝛥𝑠 becomes smaller and smaller. Consequently, the average velocity becomes instantaneous velocity and is defined as, d𝑟 ∆𝑠 d𝑠 = 𝑟󰇗 𝑣 = lim = = 𝑠󰇗 or Δ𝑡→0 Δ𝑡 d𝑡 d𝑡 Acceleration. The average acceleration of the particle during the time interval 𝛥𝑡 is defined as, ∆𝑣 𝑎𝑎𝑣𝑔 = Δ𝑡 The instantaneous acceleration at time 𝑡 is, ∆𝑣 d𝑣 = 𝑣󰇗 = 𝑠󰇘 = 𝑟󰇘 𝑎 = lim = Δ𝑡→0 Δ𝑡 d𝑡 Using the chain rule,

Note that 𝒔(𝒕), 𝒗(𝒕), and 𝒂(𝒕) are all vectors. Deceleration means that the speed is decreasing and this means that the acceleration

𝑎 is in the opposite direction of 𝑣, not necessarily negative!! In general, the three relations as, 𝑑𝑠 𝑑𝑣 𝑑𝑣 𝑣= 𝑎= 𝑎=𝑣 𝑑𝑡 𝑑𝑡 𝑑𝑠 Three possible solution types: 1. If 𝑎 is function of time (𝑡), then use 𝑣 = ∫ 𝑎 𝑑𝑡 and then 𝑠 = ∫ 𝑣 𝑑𝑡. 2. 3.

If 𝑎 is function of position (𝑠), then use ∫ 𝑣 𝑑𝑣 = ∫ 𝑎 𝑑𝑠 If 𝑎 is function of velocity (𝑣 ), then use 1

∫ 𝑑𝑡 = ∫ 𝑎 𝑑𝑣

or

𝑣

∫ 𝑑𝑠 = ∫ 𝑎 𝑑𝑣

Signe Convention: •

Consider the motion of a particle along a straight line.

•

Denote the position of the particle by s, the velocity by v, and the acceleration by a.

•

The signs of these quantities at an instant are not, in general, related to one another. Positive position does not necessarily mean positive velocity and vis versa.

•

For example, if a > 0 we cannot say that v must also be positive.

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Dr. M. Majeed

Chapter 12 Introduction and Rectilinear Motion Section 12.1-2

Dynamics ENG-203

Spring 2020

Special Case: Constant Acceleration (a=constant=ac) When the acceleration is constant the instantaneous velocity and acceleration equations can be integrated directly.

Velocity as Function of Time. Assuming that initially 𝑣 = 𝑣0when 𝑡 = 0, the acceleration is written as, d𝑣 → 𝑎𝑐 d𝑡 = d𝑣 d𝑡

𝑎𝑐 =

Integrating both side yields,

𝑣

𝑡

∫ d𝑣 = ∫ 𝑎 𝑐 d𝑡 0

𝑣0

which result in,

𝑣 = 𝑣0 + 𝑎𝑐 𝑡 Constant Acceleration

Position as a Function of Time. Assuming that initially 𝑠 = 𝑠0when 𝑡 = 0, the velocity is written as, 𝑣=

Integrating both side yields,

d𝑟 → 𝑣 d𝑡 = 𝑑𝑠 d𝑡 𝑡

𝑠

∫ d𝑠 = ∫ 𝑣d𝑡 𝑠0

but we know from the previouse results that,

therefore,

0

𝑣 = 𝑣0 + 𝑎𝑐 𝑡 𝑠

𝑡

∫ d𝑠 = ∫ [𝑣0 + 𝑎𝑐 𝑡]d𝑡 𝑠0

Leading to,

𝑡0

𝑠 = 𝑠0 + 𝑣0 𝑡 +

1 𝑎 𝑡2 2 𝑐

Constant Acceleration Velocity as Function of Position. The following differential relation may be obtained by multiplying both sides by 𝑑𝑠 or by differential 𝑑𝑡 from the relation,

𝑎=

d𝑠 d𝑣 d𝑣 d𝑣 = 𝑣 d𝑣 𝑑𝑠 = ⟹ 𝑎 d𝑠 = d𝑡 d𝑡 d𝑡

to get, Initially 𝑣 = 𝑣0 at 𝑠 = 𝑠0when 𝑡 = 0

Leading to

𝑎 d𝑠 = 𝑣 d𝑣 𝑠

𝑣

𝑠0

𝑣0

∫ 𝑎𝑐 d𝑠 = ∫ 𝑣d𝑣

𝑣 2 = 𝑣02 + 2𝑎𝑐 (𝑠 − 𝑠0 ) Constant Acceleration Important Points • • •

Always start by establishing a position coordinate s along the path and specifying its fixed origin and positive direction. Negative signs of 𝑠,𝑣 , and 𝑎 imply negative directions. Remember never apply constant acceleration equations unless you are absolutely certain that the acceleration is constant. Average speed is defined as 𝑣𝑠𝑝𝑒𝑒𝑑 = 𝑆𝑇 /Δ𝑡

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Dr. M. Majeed

Examples

Introduction and Rectilinear Motion Section 12.1-2

Dynamics ENG-203

Spring 2020

12–2. The acceleration of a particle as it moves along a straight line is given by 𝑎 = (4𝑡 3 -1) m/s2, where 𝑡 is in seconds. If 𝑠 = 2 m and 𝑣 = 5 m/s when 𝑡 = 0, determine the particle’s velocity and position when 𝑡 = 5 s. Also, determine the total distance the particle travels during this time period.

12-5 A particle moves along a straight line such that its position is defined by 𝑠 = (𝑡 2 − 6𝑡 + 5) m. Determine the average velocity, the average speed, and the acceleration of the particle when 𝑡 = 6 s.

-6D:\Dropbox\Teaching\ENG203 Dynamics Summer 2019\Dynamics ClassNotes Presentations\Chapter 12\D12.1-2 Introduction

Dr. M. Majeed

Examples

Introduction and Rectilinear Motion Section 12.1-2

Dynamics ENG-203

Spring 2020

12–6. A stone A is dropped from rest down a well, and in 1 s another stone B is dropped from rest. Determine the distance between the stones another second later.

12-18. A particle is moving along a straight line such that when it is at the origin it has a velocity of 4 m/s. If it begins to decelerate at the rate of 𝑎 = (−1.5𝑣 1/2 ) m/s2, where 𝑣 is in m/s, determine the distance it travels before it stops. How much time does it take?

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Dr. M. Majeed

Examples

Introduction and Rectilinear Motion Section 12.1-2

Dynamics ENG-203

Spring 2020

12-19. The acceleration of a rocket traveling upward is given by 𝑎𝑝 = 6 + 0.02𝑠𝑝. Determine the rocket’s velocity when 𝑠𝑝 = 2 km and the time needed to reach this altitude. Initially, 𝑣𝑝 = 0 and 𝑠𝑝 = 0 when t = 0.

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Dr. M. Majeed...