Statics

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Statics
It doesnt move, it doesnt spin
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A uniform beam has a mass of 100 kg and a length
of 10 meters. It is supported by a pin that
attaches it to a vertical wall and a cable as
shown. A 30-kg mass hangs from the cable at a
point 3.0 meters from its outer end. What is the
tension in the cable and the force exerted on the
beam by the pin?
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Pick a point about which to compute torques.
Assign CCW direction as positive.
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Let the components of the force exerted by the
pin be Px and Py. Set forces in the x and y
directions equal to zero.
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A ladder has a length of 5.0 meters and a mass of
30 kg. A 80-kg workman leans the ladder against
a frictionless wall such that the base of the
ladder is 1.5 meters from the wall. He then
climbs ¾ up the ladder. What is the force the
ladder exerts on the wall and the floor?
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Sum torques about the base of the ladder.
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Fx is supplied by friction. What is the minimum
coefficient of friction?
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Since the force of tension is tangential to the
pulley the torque it creates is simply Tr. If
the rope does not slip then the linear velocity
and acceleration of a point on the edge of the
pulley are the same as a point on the rope.
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For a pulley with some non-zero moment of inertia
the tensions are not always equal. If they were
the same the pulley could not accelerate.
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Work Kinetic Energy

• Recall the Work/Kinetic Energy Theorem WNET
  ?K
• This is true in general, and hence applies to
  rotational motion as well as linear motion.
• So for an object that rotates about a fixed axis

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Example Disk String

• A massless string is wrapped 10 times around a
  disk of mass M 40 g and radius R 10 cm. The
  disk is constrained to rotate without friction
  about a fixed axis though its center. The string
  is pulled with a force F 10 N until it has
  unwound. (Assume the string does not slip, and
  that the disk is initially not spinning).
• How fast is the disk spinning after the string
  has unwound?

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Disk String

• The work done is W ?? Dq
• The torque is ?? RF (since ? 90o)
• The angular displacement Dq is2? rad/rev x 10
  rev.

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Disk String
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ExampleWork Energy

• Strings are wrapped around the circumference of
  two solid disks and pulled with identical forces
  for the same distance. Disk 1 has a bigger
  radius, but both have the same moment of inertia.
  Both disks rotate freely around axes though
  their centers, and start at rest.
• Which disk has the biggest angular velocity after
  the pull ?

w2
w1
(1) disk 1 (2) disk 2 (3) same
F
F
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Example Solution

• The work done on both disks is the same!
• W Fd

• The change in kinetic energy of each will
  therefore also be the same since W DK.

w2
w1
F
F
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A 30-kg mass is attached to a massless rope that
is wound around a cylinder of radius 10 cm and
mass 50-kg. The mass is released from rest. The
is the acceleration of the mass and the tension
in the rope?
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Use Newtons 2nd Law for the mass.
Compute I for the cylinder
Use Newtons 2nd Law of rotation for the cylinder
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We need to link the translational motion of the
mass to the rotational motion of the cylinder.
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How does a real pulley with non-zero moment of
inertia change the dynamics of the Atwood machine?
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