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4.1 Rotational kinematics4.2 Moment of inertia4.3 Parallel axis theorem 4.4 Angular momentum and
rotational energy
CHAPTER 4: ROTATIONAL MOTION
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Part 1
Rotational kinematics
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Riview of rotations• Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one complete revolution every two seconds.• Klyde’s angular velocity is:
(a)(a) the same as Bonnie’s (b)(b) twice Bonnie’s(c)(c) half Bonnie’s
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REVIEW of ANGLE VELOCITY• The angular velocity of any point on a solid object rotating about a fixed axis is the same.
– Both Bonnie & Klyde go around once (2pi radians) every two seconds.
Their “linear” speed v will be different since v = r.
BonnieKlyde V21V
How about their “linear” speed v ?The same or different? ?
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Review: Rotational Variables.
• Rotation about a fixed axis:– Consider a disk rotating about
an axis through its center:
• First, recall what we learned aboutUniform Circular Motion:
(Analogous to )
dtd
dtdxv
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Rotational Variables...
• Now suppose can change as a function of time:• We define the angular acceleration:
2
2
dtd
dtd
Consider the case when
is constant. We can integrate this to find and as a function of time:
t0 constant
200 t
21t
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Rotational Variables...
• Recall also that for a point at a distance R away from the axis of rotation:– x = R (distance in the circle)– v = R
And taking the derivative of this we find:– a = R
R
v
x
t0 constant
200 t
21t
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Example: Wheel And Rope
• A wheel with radius R = 0.4 m rotates freely about a fixed axle. There is a rope wound around the wheel. Starting from rest at t = 0, the rope is pulled such that it has a constant acceleration a = 4 m/s2. How many revolutions has the wheel made after 10 seconds? (One revolution = 2 radians)
aa
R
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Solution
• Use a=R to find : = a / R = (4 m/s2 )/ 0.4 m = 10 rad/s2
• Now use this equations just as you would use the kinematic equations from the beginning of the semester.
times80radrot
21 xrad 500rev
200 t
21t = 0 + 0(10) + (10)(10)2 = 500 rad
21
aa
R
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Part 2
Moment of Inertia
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Rotation & Kinetic Energy
• Consider the simple rotating system shown below. (Assume the masses are attached to the rotation axis by massless rigid rods).
• The kinetic energy of this system will be the sum of the kinetic energy of each piece:
rr1
rr2rr3
rr4
m4
m1
m2
m3
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Compute: Kinetic Energy of Rotation system
• So: but vi = ri K m vi ii
12
2
K m r m ri ii
i ii
12
12
2 2 2
which we write as:
K 12
2I
I m ri ii
2
Define the moment of inertiamoment of inertiaabout the rotation axis
rr1
rr2rr3
rr4
m4
m1
m2
m3
vv4
vv1
vv3
vv2
I has units of kg m2.
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Rotation & Kinetic Energy...
• The kinetic energy of a rotating system looks similar to that of a point particle:
Point ParticlePoint Particle Rotating SystemRotating System
I m ri ii
2
K 12
2I K mv12
2
v is “linear” velocitym is the mass.
is angular velocityI is the moment of inertiaabout the rotation axis.
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Moment of Inertia
• Notice that the moment of inertia I depends on the distribution of mass in the system.• The further the mass is from the rotation axis, the bigger the moment of inertia.
* For a given object, the moment of inertia will depend on where we choose the rotation axis (unlike the center of mass).* In rotational dynamics, the moment of inertia I appears in the same way that mass m does when we study linear dynamics!
K 12
2I I m ri ii
2 So where
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Calculating Moment of Inertia
• We have shown that for N discrete point masses distributed about a fixed axis, the moment of inertia is:
I m ri ii
N2
1where ri is the distance from the mass i to the axis of rotation.
Example: Calculate the moment of inertia of four point masses (m) on the corners of a square whose sides have length L, about a perpendicular axis through the center of the square:
mm
mm
L
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Calculating Moment of Inertia...
• The squared distance from each point mass to the axis is:
2L
2L2r
222
mm
mm
Lr
L/2
2Lm4
2Lm
2Lm
2Lm
2LmrmI
22222N
1i
2ii
so
I = 2mL2
Using the Pythagorean Theorem
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Learning check
• Now calculate I for the same object about an axis through the center, parallel to the plane (as shown):
mm
mm
L
r
4Lm4
4Lm
4Lm
4Lm
4LmrmI
22222N
1i
2ii
I = mL2
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Calculating Moment of Inertia...
• Finally, calculate I for the same object about an axis along one side (as shown):
mm
mm
L
r
2222N
1i
2ii 0m0mmLmLrmI
I = 2mL2
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Calculating Moment of Inertia...
• For a single object, I clearly depends on the rotation axis!!
L
I = 2mL2I = mL2
mm
mm
I = 2mL2
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Check: Moment of Inertia
• A triangular shape is made from identical balls and identical rigid, massless rods as shown. The moment of inertia about the a, b, and c axes is Ia, Ib, and Ic respectively.– Which of the following is correct:
(a)(a) Ia > Ib > Ic
(b)(b) Ia > Ic > Ib
(c)(c) Ib > Ia > Ic
a
b
c
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Solution
a
b
c
m
m m
L
L
Masses m and lengths L as show:
Ia m L m L mL 2 2 82 2 2
Calculate moments of inerta:
Ib mL mL mL mL 2 2 2 23
Ic m L mL 2 42 2
So (b) is correct: Ia > Ic > Ib
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Calculating Moment of Inertia... For a continuous solid object
• For a discrete collection of point masses we found:
• For a continuous solid object we have to add up the mr2 contribution for every infinitesimal mass element dm.
• We have to do anintegral to find I :
I m ri ii
N2
1
r
dm
I r dm2
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Learn by heartMoments of Inertia
• Some examples of I for solid objects:Thin hoop (or cylinder) of mass M and radius R, about an axis through its center, perpendicular to the plane of the hoop.
I MR 2
R
Thin hoop of mass M and radius R, about an axis through a diameter.
I 12
2MRR
2222 MRdmRdmRdmrI
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Moments of Inertia...
• Some examples of I for solid objects:
Solid sphere of mass M and radius R, about an axis through its center.
I 25
2MR
R
I 12
2MR
RSolid disk or cylinder of mass M and radius R, about a perpendicular axis through its center.
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Moment of Inertia
• Two spheres have the same radius and equal masses. One is made of solid aluminum, and the other is made from a hollow shell of gold.– Which one has the biggest moment of inertia about an axis through its center?
same mass & radius
solid hollow
(a) solid aluminum (b) hollow gold (c) same
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Hint
• Moment of inertia depends on mass (same for both) and distance from axis squared, which is bigger for the shell since its mass is located farther from the center.– The spherical shell (gold) will have a bigger moment of inertia.
same mass & radius
ISOLID < ISHELL
solid hollow
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Moments of Inertia...
• Some examples of I for solid objects (see also Tipler, Table 9-1):
Thin rod of mass M and length L, about a perpendicular axis through its center.
I 1
122ML
L
Thin rod of mass M and length L, about a perpendicular axis through its end.
I 13
2ML
L
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Part 3
Parallel Axis Theorem
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Parallel Axis Theorem
• Suppose the moment of inertia of a solid object of mass M about an axis through the center of mass, ICM, is known.• The moment of inertia about an axis parallel to this axis but a distance D away is given by:
IPARALLEL = ICM + MD2
So if we know ICM , it is easy to calculate the moment of inertia about a parallel axis.
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Parallel Axis Theorem: Example
• Consider a thin uniform rod of mass M and length D. Figure out the moment of inertia about an axis through the end of the rod.
IPARALLEL = ICM + MD2
ICM ML1
122We know
IEND ML M L ML
112 2
13
22
2So
which agrees with the result on a previous slide.
L
D=L/2 Mx
CM
ICMIEND
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Part 4
Angular momentum and rotational energy
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Angular momentum
Defination:
CM2
ii
i
ii
iii
i
Irm
])vm[xr(LLPxrL
Example: compute L for the system m=100g, a=10cm, =(2/5) rad/s
a
mL=2ma2. =2x0.1x 10-4 x0.1
=2 10-6.
r
v
L
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Complete Motion by linear + rotation
• The total kinetic energy of a system of particles include 2 parts
2CM
2iiNET MV
21um
21K
KR KCM For a solid object rotating about its center of mass, we
now see that the first term becomes:
2CM
2CMTOT MV
21I
21K
2iiR um
21K Substituting ii ru
2ii
2R rm
21K but CM
2ii rm I
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Connection with CM motion...
• So for a solid object which rotates about its center or mass and whose CM is moving:
2CM
2CMNET MV
21I
21K
VCM
We will use this formula more in coming lectures.
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Similarity Between Linear and Rotational MotionsAll physical quantities in linear and rotational motions show striking similarity.
Quantities Linear RotationalMass Mass Moment of Inertia
Length of motion Distance Angle (Radian)Speed
AccelerationForce Force TorqueWork Work WorkPower
MomentumKinetic Energy Kinetic Rotational
2I mr
rvt
t
vat
t
maF IcosW Fd
vFP P
2
21 mvK 2
21 IKR
LM
W
vmp IL
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Review of today’s lecture
• Rotational Kinematics – Analogy with one-dimensional kinematics
• Kinetic energy of a rotating system – Moment of inertia – Discrete particles Continuous solid objects
• Parallel axis theorem
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Problem
Use the law of energy conservation to compute the velocity of the ball when it hits to the ground.
Hint : There are 2 motions
(frictionless case)
H=3m
V0=0, 0=0
M=2kg, r=4cm
A=45 Deg
Change the ball by Cylinder have the same M and r