Chapter 5: Newtons Laws of Motion

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Chapter 5 Newtons Laws of Motion

• We will study classical motion
• No quantum mechanics
• No relativity
• We introduce the concept of force
• We define force in terms of the acceleration of a
  standard body

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• Intuitively, we know that force is a push or
  pull. Forces come in different classes (types)
• Contact
• Macroscopic forces of contact friction,
  viscosity, the contact force from the floor
  supporting my feet).
• Field (originally described as action-at-a-distanc
  e)
• Examples Gravity, Electromagnetism
• Force F is a vector quantity You push or pull
  in a specific direction

F
If force has direction, what is its measure?
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The Empirical Feel of Forces

• We have a direct sensation of the forces that act
  on our body.
• As I stand on the floor, I feel my shoes pushing
  up on my feet. The nerves in the soles of my
  feet transmit this feeling to my brain.
• The nerves in our joints also give us a sense of
  the weight of our bodies.
• If you stub your toe (or worse) you feel the
  force against your toe.

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What is the connection between Force and Motion?
(Dynamics)

• The ancient Greeks, especially Aristotle, had a
  very elegant philosophy of nature
• Four elements Earth, Air, Fire, Water
• Two Forces Gravity, Levity
• Gravity pulled earth and water down, Levity
  pushed Fire and Air up.
• To Aristotle (and perhaps to our common sense)
  everything tended to its natural state. For
  material objects (earth water) the natural
  state was at rest.
• To use modern language, friction was seen as part
  of the fabric of space time.
• It was Galileo who suggested that friction was
  not essential, but rather subject to
  technological manipulation (and ideally
  elimination).

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Consider the motion of an egg (at rest or a
projectile).

• The egg is perfectly happy at any position
• Thus position is not crucial to dynamics (this
  contradicts Aristotle).
• The egg is perfectly happy at rest, or in motion.
• Thus velocity is not crucial to dynamics (this
  contradicts our common sense that velocity is THE
  thing).
• If I catch the egg gently, nothing dramatic
  happens.
• Thus CHANGE in Velocity is NOT crucial either!
• What happens if I drop the egg and it goes splat?
• It is the RATE of CHANGE of VELOCITY
  (Acceleration) that is large when the egg goes
  splat.
• Dynamics links the Force from the floor to the
  Acceleration of the egg.

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• In 1686, Newton presented his
• Three Laws of Motion
• Newtons First Law
• An object at rest remains at rest, and an object
  in motion continues in motion with constant
  velocity, unless it experiences a net force.

• Velocity constant (i.e. acceleration 0) if
  there is no force (or if all forces add to zero).
• Remember
• Velocity constant, does not mean velocity 0.
• Velocity constant means constant magnitude AND
  direction

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Examples

• An object that is moving and that continues to
  move with constant velocity without any force
  acting on it.
• A hockey puck sliding (almost without friction)
  across the ice
• 2. An object at rest that remains at rest.
• What about pushing a chair?
• If the floor pushes just as hard (friction) the
  net force (vector sum) is zero.
• What happens when you turn a corner quickly in
  your car?
• The car would continue straight ahead unless the
  friction from the road pushes inwards to guide
  the car around the circle (centripetal force).

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• We know from experience that different objects
  resist a change in motion differently.
• Example
• push a door
• push a semi-trailer
• ? Not the same response!

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Inertia

• The tendency of an object to resist a change in
  its velocity is called inertia.
• The measure of inertia is mass.
• SI units measure mass as multiples of the
  standard kilogram (kg1000g) stored at the
  International Bureau of Weights and Measures in
  Sèvres, France.
• Newtons First Law If F0, then a0.
• What if F ? 0?

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Newtons Second Law

• The acceleration of an object is directly
  proportional to the resultant force acting on it
  and inversely proportional to its mass. The
  direction of the acceleration is the direction of
  the resultant force.

F ma
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FNet m aImplicit and explicit meaning

• Force is a vector
• The net force is the vector sum of all forces
  acting on the object m.
• Mass is a scalar
• The value of the mass of an object does not
  change with the direction of the acceleration.
• The equation Fma is also a definition of mass.
• Mass is invariant
• If two objects are put together (or separated) ,
  the mass of the combined object is simply the
  arithmetic sum of the two masses m m1m2.
• Chemical combination, welding, cutting does not
  change mass.
• Einstein corrected this, but Relativistic effects
  are small (10-9) for ordinary matter.
• Force can be quantified by measuring the
  acceleration it produces on a standard kilogram
  (or any multiple there-of).

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Example

• An object of mass 5 kg undergoes an acceleration
  of a (8 m/s2) y 8 m/s2 in y direction
• What is the force on that object?
• F ma
• (5 kg)(8 m/s2) y 40 kg?m/s2 y
• y vector on unit length (no dimensions) in y
  direction.
• The force is in the same direction as the
  acceleration.

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Units

• The SI unit of Force is the Newton defined as
• 1 N 1 kg?m/s2
• For comparison
• 1 lb. 4.448 N
• Notice pounds and kilograms do not directly
  convert.
• The British unit of mass is the slug (dont ask,
  1 lb 1slug 1 ft/s2 ).
• The force of gravity (near Earths surface)
  acting on a 1 kg mass is 2.2 lb.
• (1.0 kg) (g) 2.2 lb. 9.81 N
• Do not confuse ggram with g9.8m/s2acceleration
  due to gravity.

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Walker, Problem 8, page 133
A catcher stops a 92 mi/h pitch in his glove,
bringing it to rest (with uniform acceleration)
in 0.15 m. If the force exerted by the catcher
is 803 N, what is the mass of the ball?
Chapter 2 v2 v02 2a(x-x0) a (v2 - v02)/
2 (x-x0)
Chapter 5 F ma m F/a (803 N) / (5641
m/s2) 0.142kg
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Newtons Third Law

• If object 1 exerts a force F on object 2, then
  object 2 exerts a force F on object 1.
• Forces come in pairs.
• The force pairs act on different objects.
• The forces have the same magnitude but opposite
  direction.

Example I push on the wall with a force of 20
N. The wall pushes back on me with a force of 20
N in the opposite direction.
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Walker, Problem 17, page 133
A force of magnitude 7.50 N pushes three boxes
with masses m1 1.30 kg, m2 3.20 kg, and m3
4.90 Kg, as shown in the Figure. (crucial
assumption omitted in Walker problem 16 17 no
friction!) Find the contact force between (a)
boxes 1 and 2, and (b) between boxes 2 and 3.
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Free Body Diagrams for relevant subsystems
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Free Body Diagrams for each mass
F1,2
W1 m1g
N1
F1,2 Force of Block 2 pushing on Block
1. Notice that force 7.50 N does NOT push on m1
Fy,Net N1 - m1g Fy,Net 0
Fx,Net F1,2 Fx,Net m1a
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Free Body Diagrams for each mass
F2,1
F2,3
W2 m2g
N2
Fx,Net F2,1- F2,3 Fx,Net m1a
F2,3 force of m3 pushing on m2. F2,1 force of
m1 pushing on m2.
Fy,Net N2 - m2g Fy,Net 0
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The Vector Nature of Forces

• In the formula F ma, F is the total (net) force
  acting on the object. We must consider the
  vector sum of all forces acting on an object. We
  can also consider each dimension separately

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Walker, Problem 19, page 134
A farm tractor tows a 4400-kg trailer up a 21
incline at a steady speed of 3.0 m/s. What force
does the tractor exert on the trailer? (Ignore
friction.)
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Weight

• The weight of any object on the Earth is the
  gravitational force exerted on it by the Earth
• W mg
• Note
• Weight is a force (and therefore a vector).
• Weight is not equivalent to mass.
• Can a persons weight be zero?
• When we say we want to lose weight, what do we
  really mean?

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Apparent Weight
Our sensation of weight comes from the force of
the floor pushing up on us. We can feel light or
heavy if the floor is accelerating down or up.
The upward force of the floor on our feet is
known as apparent weight Wa.
It is your apparent weight that is measured on a
scale.
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Apparent Weight
An elevator is moving upward at constant velocity
(acceleration 0)
It is your apparent weight that is measured on a
scale.
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Apparent Weight and acceleration.

• As part of a physics experiment, you stand on a
  bathroom scale in an elevator. Though your
  normal weight is 610 N, the scale at the moment
  reads 730 N.
• The acceleration of the elevator is
• upward,
• downward, or
• zero?

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Apparent Weight and velocity.
As part of a physics experiment, you stand on a
bathroom scale in an elevator. Though your
normal weight is 610 N, the scale at the moment
reads 730 N. Since your apparent weight is
greater than gravity, you are accelerating upward.

• Which statement could be true
• The elevator is moving upward at constant
  velocity
• The elevator is moving upward at but slowing down
• The elevator is moving downward, but slowing down
• None of the above

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Normal Forces

• Normal means perpendicular.

box
The normal force is a contact force and is
perpendicular to the surface between the two
objects in contact. The table and the box are
compressing each others atoms slightly, like
springs. The box pushes down on the table and the
table pushes up on the box. These two forces are
reaction pairs.
N
-N
table
If you lean against the wall, the normal force
from the wall is horizontal. When the cart rolls
down the incline in your physics lab, the normal
force is perpendicular to the incline
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Walker, Problem 39, pg. 135
A 9.0-kg child sits in a 2.3-kg high chair (a)
Draw a free-body diagram for the child, and find
the normal force exerted by the chair on the
child. (b) Draw a free-body diagram for the
chair, and find the normal force exerted by the
floor on the chair.
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Chair
Child
N1
m9.0kg
mg
Mg
Nseat
N2
M2.3kg
Action-Reaction Pairs Nseat ? ? N1
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Quiz 3

• While ice-skating, You are pushing a box of mass
  m20kg across the ice (neglect friction between
  the box and the ice).
• You are pushing with a force P that is directed
  30 degrees below the horizontal.
• The ice pushes up on the box (otherwise the box
  would accelerate down into the ice). We will
  call this force N, the Normal (perpendicular)
  force.
• Is the force N
• greater than,
• equal to, or
• less than (in magnitude)
• the force mg of gravity on the box?

m
P
mg
N
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Free Fall Lab
Leading edge of dark bands, 5cm apart
Average velocity (5cm) / (time between bands)
a(fit) 9.75 m/s2 c20.004 m2/s2
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Free Fall Lab
Leading edge of dark bands, 5cm apart
Average velocity (5cm) / (time between bands)
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Free fall lab, best estimate of instantaneous
velocity from average velocity
Note tDeltatime between bands and v are
tabulated at the time half-way between times
LogicVolts0
t v
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Motion on an InclineFrictionless
y
x

• Cart rolls without friction on incline
• Find the acceleration of the cart (as a function
  of q).
• Draw a coord system parallel to incline
• X-component of gravity
• Wx mg cos(270-q)
• Wy -mg sin(270-q)
• -mg cos(270-q) max
• N-mg sinq m ay
• Y-component of acceleration must be zero

q
Free Body Diagram
N
mg