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10.4 Rotational Kinetic Energy
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An object rotating about some axis with an
angular speed, , has rotational kinetic
energy even though it may not have any
translational kinetic energy
Each particle has a kinetic energy of
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Ki = 1/2 mivi2
Since the tangential velocity depends on the
distance, r, from the axis of rotation, we can
substitute vi = i r
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Fig 10.6
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Rotational Kinetic Energy, cont
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The total rotational kinetic energy of the
rigid object is the sum of the energies of
all its particles
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Where I is called the moment of inertia
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Rotational Kinetic Energy, final
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There is an analogy between the kinetic
energies associated with linear motion (K =
1/2 mv 2) and the kinetic energy associated
with rotational motion (KR= 1/2 I2)
Rotational kinetic energy is not a new type of
energy, the form is different because it is
applied to a rotating object
The units of rotational kinetic energy are
Joules (J)
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Moment of Inertia
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The definition of moment of inertia is
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The dimensions of moment of inertia
are ML2 and its SI units are kg.m2
We can calculate the moment of inertia
of an object more easily by assuming it
is divided into many small volume
elements, each of mass Dmi
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Moment of Inertia, cont
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We can rewrite the expression for I in terms
of Dm
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With the small volume segment assumption,
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If r is constant, the integral can be evaluated
with known geometry, otherwise its variation
with position must be known
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Moment of Inertia of a Uniform
Solid Cylinder
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Divide the cylinder
into concentric
shells with radius r,
thickness dr and
length L
Then for I
Fig 10.8
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Fig 10.7
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10. 5 Torque
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The moment arm, d,
is the perpendicular
distance from the
axis of rotation to a
line drawn along the
direction of the force
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d = r sin 
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10.5 Definition of Torque
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Torque, t, is the tendency of a force to
rotate an object about some axis
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Torque is a vector
t = r F sin  = F d
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F is the force
 is the angle between the force and the
horizontal (the line from the axis to the position
of the force)
d is the moment arm (or lever arm)
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Torque, cont.
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The horizontal component of the force
(F cos ) has no tendency to produce a
rotation
Torque will have direction
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If the turning tendency of the force is
counterclockwise, the torque will be
positive
If the turning tendency is clockwise, the
torque will be negative
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Torque Unit
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The SI unit of torque is N.m
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Although torque is a force multiplied by a
distance, it is very different from work and
energy
The units for torque are reported in N.m
and not changed to Joules
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Torque as a Vector Product
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Torque is the vector
product or cross
product of two
other vectors
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Vector Product, General
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Given any two vectors,
and
The vector product
is defined as a
third vector,
whose
magnitude is
The direction of C is
given by the right-hand
rule
Fig 10.13
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Properties of Vector Product
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The vector product is not commutative
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If is parallel (q = 0o or 180o) to
then
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If
This means that
is perpendicular to
then
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Vector Products of Unit
Vectors
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The signs are interchangeable
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For example,
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Net Torque on an object
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The force F1 will tend to
cause a
counterclockwise
rotation about O
The force F2 will tend to
cause a clockwise
rotation about O
tnet = t1 + t2 = F1d1 –
F2d2
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10.6 Force vs. Torque
Forces can cause a change in
linear motion, which is
described by Newton’s
Second Law F = Ma.
Torque can cause a change in
rotational motion, which is
described by the equation t = I a.
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The Rigid Object In Equilibrium
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The net external force must be equal zero

F = 0
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The net external torque about any axis must
be equal zero

t
= 0
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Fig 10.16(b) & (c)
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10.7 Rotational motion of a
rigid object under a net torque
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The magnitude of the torque produced by a
force around the center of the circle is
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The tangential acceleration is related to the
angular acceleration
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t = Ft r = (mat) r
St = S(mat) r = S(mra) r = S(mr 2) a
Since mr 2 is the moment of inertia of the
particle,
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St = Ia
The torque is directly proportional to the angular
acceleration and the constant of proportionality is
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the moment of inertia
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Fig 10.18(a) & (b)
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Work in Rotational Motion
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Find the work done by a
force on the object as it
rotates through an
infinitesimal distance ds = r
dq
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The radial component of the
force does no work
because it is perpendicular
to the displacement
Fig 10.19
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Work in Rotational Motion,
cont
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Work is also related to rotational kinetic
energy:
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This is the same mathematical form as the
work-kinetic energy theorem for translation
If an object is both rotating and
translating, W = DK + DKR
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Power in Rotational Motion
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The rate at which work is being done in
a time interval dt is the power
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This is analogous to P = Fv in a linear
system
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