CE 201 Statics

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CE 201 - Statics

• Lecture 10

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FORCE SYSTEM RESULTANTS

• So far, we know that for a particle to be in
  equilibrium, the resultant of the force system
  acting on the particle is to be equal to zero.
  That is
• ? F 0
• In a later stage, we are going to see that ? F
  0 is required for a particle to be in
  equilibrium, but it is not sufficient to make
  that particle in equilibrium since we know that
  forces could create momentum which will tend to
  rotate the particle.
• In the coming lectures, moments about a point or
  axis will be discussed.

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MOMENT OF A FORCE SCALAR FORMULATION

• A moment of a force about a point or axis is an
  indication or measure of the tendency of the
  force to rotate the particle about that point or
  axis.
• Fx is perpendicular to the handle of the wench
  and located a distance (dy) from point (O). Fx
  tends to turn the pipe about the z-axis. The
  turning effect will be larger if Fx or dy gets
  larger. This turning effect is sometimes
  referred to as TORQUE, but most frequently it is
  called MOMENT (M0)z

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• If Fz is applied to the wrench at dy from O. Fz
  will tend to rotate the pipe about the x-axis and
  (MO)x is produced.
• The force Fz and distance dy lie in the (y-z)
  plane, which is perpendicular to the x-axis ( the
  moment axis)

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• If Fy is applied to the wrench, no moment will be
  produced at point (O) since the line of action of
  Fy passes through point (O).

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• Generally, the moment of force F about point (O)
  which lies in a plane perpendicular to the moment
  axis, has a magnitude and direction. Therefore,
  the moment is a vector quantity.
• Magnitude
• MO F ? d
• Where d is called the moment arm, which is
  perpendicular distance from (O) to the line of
  action of force F.

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• Direction

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Resultant Moment of a System of Coplannar Forces

• F1, F2, and F3 lie in the (x-y) plane. Their
  distances from point (O) are d1, d2, and d3,
  respectively.
• MO ? (F ? d)

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CROSS PRODUCT

• If we have two vectors A and B, then the CROSS
  PRODUCT of the two vectors is
• C A ? B
• (C is equal to A cross B)
• The magnitude of C is
• C AB sin ?
• The direction of C is perpendicular to
• the plane containing A and B.
• Then
• C A ? B ( AB sin ? ) uC

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Laws of Operation

• (1) Commutative Law
• A ? B ? B ? A BUT A ? B - B ? A

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Laws of Operation

• (2) Multiplying by a Scalar
• a (A ? B) (a A) ? B A ? (aB) (A ? B) a
• (3) Distributive Law
• A ? (B D) (A ? B) (A ? D)

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Cartesian Vector Formulation

• Use C AB sin (?) to find the cross product of
  two vectors.
• i?j k i?k -j i?i 0
• j?k i j?i -k j?j 0
• k?i j k?j -i k?k 0
• i?j ij sin 90 (1)(1)(1) 1
• i?i ii sin (0) 0

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• If A and B are two vectors, then
• A ? B (Ax i Ay j Az k) ? (Bx i By j Bz
  k)
• AxBx (i ? i) AxBy (i ? j) AxBz (i
  ? k)
• AyBx (j ? i) AyBy (j ? j) AyBz (j
  ? k)
• AzBx (k ? i) AzBy (k ? j) AzBz (k
  ? k)
• (AyBz AzBy) i (AxBz AzBx) j
  (AxBy AyBx) k
• This equation can be written as
• i j k
• A ? B Ax Ay Az
• Bx By Bz
• This is nothing but a determinant whose first raw
  consists of unit vectors i, j, and k and its
  second and third raws represent the x, y, and z
  components of the two vectors A and B.

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