Macroscopic momentum balance 3: accelerating coordinates and

angular momentumCHEE 3363Spring 2014Handout 11

�Reading: Fox 4.5 and 4.7

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Learning objectives for lecture

1. Apply conservation of linear momentum to accelerating systems.�

2.

volumes.

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Accelerating coordinates 1Previous formulation of conservation of linear momentum only good for inertial reference frames�Need additional terms to describe conservation of linear momentum when the CV itself is accelerating�Consider: coordinates in accelerating control volume (x,y,z,t) and inertial system (X,Y,Z)

Z

X

Y x

yz

Inertial reference

frame

Moving reference

frame

We will only consider rectilinear acceleration (NO ROTATION)�3

Accelerating coordinates 2

Substitute acceleration in reference frame into Newton’s 2nd law:

Rewriting:

Apply Reynolds Transport Theorem:

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Problem: Cart of initial mass M0 pulled by liquid jet (density ρ) issuing horizontally from tank through opening of area A at constant speed V (neglect friction with track).

Control volume: choose accelerating CV as shownCalculate: speed of cart as it accelerates from rest U(t)y

Calculate x component of momentum equation:

Continuity:

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y

NB: can often neglect the acceleration of u in the CV frame

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Problem: Vertically-directed rocket with initial mass M0 starts from rest, ejects gas with speed Ve at a constant rate mr through an opening of area A, no air resistance.

Control volume: choose accelerating CV as shownCalculate: speed of rocket as it accelerates from rest

Calculate y component of momentum equation:

Assumptions:

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y

x CS at speed V

Ve Y

X

.

0 10 20 30

1000

2000

3000

4000

5000

Time (s)

V (m

/s)

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y

x CS at speed V

Ve Y

X

Macroscopic angular momentum balanceStart with conservation of angular momentum and assume inertial reference frame:

Apply Reynolds Transport Theorem:

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m 4.190 [3Problem: Lawn sprinkler with arms of length R rotating at rate ω; water (density ρ) emerges at vrel at angle α; friction torque at pivot Tfrate Q.

Control volumeCalculate: torque to hold stationary T

Assumptions:

rz

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For jet:

Leaving sprinkler:

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m 4.190 [3

r

Total velocity including component due to sprinkler rotation:

Summary: key equations

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F −

∫

CV

afrρ dV =∂

∂t

∫

CV

vxyzρ dV +

∫

CS

vxyzρvxyz·dA

Linear momentum, accelerating coordinate system:

r × Fs +

∫

M(sys)

r × g dm + Tshaft =∂

∂t

∫

CV

r × vρ dV +

∫

Cs

r × vρv·dA