Statics

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Statics Elasiticity
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Introduction

• Statics a special case of motion.
• Net force and the net torque on an object, or
  system of objects, are both zero.
• The system is not acceleratingit is either are
  rest or its CM is moving at a constant velocity.

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StaticsThe Study of Forces in Equilibrium

• Gravity acts on all objects except those is deep
  space, where it still acts, but can be considered
  negligible for most purposes.
• If the net force on an object is zero, then other
  forces must be acting on it to counteract gravity.

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The book is in equilibrium.
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The Conditions for Equilibrium

• First condition for equilibrium
• F 0

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Example 1

• Pull-ups on a scale.
• A 90-kg weakling cannot do even one pull-up. By
  standing on a scale, he can determine how close
  he gets. His best effort results in a scale
  reading of 23 kg. What force is he exerting?

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Example 2

• Chandelier cord tension.
• Calculate the tensions F1 and F2 in the two
  cords, which are connected to the cord supporting
  the 200-kg chandelier.

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• Net force is zero.
• Net torque is not, so body will rotate.

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The Conditions for Equilibrium

• Second condition for equilibrium
• 0
• If the forces that cause the torque act in the xy
  plane, then the torque is calculated about the z
  axis.

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Conceptual Example 3

• A lever.
• The bar in the figure is being used as a lever
  to pry up a rock. The small rock acts as a
  fulcrum. The force FP required a the long end of
  the bar can be quite a bit smaller than the
  rocks weight Mg, since it is the torques that
  balance in the rotation about the fulcrum. If,
  however, the leverage isnt quite good enough,
  and the rock isnt budged, what are the two ways
  to increase leverage?

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3 Solving Statics Problems

1. Choose a body at a time for consideration. Draw
   a free-body diagram showing all force acting on
   that body and the points at which these forces
   act.
2. Choose a convenient coordinate system and resolve
   the forces into their components.
3. Using letters to represent the unknowns, write
   down equations for SFx 0, SFy 0, St 0.
4. For the St 0 equation, choose any axis
   perpendicular to the xy plane. Give each torque
   a or sign.
5. Solve these equation for the unknowns. If an
   unknown force comes out negative, it means the
   direction you originally choose for that force is
   actually the opposite.

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Example 4

• Tower crane.
• A tower crane must always be carefully balanced
  so that there is no net torque tending to tip it.
  A particular crane at a building site is about
  to lift a 2800-kg air conditioning unit. The
  cranes dimensions are given in the figure. (a)
  Where must the cranes 9500-kg counterweight be
  placed when the load is lifted from the ground?
  (b) Determine the maximum load that can be lifted
  with this counterweight when it is placed at its
  full extent.

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Example 5

• Forces on a beam and supports.
• A uniform 1500-kg beam, 20.0 m long, supports a
  15,000-kg printing press 5.0 m from the right
  support column. Calculate the force on each of
  the vertical support columns.

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A cantilever
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Example 6

• Beam supported by a pin and cable.
• A uniform beam, 2.20 m long with a mass m 25.0
  kg, is mounted by a pin on a wall. The beam is
  held in a horizontal position by a cable that
  makes an angle q 30.0o. The beam supports a
  sign of mass M 280 kg suspended from its end.
  Determine the components of the force FP that the
  pin exerts on the beam, and the tension FT in the
  supporting cable.

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Example 7

• Force exerted by biceps muscle.
• How much force must the biceps muscle exert when
  a 5.0-kg mass is held in the hand (a) with the
  arm horizontal? (b) when the arm is at a 30o?
  Assume that the mass of forearm and hand together
  is 2.0 kg and their CG is as shown.

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Example 8

• Ladder.
• A ladder of length L12 m leans against a wall
  at a point 9.3 m above the ground. The ladder
  has a mass m 45.0 kg and its center of mass is
  L/3 from the lower end. A firefighter of mass
  M72 kg climbs the ladder until her center of
  mass id L/2 from the lower end. What then are the
  magnitudes of the forces on the ladder from the
  wall and the pavement?

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Stability and Balance

• A body in static equilibrium, if left
  undisturbed, will undergo no translational or
  rotational acceleration since the sum of all
  forces and the sum of all torques acting on it
  are zero.
• If object displaced slightly, three possible
  outcomes
• Stable Equilibrium
• Unstable Equilibrium
• Neutral Equilibrium

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Equilibrium

• In general, an object whose CG is below its point
  of support will be in stable equilibrium
• A body whose CG is above its base of support will
  be stable if a vertical line projected downward
  from the CG falls within the base of support.

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Levers and Pulley

• Mechanical Advantage of a lever

• Fl Fl

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Levers and Pulley

• Mechanical Advantage of a pulley

• F/F x/x

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Elasticity and Elastic Modulii Stress and Strain

• Study effects of forces on objects in
  equilibrium.
• If such forces are strong enough, the object will
  break, or fracture.
• If the amount of elongation, DL, is small
  compared to the length of the object, experiment
  shows that DL is proportional to the weight or
  force exerted on the object.
• F k DL

Sometimes called Hooks law.
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Elastic Limit

• This law is found to be valid for almost any
  solid material.
• But valid only up to a point.
• If the force is too great, the object stretches
  excessively and eventually breaks.
• The maximum force that can be applied without
  breaking is called the ultimate strength.

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Elastic Modulus

• The amount of elongation depends not only on the
  force, but also on the material. Experiment
  shows
• DL L0
• where L0 is the original length of the object, A
  is the cross-sectional area, and DL is the change
  in length due to the applied force, and Y is a
  constant of proportionality known as the elastic
  modulus, or Youngs modulus.

1 Y
F A
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Stress and Strain

• stress
• strain
  
• Y

force
F A
area
DL L0
change in length
original length
stress strain
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Example 9

• Tension in a piano wire.
• A 1.60-m-long steel piano wire has a diameter of
  0.20 cm. How great is the tension in the wire if
  it stretches 0.30 cm when tightened?

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Tensile Stress

• Tensile stress the rod in the figure is under
  tension or tensile stress.
• Not only is the force pulling down on the rod at
  its lower end, but since the rod is in
  equilibrium the support at the top is exerting an
  equal upward force.
• Tensile stress exists throughout the material.

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Compressive Stress

• Compressive stress is the exact opposite.
  Instead of being stretched, the material is
  compressed the forces act inwardly on the body.
  

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Shear Stress

• An object under shear stress has equal and
  opposite forces applied across its opposite faces.

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Calculating Shear Strain

• Dx
  h
• where S is called the shear modulus.

F A
1 S
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Bulk Stress

• If an object is subject to forces from all sides,
  its volume will decrease.
• A common example is a body submersed in a liquid.
• -
  DP
• or B -

DV V0
1 B
DP
DV/V0
where pressure is force per unit area, and B is
the bulk modulus.
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Fracture

• If the stress on a solid object is too great, the
  object fractures or breaks.

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Example 10

• When water freezes, it expands by 9.00. What
  pressure increase would occur inside your
  automobile engine block if the water froze? The
  bulk mo9dulus for ice is 2.00 x 109 N/m2.

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