Applying Newtons Laws

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Chapter 5

• Applying Newtons Laws

2
Overview

• Newtons three laws of motion (the basis for
  classical mechanics) can be stated very simply.
• Applying them to situations, however, can be very
  tricky indeed.
• We will begin with objects in equilibrium, then
  move onto objects not in equilibrium.
• We will study friction and its effect on an
  objects motion.
• We will study object in uniform circular motion.

3
Objectives

• SC.AP1.50.02
• The student will apply Newtons Laws of Motion
  (including friction and centripetal force) to
  static equilibrium, dynamics, and multi-body
  systems. (CS 5.12.1, CLG 5.1.3)

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Particles in Equilibrium

• Equilibrium ? when an object is at rest or at a
  constant velocity in an inertial referance frame
• Vector Form ?
• We will most often use the component form
• Some equilibrium problems may seem complicated,
  but it is important to remember that all problems
  dealing with particles in equilibrium are done
  the same way.

5.1
5.2
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Problem Solving Strategy (PSS) Equilibrium of a
Particle

• Identify
• The problem must satisfy equilibrium
• Use Newtons third law to describe forces on more
  than one body
• Identify target variable (why you are doing
  problem)
• Set Up
• Draw a sketch of the situation, showing
  dimensions and angles
• Draw a free body diagram for objects in
  equilibrium. For now, use a force diagram (dot
  and forces on it)

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PSS Particles in Equilibrium

• Include all known or calculated information, such
  as angles, magnitudes and weight. Be sure to
  accurately represent the direction each force is
  acting and to label each force appropriately.
• Only include forces that act on the body, do not
  include forces that are exerted from the body.
  Equations 5.1 and 5.2 are only valid for forces
  acting on the body. Ask, What other body causes
  that force?
• Choose a set of coordinate axis so that most of
  your forces lie along the axis.

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PSS Particles in Equilibrium

• Execute
• Find the components of each force along each of
  the bodys coordinate axis. Draw a wiggly line
  thru each force vector thats been replaced by
  its components.
• Set the algebraic sum of the x-components equal
  to zero, do the same for the y-components in
  another equation.
• If there are two or more bodies, rinse wash and
  repeat the previous steps for each body.
• Make sure you have the same number of equations
  as you do unknowns.

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PSS Particles in Equilibrium

• Evaluate
• Look at your results and ask whether they make
  sense. Try to think of special cases where you
  can guess what the results should be. Does your
  result match expectations?

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Tension on a massless rope

• A gymnast with mass mG 50.0kg suspends herself
  from the lower end of a hanging rope. The upper
  end of the rope is attached to the gymnasium
  ceiling.
• What is the gymnasts weight?
• What force does the rope exert on her?
• What is th tension at the top o the rope?

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Tension in a rope with mass
Suppose that in the previous problem, the weight
of the rope is not negligible but is 120 N. Find
the tension at each end of the rope.
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Two-dimentional equilibrium

• A car engine with weight w hangs from a chain
  that is linked at ring O to two other chains, one
  fastened to the ceiling at an angle of 60o and
  the other fastened horizontally to the wall.
• Find the tensions in each of these three chains,
  assuming w is given and the weights of the ring
  and chains are negligible.

12
An inclined plane

• A car rests on the slanted tracks of a ramp
  leading to a car-transporter trailer at an angle
  . Only a cable attached to the car and to the
  frame of the trailer prevents the car from
  rolling backward off the trailer. (the cars
  brakes and transmission lock are both released.)
• If the weight of the car is w, find the tension
  in the cable and the force with which the tracks
  push on the cars tires.

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Tension over a frictionless pulley

• Blocks of granite are being hauled up a 15o
  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 both the block and cart) is
  pulled uphill on steel rails by a dirt-filled
  bucket (weight w2, including both the dirt and
  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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Dynamics of Particles

• Weve covered the most common types of
  equilibrium problems, so now we are ready to
  explore what happens if the velocity is not
  constant, dynamics.
• In dynamics we apply Newtons second law to an
  accelerating body ?
• We will use the component form
• The problem solving strategy is very similar to
  that for particles in equilibrium.
• Be sure to understand that ma is not a force, it
  is equal to the magnitude of a force.

5.3
5.4
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PSS Dynamics of Particles

• Identify
• You have to used Newtons second law for any
  problem dealing with forces accelerating a body.
• Identify the target variable. If it is anything
  other than a force or acceleration, you need to
  identify another concept to use as well
• Set Up
• Sketch the situation.
• Draw a force diagram for each body.
• Label each force with symbol and numerical
  magnitude. This includes weight calculation if
  the objects mass is given.

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PSS Dynamics of Particles

• Choose appropriate x- and y-coordinate axes for
  each object, they may differ for each object.
• Identify any equation you may need in addition to
  Newtons second law, one equation for each
  unknown variable. If more than one body is
  involved, there may be a relationship among their
  motions. (i.e. same acceleration)
• Execute
• For each object, calculate the components of each
  force. Be sure to draw a wiggly line thru the
  original force vector.

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PSS Dynamics of Particles

• For each object, write a separate equation for
  each component of Newtons second law.
• Make a list of your know and unknown variables,
  check that your target variable is one of them.
• Check that you have one equation for each
  unknown.
• Do the math, solving for the target variable(s).
• Evaluate
• Does your answer have the correct units,
  algebraic sign? Consider extreme cases. Do the
  results match expectations? Ask, Does this
  result make sense?

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Straight line motion with a constant force

• An iceboat is at rest on a perfectly frictionless
  horizontal surface. A steady wind is blowing
  (along the direction of the runners) so that 4.0s
  after the iceboat is released, it attains a
  velocity of 6.0 m/s.
• What constant horizontal force Fw does the wind
  exert on the iceboat? The mass of the iceboat
  and rider is 200 kg.

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Straight-line motion with a time-varying force

• Lets again consider the iceboat moving on a
  frictionless surface, as in the previous example.
  But now lets suppose that once the iceboat
  starts to move, its position as a function of
  time is
• Find the force Fw exerted by the wind as a
  function of time in this case.
• What is this force at time t 3.0s?
• For what times is the force zero? Positive?
  Negative?

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Straight-line motion with friction

• Suppose the wind is once again blowing steadily
  in the x-direction as in the previous problem,
  so that the iceboar has a constant acceleration
  ax 1.5 m/s2. Now, however, there is a constant
  horizontal friction force with magnitude 100N
  that opposes the motion of the iceboat.
• In this case, what force Fw must the wind exert
  on the iceboat?

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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 slows to a stop with constant
  acceleration in a distance of 25.0 m.
• Find the tension T in the supporting cable while
  the elevator is being brought to a rest.

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Apparent weight in an accelerating elevator

• A 50.0 kg woman stands on a bathroom scale while
  riding in the previous problems elevator. What
  is the reading on the scale?

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Apparent Weight

• In general
• Apparent Weight (n) m( g ay)
• Where ay is the acceleration of the object.
• When ay is positive, the apparent weight is
  larger (you would feel pushed against the bottom
  of the elevator)
• When ay is negative, the apparent weight is less.
  Take this to the extreme, when ay g, the
  object is weightless.

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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 its acceleration?

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Two bodies with the same acceleration

• You are pushing a 1.00 kg food tray through the
  cafeteria line with a constant 9.0 N force. As
  the tray moves, it pushes in turn on a 0.50 kg
  carton of milk. The tray and carton slide on a
  horizontal surface that is so greasy that
  friction can be neglected.
• Find the acceleration of the system and the
  horizontal force that the tray exerts on the
  carton.

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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.
  The glider is connected to a lab weight with
  mass m2 by a light, flexible, nonstretching
  string that passes over a small frictionless
  pulley.
• Find the acceleration of each body and the
  tension in the string.

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Frictional Forces

• We have seen several problems that have involved
  forces such as the normal force or friction
  force, contact forces.
• These forces require direct contact with the body
  to affect its motion.
• In this section we will concern ourselves with
  the force of friction and air resistance or drag.
• Frictional forces always act to oppose an objects
  motion.

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Moving Mass Demo Force Before and After Motion

• Analyze the applied force before motion.
• Analyze the applied force during constant
  velocity
• Where is force maximized?
• How does the constant velocity force compare to
  the maximum force before the motion?
• These forces are forces required to overcome
  friction.
• Before motion is static friction, after is
  kinetic friction.

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Kinetic and Static Friction

• When a body slides or rests on a surface, we can
  represent the contact force on the body by the
  components of force perpendicular and parallel to
  the surface.
• The perpendicular force is the normal force, the
  parallel is the friction force.
• By definition

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Kinetic Friction Force

• Acts on a body from the surface it is moving
  across.
• Increases when normal force increases
• Therefore it is directly proportional to the
  objects mass or the acceleration due to gravity
• The k denotes kinetic friction, and the Greek
  letter mu is the coefficient of kinetic friction.
• The coefficient of kinetic friction varies for
  each object and surface.
• A coefficient has no units.

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Underlying cause of Friction

• Page 172, below CAUTION, second full paragraph.
• Microscopic electrical attraction forces
• Kinetic Friction always varies, as number of
  bonds varies.
• Lubrication helps to limit these electrical
  attractions

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Static Friction Force

• Occurs when there is no relative motion.
• Like kinetic friction, it is proportional to the
  normal force and hence, mass or acceleration due
  to gravity.
• Notice that the force is less than or equal to
  the product of the coefficient of static friction
  and the normal force.
• Where would the force of static friction be
  maximized?
• Think back to the friction demo.

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Friction in horizontal motion

• A delivery company has just unloaded a 500 N
  crate full of home exercise equipment in your
  driveway. You dind that to get it started moving
  toward your garage, you have to pull with a
  horizontal force of magnitude 230 N. Once it
  breaks loose and starts to move, you can keep
  it moving at a constant velocity with only 200N.
  
• What are the coefficients of static and kinetic
  friction?

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Static Friction can be less than the maximum

• In the previous problem, what is the friction
  force if th crate is at rest on the surface and a
  horizontal force of 50 N is applied to it?

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Minimizing Kinetic Friction

• Suppose you try to move the 500 N crate by tying
  a rope around it and pulling upward on the rope
  at an angle of 30o above the horizontal. Assume
  the coefficient of kinetic friction is 0.40.
• How hard do you have to pull to keep the crate
  moving with constant velocity?
• Is this easier or harder than puling
  horizontally?

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Toboggan ride with friction I

• Consider a toboggan sliding down a slope with a
  constant velocity. Derive an expression for the
  slope angle in terms of w and .

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Toboggan ride with friction II

• What if the same toboggan with the same
  coefficient of friction is sliding downhill, but
  on a steeper slope? This time the toboggan
  accelerates. Derive an expression for the
  acceleration in terms of g, w, , and the angle
  of the slope, .

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Motion with Rolling Friction

• A typical car weighs about 12,000 N. If the
  coefficient of rolling friction is µr 0.015,
  what horizontal force is needed to make the car
  move with constant speed on a level road?

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Fluid Resistance and Terminal Speed

• The moving body exerts a force on the fluid to
  push it out if its way, by Newtons 3rd Law the
  fluid exerts an equal and opposite force back on
  the body, Fluid Resistance.
• The force of resistance can be classified into
  what happens at low speeds and high speeds.
• Low Speed High Speed
• k and D are proportionality constants that vary
  with the size and shape of the object and the
  density of the fluid/air.

5.7
5.8
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Effects of Fluid Resistance

• Because this force varies with velocity, the
  acceleration from this force is not constant.
• Rock and pond example on page 179.
• As the rock falls it accelerates. This increase
  in velocity also increases the fluid resistance
  force, decreasing the net force causing the
  acceleration.
• Figure 5.6 shows the acceleration, velocity and
  position vs. time graphs for the motion of the
  rock, both with and without fluid resistance.

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Effects of Fluid Resistance Analysis

• Lets consider down to be a positive velocity,
  since there are no x-components to the rocks
  motion, Newtons 2nd gives
• SFy mg (-kvy) may
• When the rock 1st starts to move, v 0 and ay
  g. As the speed increases, so does the resisting
  force until
• mg kvy 0 or vy mg/k
• This is the terminal velocity of the rock
• The terminal velocity of an object is the
  velocity when the resistance force equals the
  applied force, or when SF 0 (equilibrium)

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Effects of Fluid Resistance

• For an object falling through the air with a high
  velocity

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Terminal Speed of a Skydiver

• For a human body falling through air in a spread
  eagle position, the numerical value of the
  constant D is about 0.25 kg/m.
• Find the terminal speed of an 80 kg skydiver.

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Dynamics of Circular Motion