Dynamics Hibbeler Chapter 12 Notes

Engineering Mechanics: Dynamics

Chapter 12 Kinematics of a Particle 12.1 INTRODUCTION Mechanics 

a branch of physical science that is concerned with the state of rest and motion of bodies subjected to action of forces 1. Statics a. Concerned with the equilibrium of a body that is either at rest or moves with constant velocity 2. Dynamics a. Concerned with bodies that have accelerated motion b. Kinematics  The study of the geometry of the motion c. Kinetics  The study of the forces that cause the motion 12.2 RECTILINEAR KINEMATICS: CONTINOUS MOTION Rectilinear 

Straight-line path

Rectilinear Kinematics 

Kinematics of particles are characterized by position, velocity, and acceleration at any given instant in a straight-line motion

Position 

Location of particle along a single coordinate axis

Displacement  

Change in position

Velocity  

Average speed is total displacement divided by total time Average velocity is displacement divided by total time

1. Average velocity a. 2. Instantaneous velocity

a. Acceleration   

Acceleration is zero if velocity is constant ( ) A particle that is slowing down is decelerating A particle can have an acceleration and yet have zero velocity

1. Average acceleration a. 2. Instantaneous acceleration a. Relationship between instantaneous velocity and instantaneous acceleration    Constant acceleration (ac)   Equations can be integrated to obtain formulas that relate ac , v, s, and t 1. Velocity as a function of time ∫

a.

∫

2. Position as function of time ∫(

∫

a.

)

3. Velocity as a function of position ∫

a. (

)

∫

(

)

(

)

12.3 RECTILINEAR KINEMATICS: ERRACTIC MOTION Examples

𝒅𝒔 𝒅𝒕

𝒗

𝒅𝒔 𝐨𝐫 𝒗𝒂𝒗𝒈 𝒅𝒕

𝒅𝒗 𝒅𝒕

𝒗𝒅𝒕

𝒅𝒔

𝒗𝒅𝒕

𝒔𝟎

𝟎

𝒔 𝒕

Differentiate

𝒂

𝒅𝒔

𝒕

Integrate

𝒗

𝒔

𝒂

𝒅𝒗 𝒅𝒕

𝒗

𝒅𝒗

𝒂𝒅𝒕

𝒕

𝒅𝒗 𝒗𝟎

𝒂𝒅𝒕 𝟎

12.4 GENERAL CURVILINEAR MOTION Curvilinear 

Curved path s



Curvilinear motion can cause changes in both in magnitude and direction of position, velocity, and acceleration

Position 

Path as a function of ( )



Designated by the position vector r



( )

Displacement 

Distance along the curve

 Velocity 

Speed is the magnitude of v



v is tangent to the path



Average velocity is displacement divided by total time

1. Average velocity a. 2. Instantaneous velocity a. 3. Speed a. Acceleration 

Acceleration is zero if velocity is constant (



A particle can have an acceleration and yet have zero velocity



)



Acceleration is tangent to the hodograph (curve) and not the path

1. Average acceleration a. 2. Instantaneous acceleration a. 12.5 CURVILINEAR MOTION: RECTANGULAR COMPONENTS (I, j, k) Position  Particle at point (x, y, z) on the curved path s 1. Position vector a. 2. Magnitude of r a.

√

Velocity 1. Velocity vector ( )

a.

b. ̇ ̇ 2. Magnitude of velocity a.

( )

(

)

(

)

̇

√

Acceleration 1. Velocity vector ( )

c. ̈

d.

̈

2. Magnitude of velocity b.

√

( ) ̈

12.6 MOTON OF A PROJECTILE    

Constant downward acceleration ac ⁄ ⁄

1. Horizontal Motion a. Velocity as a function of time (



)

b. Position as a function time (

 c. Velocity as a function of position  2. Vertical Motion a. Velocity as a function of time

(

)

(



(

)

) (

)

b. Position as a function time (

 c. Velocity as a function of position 

(

)

( (

)

(

) )

)

)

12.7 CURVILINEAR MOTION: NORMAL AND TANGENTIAL COMPONENTS (n and t) Planar Motion   

Using n (normal) and t (tangent) to describe motion u is used to designate a unit vector radius of curvature is 𝜌

Velocity 

( )

 ̇

Acceleration 1.

̇ a. The tangential component of acceleration represents the time rate of change in the magnitude of the velocity.

2. a. The normal component of acceleration represents the time rate of change in the direction of the velocity b. always acts toward the center of curvature (centripetal acceleration) ̇

3.

̇

̇ ̇

  

̇

̇ ̇ ̇

̇

4. ̇

  

5. Magnitude of acceleration 

√

12.8 CURVILINEAR MOTION: CYLINDRICAL COMPONENTS (r and )

12.9 ABSOLUTE DEPENDENT MOTION ANALYSIS OF TWO PARTICLES Example 1 Position  Velocity  Acceleration  Example 2 Position   Velocity  Acceleration 

12.10 RELATIVE-MOTION OF TWO PARTICLES USING TRANSLATING AXES Position    

Absolute position of each particle, rA and rB, is measured from the common fixed origin O Particle B moves with particle A with each having their own axis The axes of B and A are permitted to translate (move) relative to the fixed axis of O The position of B is measured relative to A is denoted by the relative-position vector rB/A



Velocity  Acceleration 