Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf ·...

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Motion in a Plane Motion in a Plane Readings: Chapter 4 1 Readings: Chapter 4

Transcript of Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf ·...

Page 1: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

Motion in a PlaneMotion in a Plane

Readings: Chapter 41

Readings: Chapter 4

Page 2: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

Kinematics in Two DimensionsΔAverage velocity: avg

rvt

Δ=Δ

rΔavgv

t=Δ

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Page 3: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

Kinematics in Two Dimensions: Instantaneous Velocityy

rv Δ=Δ 0tt Δ →Δ

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Page 4: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

Kinematics in Two Dimensions

rv Δ=

dsvd

=0t

vt Δ →Δdt

(motion along a line)

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Page 5: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

Kinematics in Two Dimensions: Instantaneous Acceleration

Average acceleration:

avgvat

Δ=Δ

Instantaneous acceleration:

0t

vat Δ →

Δ=Δ

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Page 6: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

Kinematics in Two Dimensions: Instantaneous Acceleration

Motion along a line: acceleration results in change of speed (theMotion along a line: acceleration results in change of speed (the magnitude of velocity)

Motion in a plane: acceleration can change the speed (the magnitude of velocity) and the direction of velocity.

0

vat Δ

Δ=Δ 0tt Δ →Δ

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Page 7: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

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Page 8: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

A Projectile is an object that moves in two dimensions (in a plane) under the influence of only the gravitational force.

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Page 9: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

A Projectile is an object that moves in two dimensions with free fall acceleration

a g=

0xa = - motion along x-axis – with zero acceleration – constant velocity

constantxv =

29.8yma gs

= − = −

x

- motion along y-axis – free fall motion with constant acceleration

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Page 10: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

constant cosv v v θ= = =

Example: Find the distance AB

cosx v t v t θ= =

29.8 ma g= − = −

,constant cosx i x iv v v θ= = = , cosi x ix v t v t θ= =

iv v gt= − sinv v θ=29.8ya gs ,y i yv v gt

2ABt

Point A - 0i Ay y= =2t

, sini y iv v θ=

, 2AB

final i y ABt

y v t g= −Point B - 0final By y= = then , 2AB

i y ABt

v t g=

2 2 siniv v θ,2 2 sini y iAB

v vt

g gθ

= =

then

A B

then

2 2

cos

2AB i AB

i i

x v t

v v

θ

θ θ θ

= =

10

A B 2sin cos sin 2i iv v

g gθ θ θ= =

Page 11: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

2

sin 2iAB

vx θ=

g

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Page 12: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

Motion in a CircleMotion in a Circle

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Page 13: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

Kinematics in Two Dimensions: Uniform Circular Motion: Motion with Constant Speed

The speed (magnitude of velocity) is the same

Period:

2 rπ2 rTvπ

=

θ

The position of the particle is characterized by angle (or by arc length)

Arc length:

where angle is in RADIANS

s rθ=

θ13

g0

03601 57.32

radπ

= =

Page 14: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

Example:

What is the arc length for if0 0 045 90 10θ = 1r m=What is the arc length for if

0 00

245 45 0.78360 4

rad rad radπ π= = =

45 ,90 ,10θ = 1r m=

360 40 0

0

290 90 1.57360 2

rad rad radπ π= = =

0 00

210 10 0.174360 18

rad rad radπ π= = =

Arc length: s rθ=rπ

45 0.784rs mπ

= =

90 1.572rs mπ

= =

10 0.17418

rs mπ= =

14

90 2

Page 15: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

Kinematics in Two Dimensions: Uniform Circular Motion: Motion with Constant Speed

The speed (magnitude of velocity) is the same s vt=tθ ω= tθ ω

vr

ω =2 rTvπ

=

Then

2π2Tπω =

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Page 16: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

Uniform Circular Motion s vt= 2Tπω =

tθ ω=

vr

ω =

Then

cos cos( )x r r tθ ω= = ( )

sin sin( )y r r tθ ω= =

2 2 2 2 2 2 2

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2 2 2 2 2 2 2sin ( ) cos ( )x y r t r t rω ω+ = + =

Page 17: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

tθ ω=

0ω >0ω <

0ω =

Positive angle

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Positive angle

Page 18: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

Uniform Circular Motion: Acceleration

0t

vat Δ →

Δ=Δ 0t

rvt Δ →

Δ=Δ

1 1r v tΔ = Δ

2 2r v tΔ = Δ2 2r v tΔ Δ2 1 2 1

2( )v v r r

at t− Δ − Δ

= =Δ Δ( )t tΔ Δ

Direction of acceleration is the same as direction of vector (toward the center)

2 1CB r r→

= Δ − ΔThe magnitude of acceleration:

2( 2 )CB φ θΔ Δ Δ Δ Δ18

2

2 2 2 2 2

( 2 )( ) ( ) ( ) ( ) ( )CB r r r v t t va v

t t t t t rφ π α θ ω

ωΔ Δ − Δ Δ Δ

= = = = = = =Δ Δ Δ Δ Δ

Page 19: Motion in a Planein a Plane - Physics & Astronomyphysics.gsu.edu/apalkov//lecture2211_4.pdf · ΔΔΔΔΔ. Uniform Circular Motion: Acceleration v r ω= centripetal acceleration The

Uniform Circular Motion: Acceleration vr

ω =

centripetal acceleration

The magnitude of acceleration:

22va r v

rω ω= = =

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