Chapter4-F08.ppt 1911KB Jun 23 2011 10:24:02 AM

Chapter 4:Kinematics in Two

Dimensions

1.Two-Dimension Kinematics 2.Projectile Motion

3.Relative Motion

4.Uniform Circular Motion

5.Velocity and Acceleration in Uniform Circular Motion

Stop to think 4.1 P 93 Stop to think 4.2 P 97 Stop to think 4.3 P 102 Stop to think 4.4 P 107 Stop to think 4.5 P 110 Stop to think 4.6 P 113

Position and Velocity

1

1

x i y j

 

v

v

r

xi

yj

    

v

v

v

dr

dx

dy

V

i

j

dt

dt

dt



v





Instantaneous velocity

The Instantaneous velocity vector is tangent to the trajectory.

Don’t confuse these two

graphs

s

ds

V

dt



V

(

dx

)

2

(

dy

)

2

dt

dt

Acceleration

avg

V

a

t







v

v

dV

a

dt



The instantaneous acceleration

can be

Stop to think:

This acceleration will cause the

particle to:

a. Speed up and curve upward b. Speed up and curve downward c. Slow down and curve upward d. Slow down and curve downward e. Move to the right and down

Projectile Motion

object moves in two dimensions under the gravitational force.

0

x

y

a

a

g



 

A

B

Launch angle

cos

sin

ix i iy i

V

V

V

V









2

1/ 2 ( )

ix

iy

x V t

y V t

g t

  

Ex.

A ball thrown horizontally at

velocity V

_{i ,}

travels a horizontal

distance of R m before hitting the

ground. From what height was the

ball thrown?

(1)

Since ball is thrown horizontally, V_i =V_x There is no acceleration at x direction.

ie. R = V_xt, t = R/V_x

Problem 50

6sin( 15 ) /o

oy

V   m s

6cos( 15 ) /o

ox

V   m s

2

3 oy 1/ 2 y    V t  gt

Solve a quadratic equation to get t

*

ox

d



V

t

2

The maximum height and distance

of fly ball

 For projectile motion, always

remember:

g

v

h

i i

2

sin

2 2





0,

x y

a



a

 

g

g

v

R

i

sin

2



i 2

Relative Motion

 Relative position

Relative velocity

'

r

v v

 

r

R

v

ab ac cb

Uniform Circular Motion

 Period

 Angular Position

1 circumference

speed

T



T

2

r

V





(radians)

s

r





full circle

=

2 r

2 rad

r









360

1 rad

57.3

2

o

o



Angular Velocity



Average angular velocity =∆θ/∆t



Instantaneous angular velocity



The angular velocity is constant during

uniform circular motion

d

dt







t

 

  

2

T

An old-fashioned single-play vinyl record rotates 30.0 rpm . What are (a) the angular velocity in rad/s and (b) the period of the motion

?

 rpm: revolution per

minute.

1 rpm = 2π/60 (rad)/s

2

T







T

2





Velocity in uniform circular

motion

Has only a tangential Component

The magnitude of velocity is a constant

The magnitude of centripetal

acceleration

2

2

r

V

a

r

r







P184

Velocity and acceleration in Uniform

Circular Motion

The velocity has only a tangential component V

_t

(with in rad/s)

t

ds

d

V

r

r

dt

dt

 









2

(toward center of ciecle)

V

a

r

Nonuniform Circular Motion

t

dV

a

dt



Change the speed

d

with =

dt

V





r





t

a





r

Here α is angular acceleration

if is constant

f i

t

Rotational kinematics

 For constant angular acceleration

2

i f

f i

 

t



 







f i

t



  

 

2

1/ 2 ( )

f i i

t

t



   

 





2 2

2

f i