preclass notes: chapter 3 - university of toronto2015-06-22 1 © 2012 pearson education, inc. slide...
TRANSCRIPT
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© 2012 Pearson Education, Inc. Slide 1-1
PreClass Notes: Chapter 3
• From Essential University Physics 3rd Edition
• by Richard Wolfson, Middlebury College
• ©2016 by Pearson Education, Inc.
• Narration and extra little notes by Jason Harlow,
University of Toronto
• This video is meant for University of Toronto
students taking PHY131.
© 2012 Pearson Education, Inc. Slide 1-2
Outline
• 3.1 Vector Math
• 3.2 Velocity and Acceleration
in 2-dimensions
• 3.3 Relative Velocity
• 3.4 Constant Acceleration in x
and y
• 3.5 Projectile Motion
• 3.6 Motion on a Circular Path
“At what angle should this
penguin leave the water to
maximize the range of its
jump?” – R.Wolfson
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© 2012 Pearson Education, Inc. Slide 1-3
Vectors
A quantity that is fully
described by a single number is
called a scalar quantity (ie mass,
temperature, volume)
A quantity having both a
magnitude and a direction is
called a vector quantity
The geometric representation of
a vector is an arrow with the tail
of the arrow placed at the point
where the measurement is made
We label vectors by drawing a small arrow over the letter
that represents the vector, ie: r for position, v for velocity, a
for acceleration
© 2012 Pearson Education, Inc. Slide 1-4
Vector Addition
Suppose you walk 2.0 m in a direction is 30°from the x-axis.
(This might be 30°north of east, for example.)
Then you turn right a bit and walk 1.0 m parallel to x.
Your final position can be found as: 2 1r r r
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Vector Addition: Geometric Methods
It is often convenient to draw two vectors with their tails
together, as shown in (a) below
To evaluate F = D + E, you could move E over and use the
tip-to-tail rule, as shown in (b) below
Alternatively, F = D + E can be found as the diagonal of
the parallelogram defined by D and E, as shown in (c) below
© 2012 Pearson Education, Inc. Slide 1-6
The four segments are described by displacement vectors D1,
D2, D3 and D4
The hiker’s net displacement, an arrow from position 0 to 4, is
Vector addition is easily
extended to more than two vectors
The figure shows the path of a
hiker moving from initial position
0 to position 1, then 2, 3, and
finally arriving at position 4
The vector sum is found by using the tip-to-tail method three
times in succession
Addition of More than Two Vectors
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Vector Components
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Vector Components
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Vector Math: Components Method
We can perform vector addition by adding the x- and y-
components separately
This method is called algebraic addition
For example, if D = A + B + C, then
Similarly, to find R = P – Q we would compute
To find T = cS, where c is a scalar, we would compute
© 2012 Pearson Education, Inc. Slide 1-10
Unit Vectors
Each vector in the figure to the
right has a magnitude of 1, no
units, and is parallel to a coordinate
axis
A vector with these properties is
called a unit vector
These unit vectors have the
special symbols
Unit vectors establish the directions of the positive axes of the
coordinate system
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© 2012 Pearson Education, Inc. Slide 1-11
Two-Dimensional Velocity
If the velocity vector’s angle
θ is measured from the positive
x-direction, the velocity
components are
where the particle’s speed is
Conversely, if we know the velocity components, we can
determine the direction of motion:
© 2012 Pearson Education, Inc. Slide 1-12
Two-Dimensional Acceleration
The figure to the right shows
the trajectory of a particle
moving in the x-y plane
The instantaneous velocity is
v1 at time t1 and v2 at a later
time t2
We can use vector
subtraction to find a during the
time interval Δt = t2 – t1
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© 2012 Pearson Education, Inc. Slide 1-13
Two-Dimensional Acceleration
The instantaneous acceleration is the limit average
acceleration as ∆t 0.
The instantaneous acceleration vector is shown along with the
instantaneous velocity in the figure.
By definition, is the rate at which is changing at that
instant.
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Relative Motion
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Relative MotionIf we know an object’s velocity measured in one reference
frame, S, we can transform it into the velocity that
would be measured by an experimenter in a different
reference frame, S´, using the Galilean transformation of
velocity:
Or, in terms of components:
© 2012 Pearson Education, Inc. Slide 1-16
Relative Motion
• Example:
– A jetliner flies at 960 km/h relative
to the air, heading northward.
There’s a wind blowing eastward
at 190 km/h. In what direction
should the plane fly?
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© 2012 Pearson Education, Inc. Slide 1-17
Constant Acceleration: Big Idea If the acceleration is constant, then the two
components ax and ay are both constant
In this case, everything from Chapter 2 about constant-
acceleration kinematics applies to the components
The x-components and y-components of the motion can be
treated independently
They remain connected through the fact that t must be the
same for both
jaiaa yxˆˆ
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Projectile Motion
Baseballs, tennis balls, Olympic divers, etc, all exhibit
projectile motion
A projectile is an object that moves in two dimensions
under the influence of only gravity
Projectile motion
extends the idea of free-
fall motion to include a
horizontal component of
velocity
Air resistance is
neglected
Projectiles in two dimensions follow a parabolic trajectory
as shown in the photoPhoto of “Curved water” by George Rex taken May 2010, Battersea Park, London Borough of Wandsworth, https://www.flickr.com/photos/36692623@N06/4640846739
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Projectile Motion
The start of a
projectile’s motion is
called the launch
The angle θ0 of the
initial velocity v0 above
the x-axis is called the
launch angle
where v0 is the initial speed
The initial velocity vector can be broken into components
𝑣𝑥0 = 𝑣0 cos 𝜃
𝑣𝑦0 = 𝑣0 sin 𝜃
© 2012 Pearson Education, Inc. Slide 1-20
Projectile Motion
Gravity acts downward
Therefore, a projectile
has no horizontal
acceleration
Thus
The vertical component of acceleration ay is −g of free fall
The horizontal component of ax is zero
Projectiles are in free fall
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Reasoning About Projectile Motion
If air resistance is neglected,
the balls hit the ground
simultaneously
The initial horizontal velocity
of the first ball has no influence
over its vertical motion
Neither ball has any initial
vertical motion, so both fall
distance h in the same amount of
time
A heavy ball is launched exactly horizontally at height h above
a horizontal field. At the exact instant that the ball is launched, a
second ball is simply dropped from height h. Which ball hits the
ground first?
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© 2012 Pearson Education, Inc. Slide 1-23
Range of a Projectile (Penguin Question)
This distance is sometimes
called the range of a projectile:
A projectile with initial speed v0 has a launch angle of θ above
the horizontal. How far does it travel over level ground before
it returns to the same elevation from which it was launched?
The maximum distance
occurs for θ = 45°
Trajectories of a projectile launched at
different angles with a speed of 50 m/s.
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Circular Motion
Consider a ball on a roulette
wheel
It moves along a circular
path of radius r
Other examples of circular
motion are a satellite in an
orbit, or a ball on the end of a
string
Circular motion is an
example of two-dimensional
motion in a plane
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© 2012 Pearson Education, Inc. Slide 1-25
Uniform Circular Motion
To begin the study of circular
motion, consider a particle that
moves at constant speed
around a circle of radius r
This is called uniform
circular motion
The time interval to complete
one revolution is called the
period, T
The period T is related to the
speed v:
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Centripetal
Acceleration
In uniform circular motion,
although the speed is constant,
there is an acceleration because
the direction of the velocity
vector is always changing
The acceleration of uniform
circular motion is called
centripetal acceleration
The direction of the centripetal acceleration is toward the
center of the circle
The magnitude of the centripetal acceleration is constant for
uniform circular motion
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© 2012 Pearson Education, Inc. Slide 1-27
Centripetal Acceleration
The figure shows the
velocity v1 at one instant and
the velocity v2 an
infinitesimal amount of time
dt later
By definition, a = dv/dt
By analyzing the isosceles
triangle of velocity vectors,
we can show that:
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Nonuniform Circular Motion
• In nonuniform circular motion, speed and path
radius can both change.
• The acceleration has both radial and tangential
components:
r ta a a
• is perpendicular to while is tangential to .ra v ta v
2 2
t ra a a
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Nonuniform Circular Motion