Moment of a Force

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Moment of a Force

• The tendency of a force to produce rotation about
  some axis is called the moment of a force.
• The magnitude of this tendency is equal the
  magnitude of the force times the perpendicular
  distance between the axis and the line of action
  of the force (moment arm). M F d
• Unit force x distance FL lb-in, k-in, lb-ft,
  k-ft N-m, kN-m (SI unit)
• Sign conven. clockwise (-), counterclockwise ()

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Varignons Theorem or the Principle of Moments

• The algebraic summation of of the components of a
  force with respect to any point is equal to the
  moment of the original force.

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Couples

• Two parallel forces having different lines of
  action, equal in magnitude, but opposite in sense
  constitute a couple.
• A couple causes rotation about an axis
  perpendicular to its plane. M F d
• The moment of a couple is independent of the
  choice of the axis of moment (moment center)
• A couple cannot be replaced with a single
  equivalent resultant force
• A couple may be transferred to any location in
  its plane and still have the same effect

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Resolution of a force into a force and couple
acting at another point

• Any force F acting on a rigid body may be moved
  to any given point A (with a parallel line of
  action), provided that a couple M is added. The
  moment M of the couple is equal to F times the
  perpendicular distance between the original line
  of action and the new location A.

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Resultant of two parallel forces

• The magnitude of the resultant R of the parallel
  forces A and B equals the algebraic summation of
  A and B, where R A B.
• Location of the resultant R is obtained by the
  principle of moments.

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Equilibrium Equations

• Equilibrium with zero motion, both the body and
  the entire system of external forces and moments
  acting on the body are in equilibrium.
• Conditions of equilibrium the resultant of a
  force system must be zero if the force system is
  to be in equilibrium
• The algebraic sum of all forces (or components of
  forces) along any axis must be equal to zero (? F
  0)
• The algebraic sum of the moments of the forces
  about any axis or point must be equal to zero (?
  M 0)
• In two dimensional case ? Fx 0, ? Fy 0, ? M
  0

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Equilibrium of Force System

• Collinear if we assume that all the forces are
  along the horizontal axis then, ? Fx 0
• Concurrent since the action lines of all forces
  intersect at a common point, this system cannot
  cause rotation of the body on which it acts,
  therefore, only two equations of equilibrium are
  sufficient for analyzing this type of force
  system, ? Fx 0, ? Fy 0

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Equilibrium of Force System

• Parallel if we assume the forces are parallel to
  vertical axis then, ? Fy 0, ? M 0
• Nonconcurrent coplanar
• ? Fx 0, ? Fy 0, ? M 0

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The Free Body Diagram (FBD)

• To isolate a body and identify the force system
  acting on the body so that unknown forces can be
  determined is known as the FBD.
• The external forces acting on the free body may
  be direct forces due to contact between the free
  body and other bodies external to it or indirect
  forces, such as gravitational or magnetic forces,
  which act without bodily contact.

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Statistical Indeterminacy and Improper Constraints

• When the number of unknown forces exceeds the
  number of equations of equilibrium, the rigid
  body is said to be statically indeterminate.
• When the support forces are sufficient to resist
  translation in both the x and y directions as
  well as rotational tendencies about any point,
  the rigid body is said to be completely
  constrained, otherwise the rigid body is unstable
  or partially constrained.

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