FORCES AND MOTION

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FORCES AND MOTION
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Equilibrium

• Forces in 2 dimensions. An in-depth look at
  Newtons first law of motion.

• Determine the force that produces equilibrium
  when three or more forces act on an object.
• Analyze the motion of an object on an inclined
  plane with and without friction.

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Where is this used?
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Projectile Motion

• Looking at the motion of objects when fired
  horizontally or at an angle.

• Recognize that the vertical and horizontal
  motions of a projectile are independent.
• Relate the height, time in the air, and initial
  vertical velocity of a projectile using its
  vertical motion, then determine the range.
• Explain how the shape of the trajectory of a
  moving object depends upon the frame of reference
  from which it is observed.

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Where is this used?
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Uniform Circular Motion

• Study of objects moving in circles.

• Explain the acceleration of an object moving in a
  circle at constant speed
• Describe how centripetal acceleration depends
  upon the objects speed and the radius of the
  circle.
• Recognize the direction of the force that causes
  centripetal acceleration
• Explain how the rate of circular motion is
  changed by exerting torque on it.

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Where is this used?
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Lets begin with equilibrium

• Recall Newtons 1st law of motion A body at
  rest remains at rest, and a body in motion
  remains in motion, unless acted on by an external
  unbalanced force.
• Newtons 3rd law of motion To every action
  there must be an equal and opposite reaction.

• This means that there can never be a single
  isolated force.
• Acting and reacting forces, though equal in
  magnitude and opposite in direction, can never
  neutralize each other because they always act on
  different objects.
• In order for two forces to neutralize each other
  they must act on the same object.

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Some situations for the 3rd law
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Terms

• Remember that the resultant force was defined as
  a single force whose effect is the same as a
  given system of forces. Therefore, if the
  tendency of a number of forces is to cause
  motion, the resultant will also produce this
  tendency.

• A condition of equilibrium is found to exist if a
  given set of forces indicates that there is no
  resultant force and that each force is balanced
  by all remaining forces.
• Therefore, according to Newtons 1st law, a body
  in equilibrium must be either at rest or in
  motion with constant speed since there is no
  unbalanced force.

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Terms

• If the line of action of all the forces acting on
  a body lie in a single plane, the system of
  forces is coplanar.

• Noncoplanar forces are also commonplace. For
  example, you are probably sitting on a set of
  noncoplanar forces right now. The four forces
  supporting the legs of a chair are not coplanar.
• Forces not confined to a common plane require
  more effort to analyze.

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Terms

• If the lines of action of two or more forces
  intersect at a common point then the forces are
  concurrent.

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Terms

• Mechanics is concerned with the position and
  motion of matter in space and is what we are and
  have been studying.
• Mechanics is sometimes divided into two parts

• Dynamics a description of motion and a treatment
  of its causes.
• Statics a study of the physical phenomena
  associated with bodies at rest or moving at
  constant speed.

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Statics is the study of structures

• i.e. the analysis of the forces acting on and
  within skyscrapers, towers, wrenches, crutches,
  and even people. Structurally, the human body
  shares remarkable similarities with high-rise
  buildings and trestle bridges. About 500 yrs.
  ago da Vinci discovered that the bones and
  muscles of vertebrates form a system of levers.
  We move most of our body parts (fingers, arms,
  neck, back, etc) using muscles to pivot one bone
  against another.

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For example

• A foot on tiptoe. Ligaments hold the bones
  together, and tendons attach the muscles to the
  bone. Think of the foot pivoted at the ankle
  joint, with the Achilles tendon pulling upward.
• The foot can be modeled, and, as well see
  presently, the forces in the tibia and tendon can
  be computed.

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Point of intersection

• Are these forces acting on the foot coplanar?
  Yes
• Are these forces acting on the foot concurrent?
  Yes

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Now lets look at equilibrium.

• There are two parts to equilibrium which we will
  look at.
• Translational equilibrium and
• Rotational equilibrium.

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• Translational equilibrium (a.k.a.) the first
  condition of equilibrium
• For an object to be in translational equilibrium
  the sum of all the forces acting on the object
  must be zero. (Notice, this is Newtons 1st law
  of motion)

• Formal statement
• A body is in translational equilibrium if and
  only if the vector sum of the forces acting upon
  it is zero.
• Mathematically that means

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Lets begin with a problem to work.

• A block of weight 200 hangs from a cord which is
  knotted to two other cords, A and B, fastened to
  the ceiling. If cord B makes an angle of 600
  with the ceiling and cord A forms a 300 angle
  find the tension in all three cords.

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First heres the diagram.

• Notice that the forces are coplanar and
  concurrent.
• Locate the point of concurrency and draw a small
  circle around it.
• This point of concurrency will become the origin
  for our free-body diagram.

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• Now looking at the common point of intersection,
  determine in which direction the forces A, B, and
  W are acting on the point.
• Sketch theses vectors.

point
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Set up the F.B.D. with the origin of the x-y
plane at the know
y
B
A
300
600
x
W 200
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Because the object is at rest, we know that it is
now in translational equilibrium.

• -A cos 300
• B cos 600
• -A cos 300 B cos 600 0

• A sin 300
• B sin 600
• -200
• A sin 300 B sin 600 200

Two equations, two unknowns. Use your program to
solve for A and B.
A 100 B 173.21
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A boom problem

• Find the forces in each rope and in the boom.
• A boom may also be called a beam.

rope
rope
boom
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• First find the point of concurrency. (B)
• Draw a circle around this point.
• From here you need to decide how each force is
  acting.
• Notice that the force from the boom is pushing
  upward.

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Set up F.B.D.

• Notice that force vector C is in quad. 1 and
  force vector D is in quad. 3.
• Because the object is at rest, we know that we
  are in translational equilibrium.

y
C
480
350
x
D
2000 nts
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• C cos 480
• -D cos 350
• C cos 480 D cos 350 0

• C sin 480
• -D sin 350
• -2000
• C sin 480 - D sin 350 2000

Two equations, two unknowns, use your program to
find C and D
C 7282.9 nts D 5949.1 nts A 2000 nts
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A few for you

• Find angle q in the arrangement shown. Assume
  the pulley is weightless and frictionless.

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• - A cos q A sin q
• 200 cos 200 200 sin 200
• - 150

y
200 nts
A
q
x
150 nts
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Since we are looking for q a slightly different
approach is needed. In this case we will use the
fact that
-A cos q A sin q 200
cos 200 200 sin 200
-150
-A cos q 200 cos 200 0
A sin q 200 sin 200 150
A cos q 187.94
A sin q 150 200 sin 200 A sin q 81.6