ENGR 215 ~ Dynamics Sections 12.7

Lecture Example 1: Curvilinear motion

• Consider the function

• What is the acceleration along the path of travel at time, t=1sec?

jia

jiv

jir

)4( )12()(

)4( )14()(

)2( )()(

2

3

24

tt

ttt

tttt

r(t) = (t4 + t) i + (2t2) j

0.0

1.0

2.0

3.0

0.0 1.0 2.0 3.0

x

y

At time, t=1 sec.

jia

jiv

jir

)4( )12()1(

)4( )5()1(

)2( )2()1(

Velocity and Acceleration Vectors

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14

x

y

Acceleration

Velocity

Dot Product

A

ABuBB A

Aproj

zzyyxx BABABA BA

cosABBA

Tangential Acceleration

v

va Ta

Cross Product• Magnitude

• Direction

CAB uBAC sin

sinABC

Calculating a Cross Product

k

j

iBA

)(

)(

)(

xyyx

xzzx

yzzy

BABA

BABA

BABA

zyx BBB

zyx AAA

kji

BA

Centripetal Acceleration

v

avNa

Acceleration along a path

Derivation of the normal and tangential components of acceleration.

RuleProduct dt

dv

dt

dv

vdt

d

dt

d

v

tt

t

t

uua

uv

a

uv

Putting it all together…

nnnt v

dt

ds

dt

d

dt

duuu

u

1

nt

v

dt

dvuua

2

dt

dv

dt

dv tt

uua

From the calculus book…curvature

22

23

2

/

)/(1

dxyd

dxdy

Normal and Tangential Directions

• n – direction points to center of arc.

t x n = b

3-D Motion

Lecture Example 2: Find the equation of the path, y=f(x). Find the normal and tangential component of the acceleration at t=0.25s.

Lecture Example 3: At time, t = π seconds determine the velocity of the particle, and determine the tangential and normal components of the acceleration. Draw and label the acceleration vectors on the graph below. The position of the particle is expressed in meters. Trigonometric functions should be evaluated in radians.

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4

tdt

dtt

dt

dttt

dt

dsin sinsin

r(t) = (t sin t) i + (t cos t ) j 0 < t < 2 π sec

Spiral of Archimedes

Lecture Example 4: The jet plane travels along the vertical parabolic path. When at Point A it has reached a speed of 200 m/s which is increasing at a rate of 0.8 m/s2. Determine the magnitude of the acceleration of the plane when it is at Point A.

Note: The positions x and y are given in kilometers.