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Physics 207, Lecture 5, Sept. 17

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... of a merry-go-round, and a June bug sits halfway between the outer one and the ... What is the June bug's angular velocity? J. L. A. half the Ladybug's. ... – PowerPoint PPT presentation

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Title: Physics 207, Lecture 5, Sept. 17


1
Physics 207, Lecture 5, Sept. 17
  • Goals
  • Solve problems with multiple accelerations in
    2-dimensions (including linear, projectile and
    circular motion)
  • Discern different reference frames and
    understand how they relate to particle motion in
    stationary and moving frames
  • Recognize different types of forces and know how
    they act on an object in a particle
    representation
  • Identify forces and draw a Free Body Diagram

Assignment HW3, (Chapters 4 5, due 9/25,
Wednesday) Read through Chapter 6, Sections 1-4
2
Exercise (2D motion with acceleration)
Relative Trajectories Monkey and Hunter
A hunter sees a monkey in a tree, aims his gun at
the monkey and fires. At the same instant the
monkey lets go. Does the bullet
  1. go over the monkey?
  2. hit the monkey?
  3. go under the monkey?

3
Schematic of the problem
  • xB(Dt) d v0 cos q Dt
  • yB(Dt) hf v0 sin q Dt ½ g Dt2
  • xM(Dt) d
  • yM(Dt) h ½ g Dt2
  • Does yM(Dt) yB(Dt) hf?

(x,y) (d,h)
Monkey
Does anyone want to change their answer ?
What happens if g0 ? How does introducing g
change things?
hf
g
v0
q
Bullet
(x0,y0) (0 ,0) (vx,vy) (v0 cos q, v0 sin q)
4
Uniform Circular Motion (UCM) is common so we
have specialized terms
  • Arc traversed s q r
  • Tangential velocity vt
  • Period, T, and frequency, f
  • Angular position, q
  • Angular velocity, w

s
vt
r
q
Period (T) The time required to do one full
revolution, 360 or 2p radians
Frequency (f) 1/T, number of cycles per unit time
Angular velocity or speed w 2pf 2p/T, number
of radians traced out per unit time (in UCM
average and instantaneous will be the same)
5
Example Question (note the commonality with
linear motion)
  • A horizontally mounted disk 2 meters in diameter
    spins at constant angular speed such that it
    first undergoes 10 counter clockwise revolutions
    in 5 seconds and then, again at constant angular
    speed, 2 counter clockwise revolutions in 5
    seconds.
  • 1 What is the period of the initial rotation?
  • 2 What is initial angular velocity?
  • 3 What is the tangential speed of a point on the
    rim during this initial period?
  • 4 Sketch the angular displacement versus time
    plot.
  • 5 What is the average angular velocity?
  • 6 If now the turntable starts from rest and
    uniformly accelerates throughout and reaches the
    same angular displacement in the same time, what
    must the angular acceleration be?
  • 7 What is the magnitude and direction of the
    acceleration after 10 seconds?

6
Example Question
  • A horizontal turntable 2 meters in diameter
    spins at constant angular speed such that it
    first undergoes 10 counter clockwise revolutions
    in 5 seconds and then, again at constant angular
    speed, 2 counter clockwise revolutions in 5
    seconds.
  • 1 What is the period of the turntable during the
    initial rotation
  • T (time for one revolution) Dt / of
    revolutions/ time 5 sec / 10 rev 0.5 s
  • 2 What is initial angular velocity?
  • w angular displacement / time 2 p f 2 p / T
    12.6 rad / s
  • 3 What is the tangential speed of a point on the
    rim during this initial period?
  • We need more..

7
Relating rotation motion to linear velocity
  • In UCM a particle moves at constant tangential
    speed vt around a circle of radius r (only
    direction changes).
  • Distance tangential velocity time
  • Once around 2pr vt T
  • or, rearranging (2p/T) r vt
  • w r vt
  • Definition If UCM then w constant
  • So vT w r 4 p rad/s 1 m 12.6 m/s

4 A graph of angular displacement (q) vs. time
8
Angular displacement and velocity
  • Arc traversed s q r
  • in time Dt then Ds Dq r
  • so Ds / Dt (Dq / Dt) r
  • in the limit Dt ? 0
  • one gets
  • ds / dt dq / dt r
  • vt w r
  • w dq / dt
  • if w is constant, integrating w dq / dt,
  • we obtain q qo w Dt
  • Counter-clockwise is positive, clockwise is
    negative

9
Sketch of q vs. time
5 Avg. angular velocity Dq / Dt 24 p /10
rad/s
10
Next part..
  • 6 If now the turntable starts from rest and
    uniformly accelerates throughout and reaches the
    same angular displacement in the same time, what
    must the tangential acceleration be?

11
Well, if w is linearly increasing
  • Then angular velocity is no longer constant so
    dw/dt ? 0
  • Define tangential acceleration as at dvt/dt r
    dw/dt
  • So s s0 (ds/dt)0 Dt ½ at Dt2
    and s q r
  • We can relate at to dw/dt
  • q qo wo Dt Dt2
  • w wo Dt
  • Many analogies to linear motion but it isnt
    one-to-one
  • Note Even if the angular velocity is constant,
    there is always a radial acceleration.

12
Tangential acceleration?
  • 6 If now the turntable starts from rest and
    uniformly accelerates throughout and reaches the
    same angular displacement in the same time, what
    must the tangential acceleration be?
  • q qo wo Dt Dt2
  • (from plot, after 10 seconds)
  • 24 p rad 0 rad 0 rad/s Dt ½ (at/r) Dt2
  • 48 p rad 1m / 100 s2 at
  • 7 What is the magnitude and direction of the
    acceleration after 10 seconds?

13
Circular motion also has a radial (perpendicular)
component
Uniform circular motion involves only changes in
the direction of the velocity vector, thus
acceleration is perpendicular to the trajectory
at any point, acceleration is only in the radial
direction. Quantitatively (see text)
Centripetal Acceleration ar
vt2/r Circular motion involves continuous
radial acceleration
14
Non-uniform Circular Motion
For an object moving along a curved trajectory,
with non-uniform speed a ar at (radial and
tangential)
at
ar
15
Tangential acceleration?
  • 7 What is the magnitude and direction of the
    acceleration after 10 seconds?
  • at 0.48 p m / s2
  • and w r wo r r Dt 4.8 p m/s
    vt
  • ar vt2 / r 23 p2 m/s2

Tangential acceleration is too small to plot!
16
Angular motion, signs
  • Also note if the angular displacement, velocity
    and/or accelarations are counter clockwise then
    this is said to be positive.
  • Clockwise is negative

17
Exercise
A Ladybug sits at the outer edge of a
merry-go-round, and a June bug sits halfway
between the outer one and the axis of rotation.
The merry-go-round makes a complete revolution
once each second. What is the June bugs
angular velocity?
A. half the Ladybugs. B. the same as the
Ladybugs. C. twice the Ladybugs. D. impossible
to determine.
J
L
18
Circular Motion
  • UCM enables high accelerations (gs) in a small
    space
  • Comment In automobile accidents involving
    rotation severe injury or death can occur even at
    modest speeds.
  • In physics speed doesnt kill.acceleration
    does (i.e., the sudden change in velocity).

19
Mass-based separation with a centrifuge
Before
After
arvt2 / r and f 104 rpm is typical with r
0.1 m and vt w r 2p f r
How many gs?
ca. 10000 gs
bb5
20
gs with respect to humans
  • 1 g Standing
  • 1.2 g Normal elevator acceleration (up).
  • 1.5-2g Walking down stairs.
  • 2-3 g Hopping down stairs.
  • 1.5 g Commercial airliner during takeoff run.
  • 2 g Commercial airliner at rotation
  • 3.5 g Maximum acceleration in amusement park
    rides (design guidelines).
  • 4 g Indy cars in the second turn at Disney World
    (side and down force).
  • 4 g Carrier-based aircraft launch.
  • 10 g Threshold for blackout during violent
    maneuvers in high performance aircraft.  
  • 11 g Alan Shepard in his historic sub orbital
    Mercury flight experience a maximum force of 11
    g.
  • 20 g Colonel Stapps experiments on acceleration
    in rocket sleds indicated that in the 10-20 g
    range there was the possibility of injury because
    of organs moving inside the body. Beyond 20 g
    they concluded that there was the potential for
    death due to internal  injuries. Their
    experiments were limited to 20 g.
  • 30 g The design maximum for sleds used to test
    dummies with commercial restraint and air bag
    systems is 30 g.

21
A bad day at the lab.
  • In 1998, a Cornell campus laboratory was
    seriously damaged when the rotor of an
    ultracentrifuge failed while in use.
  • Description of the Cornell Accident -- On
    December 16, 1998, milk samples were running in a
    Beckman. L2-65B ultracentrifuge using a large
    aluminum rotor. The rotor had been used for this
    procedure many times before. Approximately one
    hour into the operation, the rotor failed due to
    excessive mechanical stress caused by the
    g-forces of the high rotation speed. The
    subsequent explosion completely destroyed the
    centrifuge. The safety shielding in the unit did
    not contain all the metal fragments. The half
    inch thick sliding steel door on top of the unit
    buckled allowing fragments, including the steel
    rotor top, to escape. Fragments ruined a nearby
    refrigerator and an ultra-cold freezer in
    addition to making holes in the walls and
    ceiling. The unit itself was propelled sideways
    and damaged cabinets and shelving that contained
    over a hundred containers of chemicals. Sliding
    cabinet doors prevented the containers from
    falling to the floor and breaking. A shock wave
    from the accident shattered all four windows in
    the room. The shock wave also destroyed the
    control system for an incubator and shook an
    interior wall.

22
Relative motion and frames of reference
  • Reference frame S is stationary
  • Reference frame S is moving at vo
  • This also means that S moves at vo relative to
    S
  • Define time t 0 as that time when the origins
    coincide

23
Relative Velocity
  • The positions, r and r, as seen from the two
    reference frames are related through the
    velocity, vo, where vo is velocity of the r
    reference frame relative to r
  • r r vo Dt
  • The derivative of the position equation will give
    the velocity equation
  • v v vo
  • These are called the Galilean transformation
    equations
  • Reference frames that move with constant
    velocity (i.e., at constant speed in a straight
    line) are defined to be inertial reference frames
    (IRF) anyone in an IRF sees the same
    acceleration of a particle moving along a
    trajectory.
  • a a (dvo / dt 0)

24

Central concept for problem solving x and y
components of motion treated independently.
  • Example Man on cart tosses a ball straight up in
    the air.
  • You can view the trajectory from two reference
    frames

25
Example (with frames of reference) Vector addition
An experimental aircraft can fly at full throttle
in still air at 200 m/s. The pilot has the nose
of the plane pointed west (at full throttle) but,
unknown to the pilot, the plane is actually
flying through a strong wind blowing from the
northwest at 140 m/s. Just then the engine fails
and the plane starts to fall at 5 m/s2.
What is the magnitude and directions of the
resulting velocity (relative to the ground) the
instant the engine fails?
Calculate A B
Ax Bx -200 140 x 0.71 and Ay
By 0 140 x 0.71
26
Home Exercise, Relative Motion
  • You are swimming across a 50 m wide river in
    which the current moves at 1 m/s with respect to
    the shore. Your swimming speed is 2 m/s with
    respect to the water.
  • You swim across in such a way that your path is a
    straight perpendicular line across the river.
  • How many seconds does it take you to get across?

27
Home Exercise
Choose x axis along riverbank and y axis across
river
  • The time taken to swim straight across is
    (distance across) / (vy )
  • Since you swim straight across, you must be
    tilted in the water so that your x component of
    velocity with respect to the water exactly
    cancels the velocity of the water in the x
    direction

vy 1 m/s
rivers frame
28
Home Exercise 2
Where do you land if the river flows at 2 m/s
while swimming at the same heading in the river
(i.e., q asin ½) ?
  • The time taken to swim straight across
    (distance across) / (vy )
  • time 50 m / ( 2 m/s cos q) 50/3½ seconds
  • Dist in river vx t -2 m/s sin q t -2
    (50/3½) 1/2 m -29 m
  • (upstream)
  • Dist river flows vr t 2 m/s t -2 (50/3½) m
    58 m
  • Final position -29 m 58 m 29 m down the
    shore.

29
What causes motion? (Actually changes in
motion) What are forces ? What kinds of forces
are there ? How are forces and motion related ?
30
Physics 207, Lecture 5, Sept. 17
Assignment HW3, (Chapters 4 5, due 9/25,
Wednesday) Read Chapter 5 through Chapter 6,
Sections 1-4
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