Physics 121C Mechanics Lecture 22 Energy and Angular Momentum November 29, 2004

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Physics 121C - MechanicsLecture 22Energy and
Angular MomentumNovember 29, 2004

• John G. Cramer
• Professor of Physics
• B451 PAB
• cramer_at_phys.washington.edu

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Announcements

• Homework Assignment 7 has been posted on Tycho
  and is due Wednesday, December 1 (in 2½ days!)
• Regrades from Exam 1 have been completed. The
  Exam 1 grades on Tycho have been updated, and
  the regraded papers are available from Laura
  Clement in room C136 PAB.
• Exam 3 will be held in this room on Friday,
  December 3. Check Tycho for your seat assignment
  before coming to the exam.

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Lecture Schedule (Part 3)
You are here!
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Rotational Dynamics Problems

• MODEL Model the object as a simple shape.
• VISUALIZE Draw a pictorial representation to
  clarify the situation, define coordinates and
  symbols, and list known information.
• Identify the axis about which the object
  rotates.
• Identify forces and determine their distance
  from the axis. For most problems it will be
  useful to draw a free-body diagram.
• Identify any torques caused by the forces and
  the signs of the torques.
• SOLVE The mathematical representation is based on
  Newtons second law for rotational motion

• Find the moment of inertia in Table 13.3 or, if
  needed, calculate it as an integral or by using
  the parallel-axis theorem.
• Use rotational kinematics to find angles and
  angular velocities.
• ASSESS Check that your result has the correct
  units, is reasonable, and answers the question.

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ExampleStarting an Airplane Engine
The engine of a small plane is specified to have
a torque of 60 Nm. This engine drives a 2.0 m
long propeller having a mass of 40 kg. On
startup, how long does it take to for the
propeller to reach 200 rpm? (Assume the
propeller is essentially a long rod.)
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Example An Off-Center Disc
A 10.0 cm diameter disk with a mass of 5.0
kg turns on an axel. A vertical cable attached
to the edge of the disk exerts a 100 N force,
but, initially, a pin is keeping the disk from
rotating. What is the initial angular
acceleration a of the disk after the pin is
removed?
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Constraints ofRopes and Pulleys
The linear velocity vrope of a rope wrapped
around a pulley of radius R is vropewR.
Similarly, the linear acceleration is
aropeaR. If the pulley has an angular
acceleration a and a non-zero moment of inertia
I, then the difference in the tensions DT on the
two sides of the pulley will not zero. In
particular, DT Ia/R
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Example Lowering a Block
A 2.0 kg block is attached to a massless
string that is wrapped around a 1.0 kg cylinder
4.0 cm in diameter. The cylinder rotates on an
axel through its axis of symmetry. The block is
released from rest 1.0 M above the floor.
How long does it take to reach the floor?
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Rigid Body Equilibrium
We now have two versions of Newtons 2nd Law

• If Fnet0 then the body is in translational
  equilibrium, and the center of mass is either at
  rest or moving in a straight line with a
  constant speed. However, the body may be
  rotating.
• If tnet0 then the body is in rotational
  equilibrium, and it is either not rotating or
  rotating with a constant angular speed.
  However, the body may have translational motion.
• The condition for a rigid body to be in total
  equilibrium is for both Fnet0 and tnet0.
  That is, there is no net force and no net torque.

In particular, if a rigid body is in total
equilibrium, there is no net torque about any
selected point of interest in the system.
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Rigid-Body Equilibrium Problems

• MODEL Model the object as a simple shape.
• VISUALIZE Draw a pictorial representation that
  shows all forces and distances. List known
  information.
• Pick any point you wish as a pivot point. The
  net torque about this point is zero.
• Determine the moment arms of all forces about
  this pivot point.
• Determine the sign of each torque about this
  pivot point.
• SOLVE The mathematical representation is based on
  the fact that an object in total equilibrium has
  no net force and no net torque.
• Fnet0 and tnet0.
• Write equations for SFx0, SFy0 and St0.
• Solve the three simultaneous equations.

ASSESS Check that your result is reasonable and
answers the question.
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Example Will the Ladder Slip?
A 3.0 m long ladder leans against a
frictionless wall at an angle of 60O. What
is the minimum value of ms, the coefficient of
static friction with the ground, that will
prevent the ladder from slipping?
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Clicker Question 1
You hold a meter stick with various weights
attached. Which configuration is the most
difficult for your hand to support?
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Rotational Energy
A rotating object has kinetic energy because
almost all parts of it are in motion. We can
calculate the kinetic energy of rotation by
summing over all particles
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ExampleThe Speed of a Rotating Rod
A 1.0 m rod with a mass of 200 g is hinged
at one end and connected to a wall. It is held
horizontally, then released. With what speed
does the tip of the rod hit the wall?
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Rolling Motion
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Particle in a Rolling Object
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Rotation Plus Translation
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Kinetic Energy of Rolling
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The Great Downhill Race
A sphere, a cylinder, and a hoop, all of mass
M and radius R, are released from rest and roll
down a ramp of height h and slope q. They are
joined by a particle of mass M that slides down
the ramp without friction. Who wins the
race? Who is the big loser?
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The Winners
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Vector Description of Rotation

• So far, we have been treating the angular
  velocity w as a scalar with a sign denoting
  whether it is counterclockwise () or clockwise
  (-).
• However, we can also define w as a vector, with
  its direction perpendicular to the plane of
  rotation and given by the right hand rule
• Let the fingers of your right hand curl in the
  direction of rotation
• Then your thumb points in the direction of w.

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Multiplying Vectors
Given two vectors
Dot Product (Scalar Product)
Cross Product (Vector Product)
AB is B times the projectionof A on B, or vice
versa.
(determinant)
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Vector Cross Product
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The Right-Hand Rulefor Cross Products

• The right hand rule (2nd version)
• Let the fingers of your right hand curl in the
  direction from A to B
• Then your thumb points in the direction of AB.

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ExampleCalculating a Cross Product
Vectors C and D are in the plane of the
page. Calculate CD.
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ExampleCross Product with Unit Vectors
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End of Lecture 22

• Before the next lecture, Review Chapters 11 to 13
  for the exam and bring your questions on
  Wednesday.
• Homework Assignment 7 has been posted on Tycho
  and is due Wednesday, December 1 (in 2½ days!)
• Regrades from Exam 1 have been completed. The
  Exam 1 grades on Tycho have been updated, and
  the regraded papers are available from Laura
  Clement in room C136 PAB.
• Exam 3 will be held in this room on Friday,
  December 3. Check Tycho for your seat assignment
  before coming to the exam.