kinematic of particle

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Introduction Rectilinear Motion Curvilinear Motion Kinematics of a Particle Lecture 2 Unggul Wasiwitono Mechanical Engineering Department Lecture 2 1 / 25

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Kinematic of Particle

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Page 1: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Kinematics of a ParticleLecture 2

Unggul Wasiwitono

Mechanical Engineering Department

Lecture 2 1 / 25

Page 2: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Objectives

1 To introduce the concepts of position, displacement, velocity, andacceleration.

2 To study particle motion a long a straight line and represent this motiongraphically.

3 To investigate particle motion a long a curved path using differentcoordinate systems.

4 To present an analysis of dependent motion of two particles.5 To examine the principles of relative motion of two particles using

translating axes.

Lecture 2 2 / 25

Page 3: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Introduction

1 Mechanics is a branch of the physical sciences that is concerned withthe state of rest or motion of bodies subjected to the action of forces.

1 statics1 dynamics.

2 Statics is concerned with the equilibrium of a body that is either at restor moves with constant velocity.

3 Dynamics deals with the accelerated motion of a body. The subject ofdynamics will be presented in two parts:

3 kinematics, which treats only the geometric aspects of the motion3 kinetics, which is the analysis of the forces causing the motion.

4 Particle has a mass but negligible size and shape.

Lecture 2 3 / 25

Page 4: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Rectilinear Motion

1 The kinematics of a particle is characterized by specifying, at any giveninstant, the particle’s position, velocity, and acceleration.

2 Position

3 Displacement

∆s = s′ − s

3 The displacement of a particle is a vector quantity3 The distance traveled is a positive scalar that represents the total length

of path over which the particle travels

Lecture 2 4 / 25

Page 5: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Rectilinear Motion

1 Velocity

1 If the particle moves through a displacement ∆s during the time interval∆t, the average velocity of the particle is

vavg =∆s

∆t

1 The instantaneous velocity is

v = lim∆t→0

∆s

∆t=ds

dt

1 The magnitude of the velocity is known as the speed

Lecture 2 5 / 25

Page 6: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Rectilinear Motion

1 Acceleration

1 Provided the velocity of the particle is known at two points, the averageacceleration of the particle during the time interval ∆t is defined as

aavg =∆v

∆t

1 The instantaneous acceleration is

a = lim∆t→0

∆v

∆t=dv

dt

a =d2s

dt2

Lecture 2 6 / 25

Page 7: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Rectilinear Motion

1 Differential relation involving the displacement, velocity, andacceleration along the path

a ds = v dv

2 Constant Acceleration2 Velocity as a Function of Time

ˆ v

v0

dv =

ˆ t

0

a dt

v = v0 + at

2 Position as a Function of Timeˆ s

s0

ds =

ˆ t

0

(v0 + at) dt

s = s0 + v0t+1

2at2

Lecture 2 7 / 25

Page 8: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Rectilinear Motion

1 Constant Acceleration1 Velocity as a Function of Position

ˆ v

v0

vdv =

ˆ s

s0

a ds

v2 = v20 + 2a (s− s0)

Example 1

A particle travels along a straight line with a velocity of

v =(4t− 3t2

) ms

where t is in seconds. Determine the position of the particle when t = 4s.s = 0 when t = 0

Lecture 2 8 / 25

Page 9: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Curviliear Motion: Rectangular Components

1 PositionIf the particle is at point (x, y, z) on the curved path s then its location isdefined by the position vector

r = xi + yj + zk

At any instant the magnitude of r is defined

r =√x2 + y2 + z2

And the direction of r is specified by the unit vector ur = r/r.

Lecture 2 9 / 25

Page 10: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Curviliear Motion: Rectangular Components

1 VelocityThe first time derivative of r yields the velocity of the particle.

v =drdt

=d

dt(xi) +

d

dt(yj) +

d

dt(zk)

v =drdt

= vxi + vyj + vz

The velocity has a magnitude

v =√v2x + v2

y + v2z

Lecture 2 10 / 25

Page 11: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Curviliear Motion: Rectangular Components

1 AccelerationThe second time derivative of r yields the velocity of the particle.

a =d2rdt2

= axi + ayj + az

The velocity has a magnitude

a =√a2x + a2

y + a2z

Lecture 2 11 / 25

Page 12: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Motion of a Projectile

1 Horizontal Motion

vx = v0x

x = x0 + v0xt

2 Vertical Motion

vy = v0y − gt

y = y0 + v0yt−1

2gt2

v2y = v2

0y − 2g (y − y0)

Lecture 2 12 / 25

Page 13: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Curvilinear Motion: Normal and TangentialComponents

Lecture 2 13 / 25

Page 14: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Curviliear Motion: Normal and TangentialComponents

1 Velocitythe particle’s velocity v has a direction that is always tangent to the path

v = vut

wherev = s

Lecture 2 14 / 25

Page 15: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Curviliear Motion: Normal and TangentialComponents

1 AccelerationThe acceleration of the particle is the time rate of change of the velocity.

a = v = vut + vut

whereut = θun =

s

ρun =

v

ρun

Lecture 2 15 / 25

Page 16: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Curviliear Motion: Normal and TangentialComponents

1 Acceleration

a = atut + anun

where

at = v and an =v2

ρ

the magnitude of acceleration is the positive value of a =√a2t + a2

n

Lecture 2 16 / 25

Page 17: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Curviliear Motion: Normal and TangentialComponents

Lecture 2 17 / 25

Page 18: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Procedure for Analysis

1 Coordinate System1 Provided the path of the particle is known, we can establish a set of n andt coordinates having a fixed origin, which is coincident with the particle atthe instant considered.

1 The positive tangent axis acts in the direction of motion and the positivenormal axis is directed toward the path’s center of curvature.

2 Velocity2 The particle’s velocity is always tangent to the path.2 The magnitude of velocity is found from the time derivative of the path

function.v = s

3 Tangential Acceleration.3 The tangential component of acceleration is the result of the time rate of

change in the magnitude of velocity. This component acts in the positive sdirection if the particle’s speed is increasing or in the opposite direction ifthe speed is decreasing.

Lecture 2 18 / 25

Page 19: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Procedure for Analysis1 Tangential Acceleration

1 The relations between at, v, t and s are the same as for rectilinear motion,namely,

at = v; at ds = v dv

2 Normal Acceleration2 The normal component of acceleration is the result of the time rate of

change in the direction of the velocity. This component is always directedtoward the center of curvature of the path, i.e., along the positive n axis.

2 The magnitude of this component is determined from

an =v2

ρ

2 If the path is expressed as y = f(x), the radius of curvature p at any pointon the path is determined from the equation

ρ =

[1 +

(dydx

)2]3/4d2ydx2

Lecture 2 19 / 25

Page 20: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Curvilinear Motion: Cylindrical Components

We can specify the location of the particle using a radial coordinate r and atransverse coordinate θ

1 PositionAt any instant the position of the particle

r = rur

2 VelocityThe instantaneous velocity v is obtained by taking the time derivative ofr.

v = r = rur + rur

Lecture 2 20 / 25

Page 21: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Curvilinear Motion: Cylindrical Components

ur = lim∆t→0

∆ur∆t

=

(lim

∆t→0

∆ur∆t

)uθ

ur = θuθ

Thereforev = vrur + vθuθ

wherevr = r; vθ = rθ

Lecture 2 21 / 25

Page 22: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Curvilinear Motion: Cylindrical Components

1 Accelerationthe particle’s instantaneous acceleration

a = v = rur + rur + rθuθ + rθuθ + rθuθ

with

uθ = lim∆t→0

∆uθ∆t

=

(lim

∆t→0

∆uθ∆t

)ur

uθ = −θur

we haveLecture 2 22 / 25

Page 23: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Curvilinear Motion: Cylindrical Components

1 Acceleration

a = arur + aθuθ

where

ar = r − rθ2

aθ = rθ + 2rθ

Lecture 2 23 / 25

Page 24: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Curvilinear Motion: Cylindrical Components

1 Cylindrical Coordinates

rP = rur + zuzv = rur + rθuθ + zuz

a =(r − rθ2

)ur +

(rθ + 2rθ

)uθ + zuz

Lecture 2 24 / 25

Page 25: Kinematic of Particle

Introduction Rectilinear Motion Curvilinear Motion

Procedure for Analysis

1 Coordinate System1 Polar coordinates are a suitable choice for solving problems when data

regarding the angular motion of the radial coordinate r is given to describethe particle’s motion. Also, some paths of motion can conveniently bedescribed in terms of these coordinates.

1 To use polar coordinates, the origin is established at a fixed point, and theradial line r is directed to the particle.

1 The transverse coordinate θ is measured from a fixed reference line to theradial line.

2 Velocity and Acceleration2 Once r and the four time derivatives r, r, θ, and θ have been evaluated at

the instant considered, their values can be used to obtain the radial andtransverse components of v and a.

Lecture 2 25 / 25