Review of Lecture 4

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Review of Lecture 4

• Position Displacement
• Average Instantaneous Velocity
• Average Instantaneous Acceleration
• Projectile Motion
• Vertical Horizontal Motion
• Range
• Uniform Circular Motion
• Relative Motion

2
Acceleration

• What causes an acceleration?
• The short answer is a Force
• We will now look at forces from the standpoint of
  their effects on bodies
• When dealing with classical bodies at
  non-relativistic speeds, we are talking about
  Newtonian mechanics

3
Newtons First Law

• Prior to Newton, it was thought that the natural
  state of an object was to be at rest
• It was also thought that in order for something
  to move, it had to be propelled in some fashion
• While this may apply to my cats (and perhaps cats
  in general) it of course isnt an accurate
  description of what is happening

4
Newtons First Law

• Imagine giving a hockey puck a push so that it
  goes sliding across a wood floor
• It will go some distance before slowing down and
  coming to a stop
• Now substitute some ice and try the experiment
  again
• Clearly the puck will slide farther this time

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Newtons First Law

• Now keep imagining a slipperier and slipperier
  surface
• In the limit, you can eventually imagine a
  surface that has no friction at all
• The puck in that case will keep sliding along
  forever unless something causes it to alter its
  course

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Newtons First Law

• From these observations Newton developed his 1st
  lawIf no force acts on a body, then the bodys
  velocity cannot change that is, the body cannot
  accelerate

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Newtons First Law

• If we were to push on a standard 1 kg mass so
  that we measured its acceleration as 1 m/s2,
  then by definition we are exerting a force on
  the object of 1 Newton
• If we were to push harder so that the
  acceleration was 2 m/s2, then the force would be
  2 Newtons, and so on

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Newtons First Law

• So force is somehow closely related to
  acceleration
• Given that, it should not be a surprise that,
  like acceleration, force is a vector quantity
• That means if there were several forces pulling
  on our object, we should be able to add them up
  vectorially to produce a net force or resultant
  force

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Newtons First Law

• Given that, we can now re-state Newtons 1st law
  asIf no net force acts on a body (e.g.,
  ifFnet 0), then the bodys velocity cannot
  change that is, the body cannot accelerate

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Inertial Frames of Reference

• We have already talked about frames of reference
• A frame of reference where Newtons 1st Law holds
  is called an inertial reference frame or just an
  inertial frame
• The earth is usually a perfectly good inertial
  frame for most of what we do, but there are cases
  where it will not work

11
Mass

• What exactly is mass?
• We have some feel for this by the fact that
  different bodies react differently to the same
  force (if we have some way of consistently
  applying the same force)
• The more mass a body has, the less it will
  accelerate given the same applied force

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Mass

• So there would seem to be some kind of an inverse
  relationship between force and mass that is also
  related to the acceleration
• Suppose we took a body of mass 2 kg and applied
  the same force to it that we used to define our
  1N force
• In this case we would notice that the object
  would accelerate to 0.5 m/s2

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Mass

• And conversely, if we took a 0.5 kg mass and
  applied our standard force, we would notice that
  it would accelerate to 2 m/s2
• So we can see a ratio forming herewhere the
  force is held constant

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Mass

• So back to the initial question What is mass?
• Mass is something intrinsic to an object we can
  only define it as a characteristic of an object
  that relates forces on the object to how the body
  reacts (accelerates) in response to those forces

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Newtons Second Law

• Everything we have been discussing so far
  regarding forces, masses and acceleration can be
  neatly summed up using Newtons 2nd LawThe net
  force on a body is equal to the product of the
  bodys mass and the acceleration of the body

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Newtons Second Law

• In equation form this is the famous
• While this is very simple equation, we must be
  careful in our use of it
• We must first note that force, like acceleration,
  is a vector (this must be true as mass is only a
  scalar)

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Newtons Second Law

• We must also note that we must only concern
  ourselves with the (net) forces acting upon a
  body not any forces that the body may exert on
  another body, or other forces that may be present
  in the system (but dont act upon the body in
  question)

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Newtons Second Law

• Because force is a vector quantity, the usual
  rules apply we can break forces down into
  components one for each axis of our coordinate
  system
• So we have

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Newtons Second Law

• Because we can break down our forces into
  orthogonal components, we can see that (net)
  forces along one axis have no influence on what
  is happening along another axis
• In other wordsThe acceleration component along
  a given axis is caused only by the sum of the
  force components along that same axis, and not by
  force components along any other axis

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Newtons Second Law

• Now lets look at our equation again
• From this we can see that if the (net) force is
  zero, then the acceleration must also be zero
• From this we can see that A body in motion
  continues in motion (at a constant velocity) and
  A body at rest stays at rest
• This condition is known as equilibrium the
  various forces have balanced themselves out so
  that the net force is zero

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Particular Forces(Gravitation)

• We have already encountered a particular
  acceleration that associated with gravity
• So we can now compute the force that gravity
  exerts on an object
• Substituting g for a in our equation (and
  aligning our y axis to be vertically upward), we
  can see that

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Particular Forces(Gravitation)

• We can clean this up to be
• This force is always present whether a body is in
  free fall or resting on a table
• The weight of a body is the magnitude of the
  force necessary to prevent the body from falling
  freely (as measured at the earths surface)

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Particular Forces(Gravitation)

• So we can say that
• From this we can see that a bodys weight is
  directly proportional to its mass with the
  constant of proportionality being the
  acceleration of gravity (g for the earth)

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Particular Forces(Gravitation)

• Bear in mind that weight and mass are not the
  same thing
• A body that weighs 71 N on earth (7.25 kg) would
  only weigh 12 N on the moon
• This is because while the mass of the body didnt
  change in going from the earth to the moon, the
  acceleration of gravity (g) on the moon is only
  1.7 m/s2

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Particular Forces(Gravitation)

• Note also that if my weight was measured in an
  elevator (that was accelerating), it would not be
  constant
• This is because an accelerating elevator is not
  an inertial frame

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Particular Forces(Normal Force)

• As I am standing here on the floor, my body mass
  is exerting a force against the floor
• The floor flexes (a little) due to my presence
  and in turn exerts a force upwards against me
  otherwise I would accelerate downwards
• The force exerted by the floor against me is
  called a normal force (normal because it is
  perpendicular to the surface of the floor)

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Particular Forces(Normal Force)

• Going back to the elevator example, there is of
  course a normal force exerted by the floor of the
  elevator against someone standing in the elevator
• When the elevator is accelerating, the normal
  force will be

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Particular Forces(Friction)

• When one object slides along the surface of
  another, the motion will be resisted by friction
• Friction generates a force in opposition to the
  direction of motion
• We often will assume frictionless surfaces to
  simplify things, but we will look at friction in
  more detail in Chapter 6

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Particular Forces(Tension)

• When we pull an object with a rope (cable, cord,
  etc.), the rope exerts a force on the object that
  is directed away from the object and along the
  rope
• This is called a tension force because the rope
  is said to be in a state of tension (or under
  tension) because the rope is pulled taught

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Particular Forces(Tension)

• We will often assume that the rope is massless
  and unstretchable to simplify things
• We will also sometimes employ frictionless,
  massless pulleys to change the direction of the
  force between two objects without introducing
  other effects

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Newtons Third Law

• Two bodies interact when they exert forces on
  each other
• If I push against the wall I am clearly exerting
  a force on it the wall is also clearly exerting
  a force back against me
• These forces apparently are equal otherwise one
  of us (the wall or me) would be accelerated

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Newtons Third Law

• Newtons 3rd Law states that When two bodies
  interact, the forces on the bodies from each
  other are always equal in magnitude and opposite
  in direction
• As hard as I push on the wall, the wall will push
  back just as hard (in the opposite direction)

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Newtons Third Law

• In vector notation, we have
• This is called a third-law force pair
• We will always have a third-law force pair
  whenever two bodies interact

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Next Class

• Homework Problems Chapter 54, 11, 19, 49, 51,
  74
• Read sections Chapter 6