Summary Lecture 9

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Summary Lecture 9
Systems of Particles 9.12 Rocket
propulsion Rotational Motion 10.1 Rotation of
Rigid Body 10.2 Rotational variables 10.4 Rotation
with constant acceleration
Thursday 12 2 pm PPP Extension
lecture. Room 211 podium level Turn up any time
ProblemsChap. 9 27, 40, 71, 73, 78 Chap. 10
6, 11, 16, 20, 21, 28,
28
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momentum conservation and
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Principle of Rocket propulsion In an ISOLATED
System (no external forces) Momentum is conserved
Momentum zero
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Dm
An example of an isolated system where momentum
is conserved!
We found that the impulse (Dp Fdt) given to the
rocket by the gas thrown out the back was
F dt v dm - U dm
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F dt v dm - U dm
This means Every time I throw out a mass dm of
gas with a velocity U, when the rocket has a mass
m, the velocity of the rocket will increase by an
amount dv.
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This means If I throw out a mass dm of gas with
a velocity U, when the rocket has a mass m, the
velocity of the rocket will increase by an amount
dv.
If I want to find out the TOTAL effect of
throwing out gas, from when the mass was mi and
velocity was vi, to the time when the mass is mf
and the velocity vf, I must integrate.
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An example
Mi 850 kg mf 180 kg U 2800 m s-1 dm/dt
2.3 kg s-1
Thrust dp/dt of gas
U dm/dt
2.3 x 2800 6400 N
?F ma Thrust mg ma 6400 8500 ma a
-2100/850 -2.5 m s-2
Initial acceleration F ma gt a F/m
6400/850 7.6 m s-2
Final vel.
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Rotation
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n FIXED
Rotation of a body about an
axis
RIGID
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The orientation of the rigid body is defined by
q. (For linear motion position is defined by
displacement r.)
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The unit of ? is radian (rad)
There are 2? radian in a circle
2? radian 3600 1 radian 57.30
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Angular Velocity
At time t2
At time t1
w is a vector
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Angular Velocity
w
How do we specify its angular velocity?
w is a vector
w is the rotational analogue of v
w is rate of change of q units of wrad s-1
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Angular Acceleration
a is a vector direction same as Dw. Units of a
-- rad s-2 a is the analogue of a
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• Consider an object rotating according to
• ? -1 0.6t 0.25 t2

?
e.g at t 0 ? -1 rad
? d?/dt
? - .6 .5t
e.g. at t0 ? -0.6 rad s-1
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Angular motion with constant acceleration
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An example where ? is constant
3.49 rad s-1
0
8.7 s
? -0.4 rad s-2 Q1 How long to come to rest? Q2
How many revolutions does it take?
15.3 rad 15.3/2??2.43 rev.
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That's all folks
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Relating Linear and Angular variables
Need to relate the linear variables of a point on
the rotating body with the angular variables
q and s
s qr
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Relating Linear and Angular variables
w and v
s qr
w

Not quite true.
V, r, and w are all vectors. Although magnitude
of v wr. The true relation is v w x r
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Direction of vectors
v ? x r
Grab first vector (w) with right hand.
Turn to second vector (r) .
Direction of screw is direction of third vector
(v).
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Vector Product
C A x B
A iAx jAy B iBx jBy
So C (iAx jAy) x (iBx jBy) iAx x
(iBx jBy) jAy x (iBx jBy) ixi
AxBx ixj AxBy jxi AyBx jxj AyBy
now ixi 0 jxj 0 ixj k jxi -k
So C 0 k AxBy - kAyBx
0
C ABsin?
0 - k ABsin?
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Is ? a vector?
Rule for adding vectors The sum of the vectors
must not depend on the order in which they were
added.
However ?? is a vector!
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Relating Linear and Angular variables
a and a
The centripetal acceleration of circular
motion. Direction to centre
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Relating Linear and Angular variables
The acceleration a of a point distance r from
axis consists of 2 terms
Total linear acceleration a
a
r
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The Falling Chimney
The whole rigid body has an angular acceleration a
The tangential acceleration atan distance r from
the base is atan ar
q
At the CM atan aL/2,
and at the end atan aL
But at the CM, atan g cosq (determined by
gravity)
The tangential acceleration at the end is twice
this, but the
acceleration due to gravity of any mass point is
only g cosq.
The rod only falls as a body because it is rigid
..the chimney is NOT.