Using the Clicker

1
Using the Clicker

• If you have a clicker now, and did not do this
  last time, please enter your ID in your clicker.
• First, turn on your clicker by sliding the power
  switch, on the left, up. Next, store your student
  number in the clicker. You only have to do this
  once.
• Press the button to enter the setup menu.
• Press the up arrow button to get to ID
• Press the big green arrow key
• Press the T button, then the up arrow to get a U
• Enter the rest of your BU ID.
• Press the big green arrow key.

2
Newtons First Law for Rotation

• An object at rest tends to remain at rest, and an
  object that is spinning tends to spin with a
  constant angular velocity, unless it is acted on
  by a nonzero net torque or there is a change in
  the way the object's mass is distributed.
• The net torque is the vector sum of all the
  torques acting on an object.
• The tendency of an object to maintain its state
  of motion is known as inertia. For straight-line
  motion mass is the measure of inertia, but mass
  by itself is not enough to define rotational
  inertia.

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Rotational Inertia

• How hard it is to get something to spin, or to
  change an object's rate of spin, depends on the
  mass, and on how the mass is distributed relative
  to the axis of rotation. Rotational inertia, or
  moment of inertia, accounts for all these
  factors.
• The moment of inertia, I, is the rotational
  equivalent of mass.
• For an object like a ball on a string, where all
  the mass is the same distance away from the axis
  of rotation
• If the mass is distributed at different distances
  from the rotation axis, the moment of inertia can
  be hard to calculate. It's much easier to look up
  expressions for I from the table on page 291 in
  the book (page 10-15 in Essential Physics).

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A table of rotationalinertias
5
The parallel axis theorem

• If you know the rotational inertia of an object
  of mass m when it rotates about an axis that
  passes through its center of mass, the objects
  rotational inertia when it rotates about a
  parallel axis a distance h away is

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Worksheet, part 2

• When a system is made up of several objects, its
  total rotational inertia about a particular axis
  is the sum of the rotational inertias of the
  individual objects for rotation about that axis.
• What is the systems rotational
• inertia in the first case?
• Each block has a mass of m/3, and the rod, of
  negligible mass, has a length L.

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Worksheet, part 2

• When a system is made up of several objects, its
  total rotational inertia about a particular axis
  is the sum of the rotational inertias of the
  individual objects for rotation about that axis.
• What is the systems rotational
• inertia in the first case?

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Worksheet, part 2

• In the second case, do we expect the rotational
  inertia to be larger, smaller, or the same as the
  rotational inertia in the first case?
• What is the systems rotational
• inertia in the second case?

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Worksheet, part 2

• In the second case, do we expect the rotational
  inertia to be larger, smaller, or the same as the
  rotational inertia in the first case? Larger
  the mass is farther from the axis.
• What is the systems rotational
• inertia in the second case?

10
Worksheet, part 2

• In the second case, do we expect the rotational
  inertia to be larger, smaller, or the same as the
  rotational inertia in the first case? Larger
  the mass is farther from the axis.
• What is the systems rotational
• inertia in the second case?

The parallel-axis theorem gives the same result.
11
Newtons Second Law for Rotation

• The equation is the rotational
  equivalent of .
• Torque plays the role of force.
• Rotational inertia plays the role of mass.
• Angular acceleration plays the role of the
  acceleration.

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Applying Newtons Second Law

• A constant force of F 8 N is applied to a
  string wrapped around the outside of the pulley.
  The pulley is a solid disk of mass M 2.0 kg and
  radius R 0.50 m, and is mounted on a horizontal
  frictionless axle. What is the pulley's angular
  acceleration?
• Simulation
• What should we do first?
• Why are we told that the pulley is a solid disk?

13
Applying Newtons Second Law

• A constant force of F 8 N is applied to a
  string wrapped around the outside of the pulley.
  The pulley is a solid disk of mass M 2.0 kg and
  radius R 0.50 m, and is mounted on a horizontal
  frictionless axle. What is the pulley's angular
  acceleration?
• Simulation
• What should we do first?
• Draw a free-body diagram of the pulley.
• Why are we told that the pulley is a solid disk?

14
Applying Newtons Second Law

• A constant force of F 8 N is applied to a
  string wrapped around the outside of the pulley.
  The pulley is a solid disk of mass M 2.0 kg and
  radius R 0.50 m, and is mounted on a horizontal
  frictionless axle. What is the pulley's angular
  acceleration?
• Simulation
• What should we do first?
• Draw a free-body diagram of the pulley.
• Why are we told that the pulley is a solid disk?
• So we know what to use for the rotational
  inertia.

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Applying Newtons Second Law
16
Two pulleys

• Simulation
• We take two identical pulleys, both with string
  wrapped around them. On the one on the left we
  apply an 8 N force to the string. On the one on
  the right we hang an object with a weight of 8 N.
  Which pulley has the larger angular acceleration?

17
Two pulleys

• We take two identical pulleys, both with string
  wrapped around them. On the one on the left we
  apply an 8 N force to the string. On the one on
  the right we hang an object with a weight of 8 N.
  Which pulley has the larger angular acceleration?
  
• The one on the left
• The one on the right
• Neither, they're equal

18
Two pulleys

• For the pulley on the left, the tension in the
  string is 8 N.
• Simulation
• For the system on the right, lets draw the
  free-body diagram of the weight.
• What does the free-body diagram tell us about the
  tension in the string?

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Two pulleys

• For the pulley on the left, the tension in the
  string is 8 N.
• For the system on the right, lets draw the
  free-body diagram of the weight.
• What does the free-body diagram tell us about the
  tension in the string? For the weight to have a
  net force directed down, the tension must be less
  than the force of gravity. So, the tension is
  less than 8 N.

20
Atwoods machine

• Atwoods machine involves one pulley, and two
  objects connected by a string that passes over
  the pulley. In general, the two objects have
  different masses.

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Re-analyzing the Atwoods machine

• When we analyzed Atwoods machine earlier, we
  found an expression for the acceleration of the
  weights in terms of m, M, and g, but we neglected
  the mass of the pulley. When we include the
  pulley in the analysis, we find that

• The acceleration is larger.
• The acceleration is smaller.
• The acceleration is the same.

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Whats the difference?

• The acceleration turns out to be less than what
  we found before, because we need to accelerate
  the pulley.
• In the analysis, we use two different tension
  forces. The tension on the right is larger than
  the tension on the left to give a net clockwise
  torque to accelerate the pulley clockwise.

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Analyzing the lighter object

• Sketch a free-body diagram for the lighter
  object.
• Choose a positive direction, and apply Newtons
  Second Law.

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Analyzing the lighter object

• Sketch a free-body diagram for the lighter
  object.
• Choose a positive direction, and apply Newtons
  Second Law. Lets choose positive to be up, in
  the direction of the acceleration.

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Analyzing the heavier object

• Sketch a free-body diagram for the heavier
  object.
• Choose a positive direction, and apply Newtons
  Second Law.

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Analyzing the heavier object

• Sketch a free-body diagram for the heavier
  object.
• Choose a positive direction, and apply Newtons
  Second Law. Choose positive down this time, to
  match the objects acceleration.

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Analyzing the pulley

• Sketch a free-body diagram for the pulley.
• Choose a positive direction, and apply Newtons
  Second Law for rotation.

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Analyzing the pulley

• The pulley is a uniform solid disk with mass mp
  and radius R.
• Sketch a free-body diagram for the pulley.
• Choose a positive direction, and apply Newtons
  Second Law. Choose positive clockwise, to match
  the pulleys angular acceleration.

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Combine the equations

• Lighter object
• Heavier object
• Pulley

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Combine the equations

• Lighter object
• Heavier object
• Pulley
• Add the equations
• Previous result

31
Rolling

• View 1 View 2 View 3 View 4
• We can view rolling motion as a superposition
  of pure rotation and pure translation.
• For rolling without slipping, the rotational
  speed of the outside of the wheel equals the
  translational speed.
• The net instantaneous velocity at the bottom of
  the wheel is zero, while at the top it is twice
  the translational velocity of the wheel.

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An accelerating car

• You are driving your front-wheel drive car on
  Comm. Ave. You are stopped at a red light, and
  when the light turns green you accelerate
  smoothly so that there is no slipping between
  your car tires and the road. During the
  acceleration period, in what direction is the
  force of friction from the road acting on your
  front tires? Is it static friction or kinetic
  friction?
• 1. The frictional force is kinetic friction
  acting in the direction you are traveling.
• 2. The frictional force is kinetic friction
  acting opposite to the direction you are
  traveling.
• 3. The frictional force is static friction acting
  in the direction you are traveling.
• 4. The frictional force is static friction acting
  opposite to the direction you are traveling.

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An accelerating car

• During the acceleration period, in what direction
  is the force of friction from the road acting on
  your rear tires? Is it static friction or kinetic
  friction?
• 1. The frictional force is kinetic friction
  acting in the direction you are traveling.
• 2. The frictional force is kinetic friction
  acting opposite to the direction you are
  traveling.
• 3. The frictional force is static friction acting
  in the direction you are traveling.
• 4. The frictional force is static friction acting
  opposite to the direction you are traveling.

34
An accelerating car

• Car simulation
• Lets first turn friction off. With no friction
  at all, pushing down on the accelerator makes the
  front wheels spin clockwise. They spin on the
  frictionless surface, the rear wheels do nothing,
  and the car goes nowhere.
• Friction on the front wheels opposes the
  spinning, so it must point in the direction the
  car wants to go. For the front wheels to roll
  without slipping, the friction must be static.
• If we turn on friction to the front wheels only,
  the car accelerates forward with the back wheels
  dragging along the road without spinning.
  Friction opposes this motion, so it must point
  opposite to the way the car is going. Again, it
  must be static friction.
• The static friction force acting on the front
  wheels is the force that accelerates the car
  forward. It is much larger than the friction
  force on the rear wheels, which just has to give
  the rear wheels the correct angular acceleration.

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Whiteboard