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Quiz 4 (Ch. 6)

ffii mmmm 22112211 vvvv rrrr+=+

Conservation of momentum for 2 objects colliding

if

if

mmpF

mmptF

vv

vvrrrr

rrrr

==∆=

−=∆=∆

and 0 then ,0 if

Impulse - momentum theorem

For two-dimensional collisions can be applied along x-axis and y-axis

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Chapter 7

Rotational Motion

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Concepts of Linear Motion

Displacement

Velocity

Acceleration t

t

if

∆∆

=

∆∆

=

−=∆

va

rv

rrr

vr

rv

rrr

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Linear vs. Rotational

In linear motion: displacement mostly along the straight trajectory

In rotational motion: displacement -along the circular trajectory

Velocity and acceleration are always changingCan introduce “angular” quantities for an easier description of the rotational motion

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Concepts of Rotational Motion Angular Displacement

Angular Velocity

Angular Acceleration t

t

if

∆∆

=

∆∆

=

−=∆

ωα

θω

θθθ

av

av

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Angular DisplacementAxis of rotation is the center of the diskNeed a fixed reference lineDuring time t, the reference line moves through angle θ

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Rigid BodyEvery point on the object undergoes circular motion about the point OAll parts of the object of the body rotate through the same angle during the same timeThe object is considered to be a rigid body

This means that each part of the body is fixed in position relative to all other parts of the body

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Angular Displacement, cont.

The angular displacement is defined as the angle the object rotates through during some time interval

Each point on the object undergoes the same angular displacement The unit of angular displacement is the radian

f iθ θ θ∆ = −

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The RadianThe radian is a unit of angular measureThe radian can be defined as the arc length s along a circle divided by the radius r

sr

θ =°3.57

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Conversion Between Degrees and Radians

Comparing degrees and radians

Converting from degrees to radians

°=°

= 3.572

360rad1π

]rees[deg180

]rad[ θπθ°

=

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Average Angular SpeedThe average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval

f iav

f it t tθ θ θω

− ∆= =

− ∆

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Angular Speed, cont.The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zeroUnits of angular speed are radians/sec

rad/s

Speed will be positive if θ is increasing (counterclockwise)Speed will be negative if θ is decreasing (clockwise)

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Quick Quiz 7.1-2Set of initial and final angular positions:(a) 3 rad, 6 rad; (b) −1 rad, 1 rad; (c) 1 rad, 5 rad.Which of the sets can occur only if the rigid body rotates through more than 180°?

t = 1 s. Which choice represents the lowest average angular speed?

if θθθ −=∆

t∆∆= /av θω

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Average Angular Acceleration

The average angular acceleration of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:

α

f iav

f it t tω ω ωα

− ∆= =

− ∆

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Angular Acceleration, contUnits of angular acceleration are rad/s²Positive angular accelerations are in the counterclockwise direction and negative accelerations are in the clockwise directionWhen a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration

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Angular Acceleration, final

The sign of the acceleration does not have to be the same as the sign of the angular speedThe instantaneous angular acceleration is defined as the limit of the average acceleration as the time interval approaches zero

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Quick Quiz 7.3Set of initial and final angular positions:(a) 3 rad, 6 rad; (b) −1 rad, 1 rad; (c) 1 rad, 5 rad.The object moves counterclockwise with

For which choice is the angular acceleration the highest?

tt∆∆=∆∆=

//

ωαθω

f3f2f1f

3i2i1i 0ωωωω

ωωω======

const=α

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Rotational Motion under Constant Angular Acceleration

Described by a set of equations of motion:

Similar to one-dimensional kinematics

θαωω

αωθ

αωω

∆+=

+=∆

+=

221

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2

i

i

i

tt

t

4

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Example: Problem#6

= 3600 rev/min= 0= 50 rev

-?

iω

θ∆

α

fω

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Analogies Between Linear and Rotational Motion

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Relationship Between Angular and Linear Quantities

Displacements

Speeds

Accelerations

Every point on the rotating object has the same angular motionEvery point on the rotating object does not have the same linear motion

rs θ=

tv rω=

ta rα=For circular motion: linear becomes tangential

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Quick Quiz 7.4-5The merry-go-round is rotating at a constant angular speed

Andrea’s angular speed is (a) twice Chuck’s (b) the same as Chuck’s (c) half of Chuck’s (d) impossible to determine.Andrea’s tangential speed is (a) twice Chuck’s (b) the same as Chuck’s (c) half of Chuck’s (d) impossible to determine.

rArC

ωrrr

t

C

A

=

=

v

2

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Example: Problem#12

d=2.4 cm

= 18 rad/s

= 0

= -1.9 rad/s2

s -?

iω

αfω

d

s

iω fω

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Centripetal Acceleration

An object traveling in a circle, even though it moves with a constant speed, will have an accelerationThe centripetal acceleration is due to the change in the direction of the velocity

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Centripetal Acceleration, cont.Centripetal refers to “center-seeking”The direction of the velocity changesThe acceleration is directed toward the center of the circle of motion

if vvv rrr−=∆

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Centripetal Acceleration, final

The magnitude of the centripetal acceleration is given by

This direction is toward the center of the circle

2

c

va

r=

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Centripetal Acceleration and Angular Velocity

The angular velocity and the linear velocity are related (v = ωr)The centripetal acceleration can also be related to the angular velocity

ra 2C ω=

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Total AccelerationThe tangential component of the acceleration is due to changing speedThe centripetal component of the acceleration is due to changing directionTotal acceleration can be found from these components

2C

2t aaa +=

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Vector Nature of Angular Quantities

Angular displacement, velocity and acceleration are all vector quantitiesDirection can be more completely defined by using the right hand rule

Grasp the axis of rotation with your right handWrap your fingers in the direction of rotationYour thumb points in the direction of ω

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Velocity Directions, Example

In a, the disk rotates clockwise, the velocity is into the pageIn b, the disk rotates counterclockwise, the velocity is out of the page

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Acceleration Directions

If the angular acceleration and the angular velocity are in the same direction, the angular speed will increase with timeIf the angular acceleration and the angular velocity are in opposite directions, the angular speed will decrease with time

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Forces Causing Centripetal Acceleration

Newton’s Second Law says that the centripetal acceleration is accompanied by a force

FC = maC

FC stands for any force that keeps an object following a circular path

Tension in a stringGravityForce of friction

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Problem Solving StrategyDraw a free body diagram, showing and labeling all the forces acting on the object(s)Choose a coordinate systemthat has one axis perpendicular to the circular path and the other axis tangent to the circular path

The normal to the plane of motion is also often needed

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Problem Solving Strategy, cont.

Find the net force toward the center of the circular path (this is the force that causes the centripetal acceleration, FC)Use Newton’s second law

The directions will be radial, normal, and tangentialThe acceleration in the radial direction will be the centripetal acceleration

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Example: Problem #19

m=55 kgvi=4.0 m/sr=0.8 m

a) T-?b) T/W-?

vi

r

Tr