Goals:

1
Lecture 11

• Goals

• Chapter 8
• Employ rotational motion models with friction or
  in free fall
• Chapter 9 Momentum Impulse
• Understand what momentum is and how it relates
  to forces
• Employ momentum conservation principles
• In problems with 1D and 2D Collisions
• In problems having an impulse (Force vs. time)

• Assignment
• Read through Chapter 10, 1st four section
• MP HW6, due Wednesday 3/3

2
Zero Gravity Ride

• One last reprisal of the
• Free Body Diagram
• Remember
• 1 Normal Force is ? to the surface
• 2 Friction is parallel to the contact surface
• 3 Radial (aka, centripetal) acceleration
  requires a net force

3
Zero Gravity Ride
A rider in a horizontal 0 gravity ride finds
herself stuck with her back to the wall. Which
diagram correctly shows the forces acting on her?
4
Banked Curves

• In the previous car scenario, we drew the
  following free body diagram for a race car going
  around a curve at constant speed on a flat
  track.
• Because the acceleration is radial (i.e.,
  velocity changes in direction only) we need to
  modify our view of friction.

n
Ff
mg
So, what differs on a banked curve?
5
Banked Curves (high speed)

• 1 Draw a Free Body Diagram for a banked curve.
• 2 Use a rotated x-y coordinates
• 3 Resolve into components parallel and
  perpendicular to bank

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Banked Curves (constant high speed)

• 1 Draw a Free Body Diagram for a banked curve.
• 2 Use a rotated x-y coordinates
• 3 Resolve into components parallel and
  perpendicular to bank

q
( Note For very small banking angles, one can
approximate that Ff is parallel to mar. This is
equivalent to the small angle approximation sin q
tan q, but very effective at pushing the car
toward the center of the curve!!)
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Banked Curves, high speed

• 4 Apply Newtons 1st and 2nd Laws

S Fx -mar cos q - Ff - mg sin q S Fy mar
sin q 0 - mg cos q N Friction model ? Ff
m N (maximum speed when equal)
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Banked Curves, low speed

• 4 Apply Newtons 1st and 2nd Laws

N
Ff
q
mar sin q
mar cos q
q
mg cos q
S Fx -mar cos q Ff - mg sin q S Fy mar
sin q 0 - mg cos q N Friction model ? Ff
m N (minimum speed when equal but not
less than zero!)
mg sin q
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Banked Curves, constant speed

• vmax (gr)½ (m tan q) / (1 - m tan q) ½
• vmin (gr)½ (tan q - m) / (1 m tan q) ½
• Dry pavement
• Typical values of r 30 m, g 9.8 m/s2, m
  0.8, q 20
• vmax 20 m/s (45 mph)
• vmin 0 m/s (as long as m gt 0.36 )
• Wet Ice
• Typical values of r 30 m, g 9.8 m/s2, m
  0.1, q 20
• vmax 12 m/s (25 mph)
• vmin 9 m/s
• (Ideal speed is when frictional force goes to
  zero)

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Banked Curves, Testing your understanding

• Free Body Diagram for a banked curve.
• Use rotated x-y coordinates
• Resolve into components parallel and
  perpendicular to bank

x
y
Ff
q
At this moment you press the accelerator and,
because of the frictional force (forward) by the
tires on the road you begin to accelerate in that
direction. How does the radial acceleration
change?
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Navigating a hill

• Knight concept exercise A car is rolling over
  the top of a hill at speed v. At this instant,

1. n gt w.
2. n w.
3. n lt w.
4. We cant tell about n without knowing v.

At what speed does the car lose contact?
This occurs when the normal force goes to zero
or, equivalently, when all the weight is used to
achieve circular motion. Fc mg m v2 /r ?
v (gr)½ (just like an object in orbit) Note
this approach can also be used to estimate the
maximum walking speed.
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Orbiting satellites vT (gr)½
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Locomotion how fast can a biped walk?

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How fast can a biped walk?

• What about weight?
• A heavier person of equal height and proportions
  can walk faster than a lighter person
• A lighter person of equal height and proportions
  can walk faster than a heavier person
• To first order, size doesnt matter

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How fast can a biped walk?

• What about height?
• A taller person of equal weight and proportions
  can walk faster than a shorter person
• A shorter person of equal weight and proportions
  can walk faster than a taller person
• To first order, height doesnt matter

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How fast can a biped walk?
What can we say about the walkers acceleration
if there is UCM (a smooth walker) ?
Acceleration is radial !
So where does it, ar, come from? (i.e., what
external forces act on the walker?)
1. Weight of walker, downwards 2. Friction with
the ground, sideways
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Impulse Linear Momentum

• Transition from forces to conservation laws
• Newtons Laws ? Conservation Laws
• Conservation Laws ? Newtons Laws
• They are different faces of the same physics
• NOTE We have studied impulse and momentum
  but we have not explicitly named them as such
• Conservation of momentum is far more general than
• conservation of mechanical energy

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Forces vs time (and space, Ch. 10)

• Underlying any new concept in Chapter 9 is
• A net force changes velocity (either magnitude
  or direction)
• For any action there is an equal and opposite
  reaction
• If we emphasize Newtons 3rd Law and emphasize
  changes with time then this leads to the
• Conservation of Momentum Principle

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Example 1

• A 2 kg block, initially at rest on frictionless
  horizontal surface, is acted on by a 10 N
  horizontal force for 2 seconds (in 1D).
• What is the final velocity?
• F is to the positive F ma thus a F/m 5
  m/s2
• v v0 a Dt 0 m/s 2 x 5 m/s 10 m/s (
  direction)
• Notice v - v0 a Dt ? m (v - v0) ma Dt ? m
  Dv F Dt
• If the mass had been 4 kg what is the final
  velocity?

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Twice the mass
Before

• Same force
• Same time
• Half the acceleration (a F / m)
• Half the velocity ! ( 5 m/s )

0
2
Time (sec)
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Example 1

• Notice that the final velocity in this case is
  inversely proportional to the mass (i.e., if
  thrice the mass.one-third the velocity).
• Here, mass times the velocity always gives the
  same value. (Always 20 kg m/s.)

Area under curve is still the same ! Force x
change in time mass x change in velocity
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Example

• There many situations in which the sum of the
  product mass times velocity is constant over
  time
• To each product we assign the name, momentum
  and associate it with a conservation law.
  
• (Units kg m/s or N s)
• A force applied for a certain period of time can
  be graphed and the area under the curve is the
  impulse

Area under curve impulse With m Dv Favg Dt
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Force curves are usually a bit different in the
real world
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Example with Action-Reaction

• Now the 10 N force from before is applied by
  person A on person B while standing on a
  frictionless surface
• For the force of A on B there is an equal and
  opposite force of B on A

MA x DVA Area of top curve MB x DVB Area
of bottom curve Area (top) Area (bottom) 0
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Example with Action-Reaction

• MA DVA MB DVB 0
• MA VA(final) - VA(initial) MB VB(final) -
  VB(initial) 0
• Rearranging terms

MAVA(final) MB VB(final) MAVA(initial) MB
VB(initial) which is constant regardless of M or
DV (Remember frictionless surface)
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Example with Action-Reaction
MAVA(final) MB VB(final) MAVA(initial) MB
VB(initial) which is constant regardless of M or
DV
Define MV to be the momentum and this is
conserved in a system if and only if the system
is not acted on by a net external force (choosing
the system is key) Conservation of momentum is
a special case of applying Newtons Laws
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Applications of Momentum Conservation
Radioactive decay
Explosions
Collisions
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Impulse Linear Momentum

• Definition For a single particle, the momentum
  p is defined as

p mv
(p is a vector since v is a vector)
So px mvx and so on (y and z directions)

• Newtons 2nd Law

F ma

• This is the most general statement of Newtons
  2nd Law

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Momentum Conservation

• Momentum conservation (recasts Newtons 2nd Law
  when net external F 0) is an important
  principle
• It is a vector expression (Px, Py and Pz) .
• And applies to any situation in which there is
  NO net external force applied (in terms of the x,
  y z axes).

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Momentum Conservation

• Many problems can be addressed through momentum
  conservation even if other physical quantities
  (e.g. mechanical energy) are not conserved
• Momentum is a vector quantity and we can
  independently assess its conservation in the x, y
  and z directions
• (e.g., net forces in the z direction do not
  affect the momentum of the x y directions)

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Exercise 2Momentum Conservation

• Two balls of equal mass are thrown horizontally
  with the same initial velocity. They hit
  identical stationary boxes resting on a
  frictionless horizontal surface.
• The ball hitting box 1 bounces elastically back,
  while the ball hitting box 2 sticks.
• Which box ends up moving fastest ?

1. Box 1
2. Box 2
3. same

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Lecture 11

• Assignment
• For Monday Read through Chapter 10, 1st four
  sections
• MP HW6 due Wednesday 3/3