Applications of Newton

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Physics IClassical Mechanics

• Applications of Newtons Laws

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Calculation Methods
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Simplification

• particle
• negligible magnitude compared to...
  -gtrope, chain with no weight same
  T at both ends
• vector equations become scalar equations in each
  component using diagram
• no friction

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Newtons First Law of Motion

• body in equilibrium 1 2 3
• SFx 0, SFy 0,
  SFz 0

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Newtons Second Law of Motion

• dynamics problem, accelerating bodies
• SFx max , SFy may , SFz maz
• caution is NOT a force!
• circular motion

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The constant-acceleration formulae

• 1.
• 2.
• 3.
• 4.

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Newtons Third Law of Motion

• 1. same magnitude
• 2. opposite direction
• 3. act on different bodies
• 4. no need to only be from the contact surface
• normal force, friction force, ?tension in a
  rope, grav force

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Newtons Law of Gravitation
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Systematic Problem-Solving TechniqueFree-body
Diagram

1. How many bodies? one diagram for one body
2. How many  acting-on-me  forces are there?
   any of them is negligible?
3. Choose the coordinate axes
4. Write a separate equation for each component
   number of equations number of the unknowns
5. Solve!
6. Conclusion does it make sense?
   -gt special (particular) cases,
   critical values, generalisation

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Example

• 5-1 One-dimensional equilibrium
• A gymnast has just begun climbing up a rope
  hanging from a gymnasium ceiling. She stops,
  suspended from the lower end of the rope by her
  hands. Her weight is 500 N, and the weight of the
  rope is 100 N. Analyze the forces on the gymnast
  and on the rope.

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Example

• 5-2 Two-dimensional equilibrium
• A car engine with weight w hangs from a chain
  that is linked at point O to two other chains,
  one fastened to the ceiling and the other to the
  wall. Find the tensions in these three chains,
  assuming that w is given and the weight of the
  chains themselves are negligible.

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Example

• 5-3 An inclined plane
• A car rests on the slanted tracks of a ramp
  leading to a car-transporter trailer. The cars
  brakes and transmission lock are released only a
  cable attached to the car and to the frame of the
  trailer prevents the car from rolling down the
  ramp. If the cars weight is w, find the tension
  in the cable and the force with which the tracks
  push on the car tires.
• And if the car is being pulled up the ramp at
  a constant speed?

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Example

• 5-4 Tension over a frictionless pulley
• Blocks of granite are being hauled up a 15
  slope out of a quarry. For environmental reasons,
  dirt is also being dumped into the quarry to fill
  up old holes. You have been asked to find a way
  to use this dirt to move the granite out more
  easily. You design a system in which a granite
  block on a cart with steel wheels (weight w1,
  including the cart) is pulled uphill on steel
  rails by a bucket of dirt (weight w2, including
  the bucket) dropping vertically into the quarry.
  Ignoring friction in the pulley and wheels and
  the weight of the cable, determine how the
  weights w1 and w2 must be related in order for
  the system to move with constant speed.

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Example

• 5-5 Acceleration in one dimension
• An iceboat is at rest on a frictionless
  horizontal surface. What horizontal force F do we
  need to apply ( along the direction of the
  runners) to give it a velocity of 6.0 m/s at the
  end of 4.0 s? The mass of the iceboat and the
  rider is 200 kg.
• 5-6 Suppose the motion of the iceboat is opposed
  by a constant horizontal friction force with
  magnitude 100 N. Now what force F must we apply
  to give the iceboat a velocity of 6.0 m/s at the
  end of 4.0 s?

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Example

• 5-7 Tension in an elevator cable
• An elevator and its load have a total mass of
  800 kg. The elevator is originally moving
  downward at 10.0 m/s it is brought to rest with
  a constant acceleration in a distance of 25.0 m.
  Find the tension T in the supporting cable while
  the elevator is being brought to rest.
• 5-8 Apparent weight in an accelerating elevator
• A 50.0-kg woman stands on a bathroom scale while
  riding in the elevator. What is the reading on
  the scale?
• 4 cases1 extreme case
• weightlessness excretion of water from RBC-gt
  -volume -gtmotion sickness

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Example as homework

• 5-9 Acceleration down a hill
• A toboggan loaded with vacationing students
  (total weight w) slides down a long, snow-covered
  slope. The hill slopes at a constant angle a, and
  the toboggan is so well waxed that there is
  virtually no friction. What is the toboggans
  acceleration?
• Does the acceleration depend on the total mass?
• How can we show that an object lying on a flat
  floor and a free-falling body are special cases
  of this problem?
• What has this problem to do with the famous
  Galileos experiment?

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Example as homework

• 5-10 Two bodies with the same acceleration
• A robot arm pulls a 4.0-kg cart along a
  horizontal frictionless track with a 0.50-kg
  rope, applying a horizontal force with magnitude
  F 9.0 N to the rope. Find the acceleration of
  the system and the tension at the point where the
  rope is fastened to the cart. (On earth the rope
  would sag a little to void this complication,
  suppose the robot arm is operating in a
  zero-gravity space station.)
• What is a system? How can it simplify things?
• How can we know if the accelerations are the
  same?
• When we say two vectors are the same, it means
  they are the same in bothand?

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Example as homework

• 5-11 Two bodies with the same magnitude of
  acceleration
• An air-track glider with mass m1 moves on a
  level, frictionless air track in the physics lab.
  It is connected to a lab weight with mass m2 by a
  light, flexible, nonstreching string that passes
  over a small frictionless pulley. Find the
  acceleration of each body and the tension in the
  string.
• What does a light string mean? How does it help
  to simplify our problem?
• What does a flexible string mean? How does it
  help to simplify our problem?
• What does a nonstreching string mean? How does
  it help to simplify our problem?
• Suppose one of the mass is zero at a time to
  check if you have correctly solve this problem.

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Example as homework

• 5-12 A simple accelerometer
• A lead fish-line sinker hanging from a string
  attached to point P on the ceiling of a car. When
  the system has an acceleration a toward the
  right, the string makes an angle ß with the
  vertical. In a practical instrument, some form of
  damping would be needed to keep the string from
  swinging when the acceleration changes. Given m
  and ß, what is the acceleration a?
• Why do we call it accelerometer?
• Does the mass attached to the string matter?
• What do we measure in order to know how fast the
  car is gaining speed?
• What kind of engine do you need to see the
  accelerometer going up higher than 45 in a car
  with mass of 1.5 ton?
• One says you can never see the accelerometer
  goes up to the parallel level with the
  ceilingwhat do you think? Is it true
  mathematically? physically?

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Frictional Forces

• the oil in the car engine, the tires and the
  road, air drag parachutes, nails, light bulbs,
  ice hockey, etc.
• Contact force - normal force
• - friction force

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•  When you try to slide a heavy box of books
  across the floor, the box doesnt move at all
  unless you push with a certain minimum force.
  Then the box starts moving, and you can usually
  keep it moving with less force than you needed to
  get it started. If you take some of the books
  out, you need less force than before to get it
  started or keep it moving. 
• What general statements can we make about this
  behavior?

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Kinetic and Static Friction

• Intermolecular forces (electrical) the bonds
  form and break
• Kinetic friction is not perfectly constant
• Smoothing???  cold weld 
• Then how???

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Example 5.13

• You are trying to move a 500-N crate across a
  level floor. To start the crate moving, you have
  to pull with a 230-N horizontal force. Once the
  crate  break loose  and starts to move, you can
  keep it moving at constant velocity with only
  200N. What are the coefficients of static and
  kinetic friction?

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Example 5.16

• Toboggan ride with friction I
• Let s go back to the toboggan we studied in
  Example 5.10 (Section 5.2). The wax has worn off
  and there is now a nonzero coefficient of kinetic
  friction µk . The slope has just the right angle
  to make to toboggan slide with constant speed.
  Derive an expression for the slope angle in terms
  of w and µk.

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Example 5.17

• Toboggan ride with friction II
• The same toboggan with the same coefficient of
  friction as in Example 5.16 accelerates down a
  steeper hill. Derive an expression for the
  acceleration in terms of g, a, µk, and w.

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Rolling Friction

• Tractive resistance
• 0.002 - 0.003 for steel wheels on steel rails
• 0.01 - 0.02 for rubber tires on concrete

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Fluid Resistance and Terminal Speed

• stick your hand out the window of a fast-moving
  car carefully!
• The 3rd law
• The fluid resistance force direction?
  Magnitude?
• diffefent from the kinetic friction force??
• f kv

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• Air drag
• comparing to the rolling resistance
• Terminal speed when you drop a rock into the
  water
• How does the acceleration, velocity, and
  position vary with time?

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Graphs 5.25

• How are the graphs derived?
• When does vy becoms equal to the terminal speed
  vt?

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Terminal speed in air drag
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Example 5.19

• Terminal speed of a skydiver
• For a human body falling through air in a
  spread-eagle position, the numerical value of the
  constant D in Eq. (5.8) is about 0.25 kg/m. Find
  the terminal speed for a lightweight 50-kg
  skydiver.

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Motion in a Circle

•  uniform circular motion 
•  centripetal acceleration 

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Nonuniform Circular Motion

• v is not constant.
• N.B. and

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Dynamics of Circular Motion

•  centrifugal force 

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Bank Curves and the Flight of Airplanes
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Uniform Circular Motion in a Vertical Circle

• Example 5.24
• A passenger on a carnival Ferris wheel moves in
  a vertical circle of radius R with constant speed
  v. The seat remains upright during he motion.
  Find expression for the force the seat exerts on
  the passenger at the top of the circle and at the
  bottem.

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Test Your Understanding of Section 5.4

• Satellites are held in the orbit by the force of
  our planets gravitational attraction. A satellit
  in a small-radius orbit moves at a higher speed
  than a satellite in an orbit of large radius.
  Based on this information, what you can conclude
  about the earths gravitational attraction for
  the satellite?
• (i) It increases with increasing distance from
  the earth.
• (ii) It is the same at all distances from the
  earth.
• (iii) It decreases with increasing distance from
  the earth.
• (iv) This information by itself isnt enough to
  answer the question.

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The Fundamental Forces of Nature

• How many are there?
• What are they?