Physics 207: Lecture 2 Notes

1
Lecture 11

• Goals
• Problem solving with Newtons 1st, 2nd and 3rd
  Laws
• Forces in circular motion
• Introduce concept of work
• Introduce dot product

2
Loop-the-loop 1
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
vB
mg
3
Another example of circular motionLoop-the-loop
2
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 ?
4
Loop-the-loop 2
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
5
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
6
Orbiting satellites
Net Force
mar mg mvt2 / r
gr vt2
The only difference is that g is less because you
are further from the Earths center!
7
Example, Circular Motion Forces with Friction
(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
8
Navigating a curve

• 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 (or 45 mph)

y
N
x
Fs
mg
mcar 1600 kg mS 0.5 for tire/road r 80 m
g 10 m/s2
9
A slight variation

• 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 vt2 / r Fs -ms N (at w)
• y-dir ma 0 N mg N mg

mpuck 2 kg mS 0.8 r 0.5 m g 10 m/s2
10
A slight variation

• 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
11
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?
12
Banked Curves

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

y
x
Ff
No speed
q
13
Banked Curves

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

Low speed
High speed
14
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)
15
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
16
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)

17
Hanging Pink Fuzzy Dice Problem

• You are in a car going around a horizontal curve
• of radius 40.0 m and a speed of 10 m/s. There is
  a
• 0.10 kg pink fuzzy dice at the end of a 0.10 m
  string.
• What is the angle of the die with respect to
  vertical?
• (Little g is 10 m/s2)
• x-dir SFx mar -m vT2 / r -Tx -0.1 x 2.5
  - 0.25 N
• y-dir SFy may 0 Ty - mg ? Ty 0.1 x 10
  1.0 N
• tan q -0.25 ? q 14
• But to you, if you are not paying attention, it
  may look like there is a mysterious force pushing
  the die out.a so-called centrifugal (center
  fleeing) or fictitious force.
• Observers in accelerated frames of reference do
  not see proper physics and so come to erroneous
  conclusions.

18
Drag forces (forces that oppose motion)

• Serway Jewett describe three models

• The velocity dependent model give a general
  analytic solution.
• Terminal velocity for velocity dependent drag
  forces occur where drag force equals applied
  force

19
Drag at high velocities in a viscous medium

• With a cross sectional area, A (in m2), D
  coefficient of drag (0.5 to 2.0), ? density, and
  velocity, v (m/s), the drag force is
• R ½ D ? A v2
• Example Bicycling at 10 m/s (22 m.p.h.), with
  projected area of 0.5 m2 and drag coefficient 1
  exerts a force of 30 Newtons
• At low speeds air drag is proportional to v but
  at high speeds it is v2
• Minimizing drag is often important

20
Energy Work

• Work (Force over a distance) describes energy
  transfer
• No working definition for energy.yet
• Net forces result in acceleration
• Velocity can change direction, magnitude or both
• Only net force acting along the path changes the
  speed
• Forces acting over a distance ? work.energy
  changes
• Forces acting over a time ? impulse..momentum
  changes

21
Perpendicular Forces

• I swing a sling shot over my head. The tension in
  the rope keeps the shot moving at constant speed
  in a circle.
• Is there a net force?
• Does the force act over a distance?
• What if there were a tangential force.what would
  happen?
• Only parallel forces change speed

22
Motion along a line

• The only net force is along the horizontal

• A net force acting along the path, over a
  distance, induces changes in speed (magnitude of
  the velocity)
• We call this action work
• The vector dot product simplifies the notation

23
Scalar Product (or Dot Product)

• Useful for finding parallel components

A ? î Ax î ? î 1 î ? j 0

• Calculation can be made in terms of components.

A ? B (Ax )(Bx) (Ay )(By ) (Az )(Bz )
Calculation also in terms of magnitudes and
relative angles.
A ? B A B cos q
You choose the way that works best for you!
24
Scalar Product (or Dot Product)

• Compare
• A ? B (Ax )(Bx) (Ay )(By ) (Az )(Bz )
• with A as force F, B as displacement Dr
• Notice if force is constant
• F ? Dr (Fx )(Dx) (Fy )(Dz ) (Fz )(Dz)
• Fx Dx Fy Dy Fz Dz
• So here
• Fnet ? Dr Wnet
• A Parallel Force acting Over a Distance does
  Work

25
Units

• Force x Distance Work

Newton x ML / T2
Meter Joule L ML2 / T2
cgs
Other
mks
BTU 1054 J calorie 4.184 J foot-lb 1.356
J eV 1.6x10-19 J
Dyne-cm or erg 10-7 J
N-m or Joule
26
Circular Motion

• I swing a sling shot over my head. The tension in
  the rope keeps the shot moving at constant speed
  in a circle.
• How much work is done after the ball makes one
  full revolution?

(A) W gt 0
Fc
(B) W 0
(C) W lt 0
(D) need more info
27
Definition of Work, The basics
Ingredients Force ( F ), displacement ( ? r )
Work, W, of a constant force F acts through a
displacement ? r W F ? r (Work is a scalar)
Work tells you something about what happened on
the path! Did something do work on you? Did
you do work on something? If only one force
acting Did your speed change?
28
Infinitesimal Work vs speed along a linear path
29
Work vs speed along a linear path

• If F is constant

• DK is defined to be the change in the kinetic
  energy

30
Work in 3D.

• x, y and z with constant F

31
Examples of Net Work (Wnet)

• DK Wnet
• Pushing a box on a smooth floor with a constant
  force there is an increase in the kinetic energy

Examples of No Net Work

• Pushing a box on a rough floor at constant speed
• Driving at constant speed in a horizontal circle
• Holding a book at constant height
• This last statement reflects what we call the
  system
• ( Dropping a book is more complicated because it
  involves changes in U and K, U is transferred to
  K. The answer depends on what we call the
  system. )

32
Net Work 1-D Example (constant force)

• A force F 10 N pushes a box across a
  frictionless floor for a distance ?x 5 m.

?x

• Net Work is F ?x 10 x 5 N m 50 J
• 1 Nm 1 Joule and this is a unit of energy
• Work reflects energy transfer

33
Net Work 1-D 2nd Example (constant force)

• A force F 10 N is opposite the motion of a box
  across a frictionless floor for a distance ?x 5
  m.

Finish
Start
q 180
F
?x

• Net Work is F ?x -10 x 5 N m -50 J
• Work again reflects energy transfer

34
Work 2-D Example (constant force)

• An angled force, F 10 N, pushes a box across a
  frictionless floor for a distance ?x 5 m and ?y
  0 m

Finish
Start
F
q -45
Fx
?x

• (Net) Work is Fx ?x F cos(-45) ?x 50 x
  0.71 Nm 35 J
• Work reflects energy transfer

35
Work and Varying Forces (1D)

• Consider a varying force F(x)

Area Fx Dx F is increasing Here W F ? r
becomes dW F dx
Fx
x
Dx
Finish
Start
F
F
q 0
Dx
Work has units of energy and is a scalar!
36
Fini

• Read all of Chapter 7