MID-TERM II

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MID-TERM II MOVED TO NOVEMBER 14th
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The center of mass of a system of masses is the
point where the system can be balanced in a
uniform gravitational field.
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Center of Mass for Two Objects Xcm (m1x1
m2x2)/(m1 m2) (m1x1 m2x2)/M

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Locating the Center of Mass
In an object of continuous, uniform mass
distribution, the center of mass is located at
the geometric center of the object. In some
cases, this means that the center of mass is not
located within the object.
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Suppose we have several particles A, B, etc.,
with masses mA, mB, . Let the coordinates of A
be (xA, yA), let those of B be (xB, yB), and so
on. We define the center of mass of the system
as the point having coordinates (xcm,ycm) given
by xcm (mAxA mBxB .)/(mA mB
), Ycm (mAyA mByB .)/(mA mB
).
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The velocity vcm of the center of mass of a
collection of particles is the mass-weighed
average of the velocities of the individual
particles vcm (mAvA mBvB .)/(mA
mB ). In terms of components,
vcm,x (mAvA,x mBvB,x .)/(mA mB
), vcm,y (mAvA,y mBvB,y .)/(mA
mB ).
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For a system of particles, the momentum P is the
total mass M mA mB times the velocity vcm
of the center of mass Mvcm mAvA mBvB
P It follows that, for an isolated system, in
which the total momentum is constant the velocity
of the center of mass is also constant.
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Acceleration of the Center of Mass Let acm be
the acceleration of the cener of mass (the rate
of change of vcm with respect to time) then Macm
mAaA mBaB The right side of this
equation is equal to the vector sum SF of all
the forces acting on all the particles. We may
classify each force as internal or external. The
sum of forces on all the particles is then SF
SFext SFint Macm
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CHAPTER 9 ROTATIONAL MOTION
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Goals for Chapter 9

• To study angular velocity and angular
  acceleration.
• To examine rotation with constant angular
  acceleration.
• To understand the relationship between linear and
  angular quantities.
• To determine the kinetic energy of rotation and
  the moment of inertia.
• To study rotation about a moving axis.

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• Angular displacement ? (radians, rad).
• Before, most of us thought in degrees.
• Now we must think in radians. Where 1 radian
  57.3o or 2p radians360o .
• Try to convert some common angles ( 45o, 90o,
  360o).

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Unit rad/s2
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MID-TERM II MOVED TO NOVEMBER 14th
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Relationship Between Linear and Angular Quantities
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v r? atan ra arad r?2
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Kinetic Energy of Rotating Rigid BodyMoment of
Inertia

• KA (1/2)mAvA2
• vA rA ? vA2 rA2 ?2
• KA (1/2)(mArA2)?2
• KB (1/2)(mBrB2)?2
• KC (1/2)(mCrC2)?2
• ..
• K KA KB KC KD .
• K (1/2)(mArA2)?2 (1/2)(mBrB2)?2 ..
• K (1/2)(mArA2) (mBrB2) ?2
• K (1/2) I ?2
• I mArA2 mBrB2 mCrC2) mDrD2
• Unit kg.m2

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Rotational energy
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Moments of inertia rotation
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Rotation about a Moving Axis

• Every motion of of a rigid body can be
  represented
• as a combination of motion of the center of mass
  (translation) and rotation about an axis through
  the center of mass
• The total kinetic energy can always be
  represented as the sum of a part associated with
  motion of the center of mass (treated as a point)
  plus a part asociated with rotation about an axis
  through the center of mass

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Total Kinetic Energy

• Ktotal (1/2)Mvcm2 (1/2)Icm?2

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A rotation while the axis moves
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Race of the objects on a ramp