Types of motion!

Translational motion: Object as a whole moves

along a trajectory but does not rotate

Rotational motion: Object rotates around a fixed point. Every point on the

object moves in a circle.

Combined motion: An object rotates as it moves along a trajectory.

!"#$%&'()*'+),%-'*'+%Axis of rotation through O &

! to picture. All points

move in circles about O

A rigid body is an extended

object whose size, shape, &

distribution of mass do not change as the object moves and

rotates. Example: CD "

Rigid Body Rotation!

!! .%#%!"#$%&'()*+,%&-./."0(

!! "#$$#%&'()$*+,-.*).)/%)0*1.**,2)

!! -)(/%#3).#4+5#%6)7+,/*.)/3))

%%.%/,)$*+,-.*0)/%)1&2*&"+((

!! 0%&)12)+%#%8%1&*),9*:4)4;.#-1;)

%9;*%)4;*)!"#$%&'()*$+$"!,-./$$ $

Reference Line

! here is text’s s!

Key Definitions!

!

Angle in Radians =Arc length

Radius

!

360° =2"r

r= 2" Rad

!

=l

r

Relationship between degrees and radians:!

!

1 Radian =360

2 " 3.1415926# 57.30°

Linear vs. Angular!

!!%

!!So far we have dealt with linear motion but objects can rotate as well as move!

!!Need to repeat our previous work on linear kinematics for rotational kinematics!!! Exact parallel with 1D kinematics!

!

Displacement (distance) " Displacement (angle)

Velocity " Angular velocity

Acceleration " Angular acceleration

!!A few differences though...!!! Displacement is an angle not a distance!

!! Rotate far enough and you get back to where you started!!

!! Sign gives clockwise/anti-clockwise rotation sense!

Angular position and angular displacement

" is NOT a vector.

Are vectors however (Unfortunately, by

convention only: positive if counter-

clockwise, a strange one):

Angular Velocity!

the axis of rotation!

The vector nature of Angular Velocity !

"av,z

=#$

#t

“rate of change of angular position”!

!

"z

= lim#t$0

#%

#t=d%

dt

!! Units are radians per second

(rad s-1) !! since radians have no

dimensions ([L]/[L]) angular velocity has different dimensions

than linear velocity

!

d! "

Period and Linear Speed!

!!324$+%(/)(%(/$%)+5",)#%4$,'62(7%28%(/$%#)($%'9%6/)+5$%'9%)+5,$%(/$+:%9'#%)%6'+8()+(%)+5",)#%4$,'62(7:%(/$%;$#2'1%'9%(/$%<'*'+%=2,,%>$?%

!!@/$%,2+$)#:%()+5$+*),:%4$,'62(7%'9%)%;)#*6,$%=2(/%)+5",)#%4$,'62(7%A%)%128()+6$%#%9#'<%(/$%)B28%'9%#'()*'+%28%524$+%>7?%

!

Period T =2"

#

!

Linear velocity v ="s

"t=r"#

"t

!

Linear velocity v ="r

! in Radians for a circle of radius r, arc

length ! is defined as: ! # (!/r)

Average angular velocity!

!!@/$%C'+1'+%D7$:%(/$%='#,1E8%,)#5$8(%E>25%=/$$,E%#21$:%<)F$8%'+$%#$4',"*'+%2+%0GHH8I%J/)(%28%2(8%)4$#)5$%)+5",)#%4$,'62(7K%%

LI !H%#)1M8%

NI!OM0GHH%#)1M8%

PI !HIQ%#)1M8%

RI !OMSHH%#)1M8%

DI!T'(%$+'"5/%2+9'#<)*'+%28%524$+%

Answer: D

<+./+=&*)8%1-&+.)<*&#>/4')

!!U9%)+%'>V$6(%#'()($8%8"6/%(/)(%(/$%)+5,$%2(%/)8%#'()($1%(/#'"5/%)W$#%)%*<$%(%28?%

% %.XQ(GYZ([Q%J/)(%28%2(8%)+5",)#%4$,'62(7K%

LI !AXQ(YZ%

NI!AXQ(GYZ([Q%

PI !AXS(YZ%

RI !AX(QYG(G[Q(Y6'+8()+(%

DI!T'(%$+'"5/%2+9'#<)*'+%524$+%

Answer: C

?:/%%/%1)@/,>)!!P'+821$#%(/$%(='%;'2+(8:%L%)+1%N:%'+%)%#2521%8;2++2+5%1286%)8%8/'=+I%J/26/%/)8%(/$%,)#5$8(%&"#$%&'%4$,'62(7K%

LI !%%L%/)8%(/$%,)#5$8(%)+5",)#%4$,'62(7%

NI!%%%N%/)8%(/$%,)#5$8(%)+5",)#%4$,'62(7%

PI !%%%N'(/%/)4$%(/$%8)<$%)+5",)#%4$,'62(7%

RI !%%%U(%28%2<;'882>,$%('%($,,%

A

B

Answer: C

?:/%%/%1)@/,>)!!P'+821$#%(/$%(='%;'2+(8:%L%)+1%N:%'+%)%#2521%8;2++2+5%1286%)8%8/'=+I%J/26/%/)8%(/$%,)#5$8(%,2+$)#%4$,'62(7K%

LI !%%L%/)8%(/$%,)#5$8(%,2+$)#%4$,'62(7%

NI!%%N%/)8%(/$%,)#5$8(%,2+$)#%4$,'62(7%

PI !%%N'(/%/)4$%(/$%8)<$%,2+$)#%4$,'62(7%

RI !%%U(%28%2<;'882>,$%('%($,,%

A

B

Answer: A

I thought only hamsters like this device (even my two buggies hate it!), I am wrong… Yours Truly!

Example: In this expanded version of “pet treadmill’, the speed of a honeybee clinging onto part of the outer rim of the treadmill is 1.5 m/s. If the radius of the device is 1 m, Find!(a)!Angular velocity!(b)! Period of the motion!

Solution: (a)

!

v ="r

!

1.5 =" #1

!

" =1.5 rad/s

(b)

!

T =2"

#

!

=6.283

1.5

!

T = 4.19 s

Angular Acceleration

!

"av#z =

$%z

$t=%2z#%

1z

$t

“rate of change of angular velocity”

For comparison.

Constant Angular Acceleration Kinematics

Example: Angular Acceleration!Suppose the Chevy Cavalier from your classmate starts from

rest and Uniformly accelerates to 20 m/s in 10 sec. If the outer

radius of its tires is 20 cm, assume there is no skidding on the ground, what is the angular acceleration on the wheels?

Solution: No skidding---> point of contact between the wheel and

the ground is stationary

linear velocity due to the rotation = Chevy's linear velocity

!

Final vedge ="wheelrwheel

!

20 = 0.2 "#wheel

!

"wheel

=100 rad/sec

!

Angular Acceleration : "z = ˙ ̇ # =$ f %$ i

&t

!

=100 " 0

10=10 rad/s

2

Question: If tires are bigger, how would the result change??

!

˙ ̇ " = 2nd time derivative

The “wheel of Fortune” is rotating at 0.2 rad/s when Chuck Woolery Gives it a another spin which accelerates the wheel at 2 rad/s2 for 0.2 sec. What is the final angular velocity?!

Which Equation?!

Answer: A

The “wheel of Fortune” is rotating at 0.2 rad/s when Chuck Woolery Gives it a another spin which accelerates the wheel at 2 rad/s2 for 0.2 sec. What is the final angular velocity?!

Which Equation?!

Answer: A

!

" f ="0

+#t = 0.2 + 2 $ 0.2 = 0.6 rad/s

Continue from the previous example. Suppose the wheel slows to a halt (with constant angular acceleration) in 30 sec from the angular velocity of 0.6 rad/s. How many revolution(s) did it make in that time?!

Which Equation?!

Answer: B

!

" =# +#

0

2

$

% &

'

( ) t =

0 + 0.6

2

$

% &

'

( ) * 30 = 9 rad

!

N ="

2#=

9 rad

6.283 rad/rev=1.43 rev

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%#)1M8G2)A4B,))+%1-&+.),:**0)+4)5$*)(%X%H)/,)A2%X%GIH%#)1M82))

%^)_)C/%0)4;*)+%1-&+.)0/,:&+>*$*%4)`.%/4)$+D*,)+E*.)(%X%G%82)

Use: "! = #it + ($)%t2

= (2)(2) + ($)(3)(2)2 = 11.0 rad (630º)

(b) Find the number of revolutions it makes in this time.

Convert "! from radians to revolutions:

A full circle = 360º = 2& radians = 1 revolution

11.0 rad = 630º = 1.75 rev

(c) Find the angular speed #f after t = 2 s.

#f = #i + %t = 2 + (3.5)(2) = 9 rad/s