Welcome to Physics I !!!

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Physics I95.141LECTURE 2211/29/10
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Administrative Notes

• Exam III
• In Class Wednesday, 9am
• Chapters 9-11
• 9 problems posted on Website in Practice Exam
  Section. At least 1 of these problems will be on
  EXAM III.
• If you have questions, start a discussion thread
  on Facebook, that way my response is seen by
  everyone in the class.
• Review Session Tuesday Night, 600pm, OH150

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Outline

• Work by Constant Force
• Scalar Product of Vectors
• Work done by varying Force
• Work-Energy Theorem
• Conservative, non-conservative Forces
• Potential Energy
• Mechanical Energy
• Conservation of Energy
• Dissipative Forces
• Gravitational Potential Revisited
• Power
• Momentum and Force
• Conservation of Momentum
• Collisions
• Impulse
• Conservation of Momentum and Energy
• Elastic and Inelastic Collisions2D, 3D Collisions
• Center of Mass and translational motion
• Angular quantities

• Oscillations
• Simple Harmonic Motion
• What do we know?
• Units
• Kinematic equations
• Freely falling objects
• Vectors
• Kinematics Vectors Vector Kinematics
• Relative motion
• Projectile motion
• Uniform circular motion
• Newtons Laws
• Force of Gravity/Normal Force
• Free Body Diagrams
• Problem solving
• Uniform Circular Motion
• Newtons Law of Universal Gravitation
• Weightlessness
• Keplers Laws

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Review of Lecture 21

• Discussed cross product definition of angular
  momentum and torque
• Why would we ever use cross products instead of
  simpler scalar expressions?
• 3D vectors
• Point masses not moving in uniform circle
• Conservation of Angular Momentum
• Newtons 2nd Law for rotational motion
• No external torque, angular momentum conserved.

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Oscillations (Chapter 14)

• Imagine we have a spring/mass system, where the
  mass is attached to the spring, and the spring is
  massless.

A
vmax
A
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Oscillations

• So we can say that the mass will move back and
  forth (it will oscillate) with an Amplitude of
  oscillation A.
• Can we describe what is going on mathematically?
• Would like to determine equation of motion of the
  mass. In order to do this, we need to know the
  force acting on the block.
• Force depends on position Hookes Law

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Oscillations
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Possible Solutions

• What if xAt?
• What if xAebt?

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Possible Solutions

• What if xAcos(bt)?
• What if xAcos(btF)?

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Possible Solutions

• What if
• If we start with the mass displaced from
  equilibrium by a distance A at t0, then we can
  determine x(t).

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What does motion vs. time look like?

• Plot x at t0, p/2?, p/?, 3p/2?, 2p/?, 5p/2?

x
A
t
-A
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Harmonic Motion terminology

• Displacement Distance from equilibrium
• Amplitude of oscillation max displacement of
  object from equilibrium
• Cycle one complete to-and-fro motion, from some
  initial point back to original point.
• Period Time it takes to complete one full cycle
• Frequency number of cycles in one second

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Simple Harmonic Motion

• A form of motion where the only force on the
  object is the net restoring force, which is
  proportional to the negative of the displacement.
• Such a system is often referred to as a simple
  harmonic oscillator
• The simple harmonic oscillators motion is
  described by

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What is F?
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More terminology

• So the F term is known as the phase of the
  oscillation. It basically shifts the x(t) plot
  in time.
• The term ?, which for a spring mass system, is
  equal to , is known as the angular
  frequency.

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Velocity and Acceleration for SHO

• If we know x(t), we can calculate v(t) and a(t)

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Velocity and Acceleration for SHO

• If

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Example

• A SHO oscillates with the following properties
• Amplitude3m
• Period 2s
• Give the equation of motion for the SHO

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Example

• A SHO oscillates with the following properties
• Amplitude3m
• Period 2s
• At t0s, x3m.
• Give the equation of motion for the SHO

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Example

• A SHO oscillates with the following properties
• Amplitude3m
• Period 2s
• At t0s, x1.5m
• Give the equation of motion for the SHO

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Example

• A SHO oscillates with the following properties
• Amplitude3m
• Period 2s
• At t0s, v2m/s.
• Give the equation of motion for the SHO

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Energy of SHO

• The total energy of a simple harmonic oscillator
  comes from the potential energy in the spring,
  and the kinetic energy of the mass.

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Example

• A spring mass system with m4kg and k400N/m is
  displaced 0.2m from equilibrium and released.
• A) What is the equation of motion for the mass?

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Example

• A spring mass system with m4kg and k400N/m is
  displaced 0.2m from equilibrium and released.
• B) What is the total energy of the system?

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Example

• A spring mass system with m4kg and k400N/m is
  displaced 0.2m from equilibrium and released.
• C) What is the Kinetic Energy and Potential
  Energy of the system at t2s?

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SHO and Circular Motion
y
A

• You will notice that we use the same variable for
  both angular velocity and angular frequency of a
  simple harmonic oscillator.
• If we imagine an object moving with uniform
  circular motion (angular velocity?) on a flat
  surface. Starting, at t0s, at ?0.
• We know that ?(t)?t
• We can write the x-position of the object as
• And the y-position as

?
x
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The pendulum

• A simple pendulum consists of a mass (M) attached
  to a massless string of length L.
• We know the motion of the mass, if dropped from
  some height, resembles simple harmonic motion
  oscillates back and forth.
• Is this really SHO? Definition of SHO is motion
  resulting from a restoring force proportional to
  displacement.

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Simple Pendulum
L

• We can describe displacement as
• The restoring Force comes from gravity, need to
  find component of force of gravity along x
• Need to make an approximation here for small ?

?
?x
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Simple Pendulum
L

• Now we have an expression for the restoring force
• From this, we can determine the effective
  spring constant k
• And we can determine the natural frequency of the
  pendulum

?
?x
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Simple Pendulum
L

• If we know
• We can determine period T
• And we can the equation of motion for
  displacement in x
• or ?

?
?x