Summary Lecture 11

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Summary Lecture 11
Rotational Motion 10.5 Relation between angular
and linear variables 10.6 Kinetic Energy of
Rotation 10.7 Rotational Inertia 10.8 Torque 10.9
Newton 2 for rotation 10.10 Work and Power
Tomorrow 12 2 pm PPP Extension
lecture. Room 211 podium level Turn up any time
ProblemsChap.10 6, 7, 16, 21, 28 ,
33, 39, 49
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Relating Linear and Angular variables
q and s
Need to relate the linear motion of a point in
the rotating body with the angular variables
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.
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Kinetic Energy of a rotating body
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What is the KE of the Rotating body?
1/2 MVcm2 ??
It is clearly NOT ½ MV2cm since Vcm 0
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Kinetic Energy of Rotation
Krot ½m1v12 ½m2v2 2 ½m3v3 2 But all these
values of v are different, since the masses are
at different distances from the axis.
However w (angular vel.) is the same for all.
We know that v wr.
So that Krot½m1(wr1)2½m2(wr2)2½m3(wr3)2
? ½miri2w2
Krot ½ w2 ? miri2
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Krot½m(wr1)2½m(wr2)2½m(wr3)2 . . . . . .
? ½miri2w2 ½ w2 ?miri2
Krot½ I w2
Where I is Rotational Inertia or
Moment of Inertia of the rotating body So Krot
½ I w2
(compare Ktrans ½ m v2)
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Rotational inertia
moment of inertia
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Rotational Inertia
I ? mi ri2
I is the rotational analogue of inertial mass
m
For rotational motion it is not just the value of
m, but how far it is from the axis of
rotation. The effect of each mass element is
weighted by the square of its distance from the
axis
The further from the axis, the greater is its
effect.
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Krot ½ I w2
The bigger I , the more KE is stored in the
rotating object for a given angular velocity
A flywheel has (essentially) all its mass at the
largest distance from the axis.
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Some values of rotational inertia for mass M
Mass M on end of (weightless) rod of length R
I ? mi ri2 MR2
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Some values of rotational inertia for mass M
I ? mi ri2 1/2 MR2 1/2 MR2 MR2
Same as mass M on end of rod of length R ...MR2
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Some values of rotational inertia for mass M
Mass M in a ring of radius R
Same as mass M on end of rod, Same as
dumbell...MR2
I ? mi ri2 ? mi R2 MR2
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Rotational Inertia of a thin rod about its centre
For finite bodies
I ?mi ri2
mass M
mass of the rod M ? L
M
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Some Rotational Inertia
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Parallel-axis Theorem
The rotational inertia of a body about any
parallel axis, is equal to its
R.I. about an axis through its CM,
PLUS
R.I. of its CM about a parallel axis through the
point of rotation
I ICM Mh2
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Proof of Parallel-axis Theorem
One rotation about yellow axis involves one
rotation of CM about this axis plus one
rotation of body about CM.
I Icm Mh2
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Example
R
RI of CM about suspension point, distance R away
is MR2. So total RI is 2MR2
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The Story so far...
?, ?, ? relation to linear variables vector
nature
Rotational Variables
Rotational kinematics with const. ?
Analogue equations to linear motion
Rotation and Kinetic Energy
Rotational Inertia
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That's all folks