Physics 207: Lecture 2 Notes

1
Lecture 10

• Goals
• Employ Newtons Laws in 2D problems with
  circular motion

Assignment HW5, (Chapters 8 9, due 3/4,
Wednesday) For Tuesday Finish reading Chapter
8, start Chapter 9.
2
Uniform Circular Motion
For an object moving along a curved trajectory,
with non-uniform speed a ar (radial only)
v
ar
Perspective is important
3
Non-uniform Circular Motion
For an object moving along a curved trajectory,
with non-uniform speed a ar at (radial and
tangential)
at
ar
4
Circular motion

• Circular motion implies one thing

aradial vT2 / r
5
Key steps

• Identify forces (i.e., a FBD)
• Identify axis of rotation
• Apply conditions (position, velocity,
  acceleration)

6
Example The pendulum
axis of rotation
Consider a person on a swing
When is the tension on the rope largest? And at
that point is it (A) greater than (B) the
same as (C) less than the force due to gravity
acting on the person?
7
Example Gravity, Normal Forces etc.
axis of rotation
T
T
vT
q
mg
mg
at top of swing vT 0 Fr m 02 / r 0 T
mg cos q T mg cos q T lt mg
at bottom of swing vT is max Fr m ac m vT2
/ r T - mg T mg m vT2 / r T gt mg
8
Conical Pendulum (very different)

• Swinging a ball on a string of length L around
  your head
• (r L sin q)

axis of rotation
S Fr mar T sin q S Fz 0 T cos q
mg so T mg / cos q (gt mg) mar mg
sin q / cos q ar g tan q vT2/r ? vT (gr
tan q)½
Period t 2p r / vT 2p (r cot q /g)½
9
Loop-the-loop 1
A match box car is going to do a loop-the-loop of
radius r. What must be its minimum speed vt at
the top so that it can manage the loop
successfully ?
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Loop-the-loop 1
To navigate the top of the circle its tangential
velocity vT must be such that its centripetal
acceleration at least equals the force due to
gravity. At this point N, the normal force, goes
to zero (just touching).
Fr mar mg mvT2/r vT
(gr)1/2
vT
mg
11
Loop-the-loop 2
The match box car is going to do a loop-the-loop.
If the speed at the bottom is vB, what is the
normal force, N, at that point? Hint The
car is constrained to the track.
Fr mar mvB2/r N - mg N mvB2/r mg
N
v
mg
12
Loop-the-loop 3
Once again the car is going to execute a
loop-the-loop. What must be its minimum speed at
the bottom so that it can make the loop
successfully? This is a difficult problem to
solve using just forces. We will skip it now and
revisit it using energy considerations later on
13
Example Problem
Swinging around a ball on a rope in a nearly
horizontal circle over your head. Eventually the
rope breaks. If the rope breaks at 64 N, the
balls mass is 0.10 kg and the rope is 0.10
m How fast is the ball going when the rope
breaks? (neglect mg contribution, 1 N ltlt 40 N)
Fr m vT2 / r T vT (r Fr / m)1/2 vT
(0.10 x 64 / 0.10)1/2 m/s vT 8 m/s
14
Example, Circular Motion Forces with Friction
(recall mar m vT 2 / r Ff ms N )

• How fast can the race car go?
• (How fast can it round a corner with this radius
  of curvature?)

mcar 1600 kg mS 0.5 for tire/road r 80 m
g 10 m/s2
r
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Example

• Only one force is in the horizontal direction
  static friction
• x-dir Fr mar -m vT 2 / r Fs -ms N (at
  maximum)
• y-dir ma 0 N mg N mg
• vT (ms m g r / m )1/2
• vT (ms g r )1/2 (0.5 x 10 x 80)1/2
• vT 20 m/s

y
N
x
Fs
mg
mcar 1600 kg mS 0.5 for tire/road r 80 m
g 10 m/s2
16
Another Example

• A horizontal disk is initially at rest and very
  slowly undergoes constant angular acceleration.
  A 2 kg puck is located a point 0.5 m away from
  the axis. At what angular velocity does it slip
  (assuming aT ltlt ar at that time) if ms0.8 ?
• Only one force is in the horizontal direction
  static friction
• x-dir Fr mar -m vT 2 / r Fs -ms N (at
  w)
• y-dir ma 0 N mg N mg
• vT (ms m g r / m )1/2
• vT (ms g r )1/2 (0.8 x 10 x 0.5)1/2
• vT 2 m/s ? w vT / r 4 rad/s

mpuck 2 kg mS 0.8 r 0.5 m g 10 m/s2
17
UCM Acceleration, Force, Velocity
F
a
v
18
Zero Gravity Ride
A rider in a 0 gravity ride finds herself stuck
with her back to the wall. Which diagram
correctly shows the forces acting on her?
19
Banked Curves

• In the previous car scenario, we drew the
  following free body diagram for a race car going
  around a curve on a flat track.

n
Ff
mg
What differs on a banked curve?
20
Banked Curves

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

y
x
Ff
q
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!!
21
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 accelerate in that
direction. How does the radial acceleration
change?
22
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 are on the walker?)
1. Weight of walker, downwards 2. Friction with
the ground, sideways
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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 are on the walker?)
1. Weight of walker, downwards 2. Friction with
the ground, sideways
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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 are on the walker?)
1. Weight of walker, downwards 2. Friction with
the ground, sideways (most likely from back
foot, no tension along the red line) 3. Need a
normal force as well
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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 are on the walker?)
1. Weight of walker, downwards 2. Friction with
the ground, sideways (Notice the new direction)
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How fast can a biped walk?
Given a model then what does the physics
say? Choose a position with the simplest
constraints. If his radial acceleration is
greater than g then he is in orbit Fr m ar
m v2 / r lt mg Otherwise you will lose
contact! ar v2 / r ? vmax (gr)½ vmax 3
m/s ! (And it pays to be tall and live on Jupiter)
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Orbiting satellites vT (gr)½
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Geostationary orbit
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Geostationary orbit

• The radius of the Earth is 6000 km but at 36000
  km you are 42000 km from the center of the
  earth.
• Fgravity is proportional to r-2 and so little g
  is now 10 m/s2 / 50
• vT (0.20 42000000)½ m/s 3000 m/s
• At 3000 m/s, period T 2p r / vT 2p 42000000
  / 3000 sec
• 90000 sec 90000 s/ 3600 s/hr 24 hrs
  
• Orbit affected by the moon and also the Earths
  mass is inhomogeneous (not perfectly
  geostationary)
• Great for communication satellites
• (1st pointed out by Arthur C. Clarke)