(9) Angular Kinematics

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19 Terms

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Angular kinematics

  1. units

    1. degree

    2. radian

  2. Quantities

    1. angular position and displacement

    2. angular velocity

    3. angular acceleration

  3. linear motion vs angular motion

    1. circular motion

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Units

Degree

  • 1 revolution= 360°

Radian

  • the angle with which the length of the arc becomes the same to the radius

  • 1 rad=57.30°

  • r=1rad

  • 2r=2rad

  • 3r=3rad

  • 2pi r= 2pi rad

  • 1 rev= 2pi rad= 360 degree

  • pi rad= 180 deg

  • pi/2 rad= 90 deg

  • pi/3 rad= 60 deg

  • pi/4 rad= 45 deg

  • pi/6 rad= 30 deg

pi=3.141592654

180/pi=57.29577951

<p>Degree</p><ul><li><p>1 revolution= 360<span>° </span></p></li></ul><p>Radian</p><ul><li><p>the angle with which the length of the arc becomes the same to the radius </p></li><li><p>1 rad=57.30<span>°</span></p></li><li><p>r=1rad</p></li><li><p>2r=2rad</p></li><li><p>3r=3rad</p></li><li><p>2pi r= 2pi rad</p></li><li><p>1 rev= 2pi rad= 360 degree</p></li><li><p>pi rad= 180 deg</p></li><li><p>pi/2 rad= 90 deg</p></li><li><p>pi/3 rad= 60 deg</p></li><li><p>pi/4 rad= 45 deg</p></li><li><p>pi/6 rad= 30 deg</p></li></ul><p>pi=3.141592654</p><p>180/pi=57.29577951</p>
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why radian?

simple relationship between the angle and length of the arc

  • d=r𐤃Θ

<p>simple relationship between the angle and length of the arc</p><ul><li><p>d=r<span>𐤃Θ</span></p></li></ul><p></p>
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angular location at a given instant

  • unit: rad

  • direction

    • counterclockwise= +

    • clockwise= -

<ul><li><p>unit: rad</p></li><li><p>direction</p><ul><li><p>counterclockwise= +</p></li><li><p>clockwise= - </p></li></ul></li></ul><p></p>
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angular displacement

  • change in angular position

    • net effect of angular motion

  • unit: rad

  • direction

    • counterclockwise: +

    • clockwise:-

  • zero angular displacement

    • no net rotation 𐤃Θ= Θ2-Θ1=0

    • no rotation dΘ= Θ2-Θ1=0

<ul><li><p>change in angular position</p><ul><li><p>net effect of angular motion</p></li></ul></li><li><p>unit: rad</p></li><li><p>direction</p><ul><li><p>counterclockwise: +</p></li><li><p>clockwise:- </p></li></ul></li><li><p>zero angular displacement</p><ul><li><p>no net rotation 𐤃Θ= Θ<sub>2</sub>-Θ<sub>1</sub>=0</p></li><li><p>no rotation dΘ= Θ<sub>2</sub>-Θ<sub>1</sub>=0</p></li></ul></li></ul><p></p>
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<p>example of angular displacement </p>

example of angular displacement

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Angular velocity

  • rate of change in angular position

  • angular displacement/time

  • direction

    • positive: counterclockwise rotation

    • negative: clockwise rotation

  • vector

  • unit: rad/s

<ul><li><p>rate of change in angular position</p></li><li><p>angular displacement/time </p></li><li><p>direction </p><ul><li><p>positive: counterclockwise rotation </p></li><li><p>negative: clockwise rotation </p></li></ul></li><li><p>vector </p></li><li><p>unit: rad/s</p></li></ul><p></p>
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zero angular velocity

  • zero average angular velocity: no displacement=no net motion

  • zero instantaneous angular velocity: no motion=constant angular position

<ul><li><p>zero average angular velocity: no displacement=no net motion</p></li><li><p>zero instantaneous angular velocity: no motion=constant angular position</p></li></ul><p></p>
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angular acceleration

  • rate of change in angular velocity

    • a(bar)= 𐤃ω/𐤃t= ω2-ω1/𐤃t a= dω/dt= ω2-ω1/dt

  • positive angular acceleration

    • speed-up of counterclockwise ω

    • slow-down of clockwise ω

  • negative acceleration

    • Slow-down of counterclockwise ω

    • speed up of clockwise ω

  • vector

  • unit: rad/s2

<ul><li><p>rate of change in angular velocity</p><ul><li><p>a(bar)= <span>𐤃</span><strong>ω/</strong><span>𐤃t= </span><strong>ω<sub>2</sub>-ω<sub>1</sub>/</strong>𐤃t a= d<strong>ω/dt= ω<sub>2</sub>-ω<sub>1</sub>/dt</strong></p></li></ul></li><li><p>positive angular acceleration</p><ul><li><p>speed-up of counterclockwise <strong>ω</strong></p></li><li><p>slow-down of clockwise <strong>ω</strong></p></li></ul></li><li><p>negative acceleration</p><ul><li><p>Slow-down of counterclockwise <strong>ω</strong></p></li><li><p>speed up of clockwise <strong>ω</strong></p></li></ul></li><li><p>vector</p></li><li><p>unit: rad/s<sup>2</sup></p></li></ul><p></p>
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zero angular acceleration

a(bar)= 𐤃ω/𐤃t= ω2-ω1/𐤃t =0 a= dω/dt= ω2-ω1/dt= 0

  • zero average a: no net change in ω

  • zero instantaneous a: constant ω

<p>a(bar)= 𐤃<strong>ω/</strong>𐤃t= <strong>ω<sub>2</sub>-ω<sub>1</sub>/</strong>𐤃t =0 a= d<strong>ω/dt= ω<sub>2</sub>-ω<sub>1</sub>/dt= 0</strong></p><ul><li><p>zero average a: no net change in <strong>ω</strong></p></li><li><p>zero instantaneous a: constant <strong>ω</strong></p></li></ul><p></p>
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example angular acceleration

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linear vs angular quantity differences

<p></p>
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circular motion

directions

  • tangential (T): along the tangent

  • radial (R): Along the radius, perpendicular to the tangent

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Velocity relationship

  • direction of the linear velocity: tangential

  • magnitude of the linear velocity

<ul><li><p>direction of the linear velocity: tangential</p></li><li><p>magnitude of the linear velocity </p></li></ul><p></p>
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velocity relationship cont.

  • for a given ω: v is proportional to r

    • v=rω

    • v∝r

  • for a given r: v is proportional to ω

    • v=rω v∝ω

    • to hit the ball farther, one has to swing the bat faster

<ul><li><p>for a given <strong>ω: </strong>v is proportional to r </p><ul><li><p>v=r<strong>ω</strong></p></li><li><p>v<span>∝r</span></p></li></ul></li><li><p>for a given r: v is proportional to <strong>ω</strong></p><ul><li><p><span>v=<strong>r</strong></span>ω v<span>∝</span><strong>ω</strong></p></li><li><p>to hit the ball farther, one has to swing the bat faster </p></li></ul></li></ul><p></p>
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velocity relationship cont. pt 2

  • for a give v: r and ω are inversely proportional to each other

    • v=rω ω∝1/r

    • v remains constant

    • r decreases —→ ω increases

<ul><li><p>for a give v: r and <strong>ω</strong> are inversely proportional to each other </p><ul><li><p><strong>v</strong>=rω ω∝1/r</p></li><li><p>v remains constant </p></li><li><p>r decreases —→ ω increases </p></li></ul></li></ul><p></p>
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example: Slinging and hammer throw

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example: batting

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two different kinds of acceleration

acceleration relationship

  • tangential acceleration

    • due to the change in speed (magnitude of v)(speed up or slow)

    • aT= ra= v2-v1/𐤃t

  • radial acceleration

    • due to the change in the direction of v

    • aR=rω2= v2/r

    • towards the center of rotation: centripetal acceleration

<p>acceleration relationship</p><ul><li><p>tangential acceleration</p><ul><li><p>due to the change in speed (magnitude of v)(speed up or slow)</p></li><li><p>a<sub>T</sub>= ra= v<sub>2</sub>-v<sub>1</sub>/𐤃t</p></li></ul></li><li><p>radial acceleration</p><ul><li><p>due to the change in the direction of v</p></li><li><p>a<sub>R</sub>=rω<sup>2</sup>= v<sup>2</sup>/r</p></li><li><p>towards the center of rotation: centripetal acceleration</p></li></ul></li></ul><p></p>

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