Physical Sciences 2: Lecture 1a Pre-Reading
September 3, 2015
Pre-Reading for Lecture 1a:
Vectors, Velocity, and Momentum
Read this while you wait… much of it may be review; we will cover it briefly in class.
•
One of the most important concepts in PS2 is the concept of a vector. You can think of a
vector as being like an arrow—something that has a direction, and a magnitude (or
length). Here are a bunch of vectors, with various magnitudes and directions:
You can move a vector around on the page (or anywhere really) and it will always be the
same vector—as long as it keeps the same magnitude and direction! Here are a bunch of
vectors that are all identical:
Here are some vectors that all have the same magnitude but different directions:
And here are some vectors that all have the same direction but different magnitudes:
•
We use vectors to represent physical quantities that require both a magnitude and a
direction. For instance, we use vectors to describe the velocity of an object. In physics,
the word “velocity” means both the speed and direction that an object is moving. So we
need a vector to describe velocity:
The magnitude of a velocity vector tells you the speed of an object:
how fast is it going? (Note that the magnitude is always positive.)
The direction of a velocity vector tells you the direction of motion:
in what direction is it moving?
Speed can be described using any convenient units, like miles per hour or meters per
second. Direction can be specified in many ways—for instance, “north” or “east.”
1
Physical Sciences 2: Lecture 1a Pre-Reading
•
September 3, 2015
One way to specify a vector is by using a coordinate system—typically, a set of x- and
y-axes with a coordinate grid. Here is an example:
A vector points
from the tail
to the head.
It's important to remember that the
vector is the same no matter where it is:
all four arrows drawn on this graph
represent the same vector.
To describe a vector,
you need to tell me how long it is
and in what direction it is pointing.
5 steps
(5, –2)
2 steps
Since it doesn't matter where you place
a vector, if you place its tail at the origin
of your coordinate system, then the
coordinates of the head will be equal
to the components of the vector:
the head here is at coordinates (5, –2)
For this vector, going from the
tail to the head means:
Move 5 steps in the +x direction
and 2 steps in the –y direction
so the components are (x, y) = (5, –2)
•
•
In contrast to a vector, an ordinary number is called a scalar (pronounced SKAY-ler).
We must distinguish vectors from scalars in our equations. If something is a vector, we
will write it with a little arrow above it:

a is a scalar
a is a vector
(In your textbook, vectors are distinguished by boldface type: a is a vector.)

Given a vector v , we will use subscripts to indicate its components:

For the vector v , the x-component is vx, and the y-component is vy.

So in the example pictured above, the vector v has vx = 5 and vy = –2. Note that the
components of a vector are ordinary scalars (numbers)!
•
The magnitude of a vector can be found from its components using the Pythagorean
theorem:


The magnitude of v is written as v , and is equal to vx2 + vy2
•
Since the magnitude of a vector
is a scalar, we will often use v (without an arrow!) to

represent the magnitude v . So it is extremely important to use the arrow for
vectors—if there’s no arrow, the letter alone means the magnitude of the vector!
2
Physical Sciences 2: Lecture 1a
September 3, 2015
Welcome to Physical Sciences 2
•
What will we be learning this semester in PS2?
1
Physical Sciences 2: Lecture 1a
September 3, 2015
Introduction
•
Our course philosophy and approach to teaching:
•
Learning objectives: After this lecture, you will be able to…
1. Draw arrows (vectors) to represent quantities that have both magnitude and direction.
Give examples of physical quantities that require a vector representation.
2. Describe a vector using its magnitude and angle with respect to an axis, or using its
components in a coordinate system. Convert between these two representations.
3. Use vectors to determine the mean electrical axis of the heart from an EKG.
4. Add or subtract vectors graphically, and using components. Show how unit vectors
can be used to construct a vector from its components.
5. Describe the position of an object using its position vector, and use vector subtraction
to calculate the displacement vector.
6. Calculate the average velocity of an object, and compare average velocity with the
instantaneous velocity.
7. Calculate the average speed of an object, and compare speed versus velocity.
8. Explain what momentum is and why it is important.
9. Find the momentum of a single object or a system of several objects.
10. Identify the interactions between any pair of objects.
11. Find the interactions that could be responsible for the change in the momentum of an
object.
12. Determine whether a system of objects is functionally isolated.
13. Use momentum conservation for a system of several objects to solve for various
unknown quantities (masses, speeds, angles…).
2
Physical Sciences 2: Lecture 1a
September 3, 2015
Activity 1: Vectors, Angles, and Components
•

The arrow shown below represents the velocity vector ( v ) of a baseball. The length of
the arrow represents how fast the ball is moving (its speed), while the direction of the
arrow represents the direction of the ball.
a)
Draw another arrow representing the same velocity vector, but with the tail of the vector
located at the origin.
b)
What are the x- and y-components of the vector? Does it matter where it is located?
c)
Calculate the magnitude of this vector. (The magnitude is the length of the vector.)
d)
Estimate the standard angle of this vector (defined as the angle measured from the
x-axis, such that the y-axis is defined to have a standard angle of +90°.)
3
Physical Sciences 2: Lecture 1a
September 3, 2015
Vectors in Medicine
Vectors and components are crucial tools for analyzing EKG’s…
There are many signals (leads) recorded on
an ECG. Let’s look at two:
Lead I:
(x-component)
Lead aVF:
(y-component)
The large “peak” on the ECG tells about the mean electrical axis of the heart.
When the heart contracts, an
! electrical signal propagates through the heart muscle. The mean
electrical axis is a vector H that points in the direction of this electrical signal.
The two leads
shown above give
the components
of
!
this vector H .
What is the
direction of the
mean electrical
axis for the patient
shown here?
Be careful about
the direction of the
y-axis (aVF)!
4
Physical Sciences 2: Lecture 1a
September 3, 2015
Adding and Subtracting Vectors
•
  
Suppose I tell you that c = a + b .
What does this mean mathematically?
What does this mean graphically?
•
Unit vectors are very useful. Given any coordinate system, we can define:
x̂ : a vector that points in the +x direction with unit length
•
ŷ : a vector that points in the +y direction with unit length


Show graphically that a vector with components a = (–2, 3) is equal to a = −2 x̂ + 3ŷ .
  
Now suppose that c = a − b .
What does this mean mathematically?
What does this mean graphically?
5
Physical Sciences 2: Lecture 1a
September 3, 2015
Position, Velocity,
and Speed
•
1.
2.
3.
This image shows repeated
photographs of a basketball as it
bounces (from left to right).

y
The position vector r is a
vector from the origin of the
coordinate system to the location
of as object. Draw position
vectors for the initial position
and final position of the ball.

The displacement Δr is the
difference betweenthefinal and
initial positions: Δr = rf − ri .
Draw the overall displacement
x
vector between the two positions
from question 1 above.

Δr


The average velocity v during an interval is defined as: v =
Δt
If this photograph was captured at 25 images per second, estimate the average velocity of
the basketball during the duration of this image. (A basketball has a diameter of 23 cm.)
4.
5.
The average speed v of an object is a scalar (not a vector!) defined as the total
distance traveled divided by the total time. Estimate the average speed of the basketball
during the duration of this image.

Bonus! The instantaneous velocity v at any time is defined as the average velocity over
a very short time interval (strictly speaking, in the limit that the interval approaches zero).
Draw some vectors on the image to represent the instantaneous velocity of the ball at
various times.
6
y
Physical Sciences 2: Lecture 1a
September 3, 2015
Am I getting it?
•
1.
2.
3.
I’m going to start at the origin and walk 6 meters in the +x
direction at a speed of 2 m/s. Then I’ll run back to where I
started at a speed of 6 m/s.

What is my position vector r after walking (before I turn
around)? Choose all that are correct.



a) r = 6 m
b) r = (0 m, 6 m)
c) r = (6 m, 0 m)



d) r = (6 m) x̂
e) r = (6 m) x̂ + (0 m) ŷ
f) r = 6 x̂

g) r = 6 x̂ + 0 ŷ

What is my average velocity v during the entire trip (walking and running back)?
Choose all that are correct.



a) v = 3 m/s
b) v = 0 m/s
c) v = ( 0 m/s ) x̂



d) v = ( 0 m/s ) ŷ
e) v = (3 m/s, 0 m/s)
f) v = (0 m/s, 0 m/s)

g) v = 0
What is my average speed v during the entire trip (walking and running back)?
Choose all that are correct.
a) v = 3 m/s
b) v = 4 m/s
d) v = (3 m/s, 0 m/s)
e) v = 0
c) v = 0 m/s
Bonus! For each of the following equations, write an equivalent equation (or equations!)
involving the vector components. (x and y components will be sufficient)
 

a = 2b + c
K=
1 2
mv
2

1

Fdrag = − Cd ρ A v v
2
7
x
Physical Sciences 2: Mechanics, Elasticity, Fluids, and Diffusion
Harvard University, Fall 2015
Tu, Th 9:30–11:00am, Science Center C
Website: https://canvas.harvard.edu/courses/6282
Instructors:
Dr. Logan McCarty ([email protected]; 496-9009; Science Ctr 108)
Dr. Louis Deslauriers ([email protected]; Science Ctr 117C)
Dr. Greg Kestin ([email protected]; Science Ctr 117C)
• A schedule of office hours will be posted on the course website.
Head TF:
Michael Rowan, [email protected]
Sectioning:
Please select your section and lab times on the Canvas course website. The deadline
for requesting sections is NOON on Friday, Sept. 4. Any requests after that
deadline will be considered on a space-available basis. For questions about
sectioning, please contact our Head TF, Michael Rowan ([email protected])
Course Text:
Knight, Physics for Scientists and Engineers, 3rd edition. ISBN: 978-0321752949.
We consider this text to be recommended, not required. It is really just a reference.
Online Homework:
We will be using Sapling Learning for homework assignments and additional tutorials
this semester. Go to www.saplinglearning.com to register and purchase a subscription.
One semester costs $40; a full year of access (for Physical Sciences 2 and 3) costs $60.
Course Description:
An introduction to classical mechanics, with special emphasis on the motion in fluids
of biological objects, from proteins to people. Topics covered include: momentum and
energy conservation, kinematics, Newton's laws of motion, oscillations, elasticity,
fluids, random walks, and diffusion. Examples and problem set questions will be drawn
from the life sciences and medicine.
Placement Information:
PS 2 and 3 offer a calculus-based introduction to physics, with many examples and key
topics drawn from the life sciences and medicine.
Examination Dates:
Thursday, October 2
Thursday, November 13
Course Grading:
Midterm Exam 1
Midterm Exam 2
Final Exam
Problem Sets
Laboratory
Course Participation
15%
15%
30%
15%
15%
10%
Midterm 1
Midterm 2
8:30–11:00am (note the early start time!)
8:30–11:00am
Note: There will be no makeup exams.
If you miss one exam for a valid
reason, your final exam score will
be used in place of the missed exam.
Note: Your lowest problem set will be dropped.
Note: Any missed labs will receive a grade of zero.
Graded exams will be returned in discussion sections. Any concerns about grading
errors must be noted in writing and submitted to your TF in person before leaving the
classroom. Once an exam has left the classroom its grade will be considered final.
Course participation includes in-class activities, pre-class quizzes, and other activities
designed to help you learn. We will tell you the correct answers for every problem, so
you can earn full credit for every activity. If you earn 80% of the participation points
you will get full credit for participation; anything less will be prorated accordingly.
Prerequisites:
Calculus at the level of Mathematics 1b (may be taken concurrently).
Accessible Education:
Any student receiving accommodations through the Accessible Education Office
should present their AEO letter to the Head TF by Friday, Sept. 18. Failure to do so
may prevent us from making appropriate arrangements for the first exam.
Discussion Sections:
Discussion sections (1 hour) meet on Tues. and Wed. starting Sept. 8; you will choose
your section with the online sectioning program. For questions about sectioning,
contact the Head TF ([email protected]).
Problem Sets:
Problem sets will be posted on the Sapling website each Tuesday and will be due the
following Tuesday before lecture (by 9:30am). The written portions of each
assignment must be placed in the TF boxes outside Sci Ctr 108. Late problem sets will
not be accepted, but we will drop your lowest problem set grade. Collaboration on
problem sets is allowed, but you must acknowledge your collaborators, and you may
not simply copy answers: the final write-up must be your own work. Collaboration on
examinations is not allowed.
Help Room:
Teaching Fellows will be available to help with problem sets. The Physics Help
Room in Science Center 102 is open Fridays 3pm–5pm and Mondays 4pm–10pm.
Laboratory:
Laboratory experiments are an integral part of the course: your understanding of lab
will be tested on problem sets and on exams. Lab sessions meet in Sci Ctr 115 on
Tues., Wed., and Thurs. from 2-5 and 7-10pm, and Fri. from 9am-noon during the
weeks indicated on the schedule below. You should sign up for a lab time using the
online sectioning program. Attendance and participation is required. Questions about
labs should be sent to your Lab TF.
Labs
Lab 1:
Measurement
Lab 2:
Momentum
Lab 3:
Rotation
Lab 4:
Work and Energy
Lab 5:
Oscillations
Lab 6:
Random walk
Module/Class
1a
1b
2a
2b
3a
3b
4a
4b
Exam 1
4c
5a
5b
6a
6b
7a
7b
8a
8b
9a
9b
Exam 2
9c
10a
10b
10c
Final Exam
Date
Thu Sep 3
Tue Sep 8
Thu Sep 10
Tue Sep 15
Thu Sep 17
Tue Sep 22
Thu Sep 24
Tue Sep 29
Thu Oct 1
Tue Oct 6
Thu Oct 8
Tue Oct 13
Thu Oct 15
Tue Oct 20
Thu Oct 22
Tue Oct 27
Thu Oct 29
Tue Nov 3
Thu Nov 5
Tue Nov 10
Thu Nov 12
Tue Nov 17
Thu Nov 19
Tue Nov 24
Thu Nov 26
Tue Dec 1
Thu Dec 3
Tue Dec 8
Date TBA
Lectures
Vectors, Velocity, and Momentum
Interactions and the Center of Mass
Forces Change Momentum; HW 0 due (bonus!)
Kinematics with Constant Acceleration; HW 1 due
Contact Forces; Free-Body Diagrams
Friction and Drag; HW 2 due
Conservation of Energy
Work and Kinetic Energy; HW 3 due
Midterm Exam 1: Modules 1–3, Labs 1–2
Work and Potential Energy
Introduction to Rotational Motion
Rotational Dynamics; HW 4 due
Static Equilibrium
Simple Harmonic Motion; HW 5 due
Elasticity
Fluid Statics and Buoyancy; HW 6 due
Surface Tension
Gases, Fluid Flow, Aneurysms; HW 7 due
High Reynolds Number Flows
Low Reynolds Number Flows; HW 8 due
Midterm Exam 2: Modules 4–8, Labs 3–5
Statistical Mechanics
Boltzmann Distribution, Random Walk
Brownian Motion, Diffusion, Friction; HW 9 due
Thanksgiving
Brownian Motion, Diffusion, Friction
Final Exam Review
(reading period) HW 10 due
Final Examination
Physical Sciences 2: Lecture 1a
September 3, 2015
Interactions
•
Kinesin is pulling a vesicle along a microtubule:
1.
Identify all the interactions involving the vesicle. Classify each interaction as a contact
interaction or a long-range interaction.
2.
At various times, the vesicle starts and stops moving. Can you identify some interactions
that could be responsible for these changes in its motion?
3.
What are the fundamental interactions responsible for each of the interactions you
identified above?
4.
Bonus! An airplane is flying. Identify all the interactions involving the airplane.
8
Physical Sciences 2: Lecture 1a
September 3, 2015
Activity 2: Motion
•
This multi-flash photograph shows two pool balls (6 cm in diameter) rolling on a surface
without friction. The grid lines are spaced 10 cm apart, and the balls are photographed
at a rate of 25 frames per second. Initially, ball 2 is at rest (the dark black one), and ball
1 is entering from the left. The two balls collide and move off to the right.
ball 2
ball 2
ball 1
ball 1
1.

Estimate the velocity ( v ) of:
a) ball 1 before the collision
b) ball 1 after the collision
c) ball 2 after the collision
Activity continues on next page…
9
Physical Sciences 2: Lecture 1a
September 3, 2015
Activity 2 (continued)
2.
Identify all the interactions involving these balls.
3.
The mass of a pool ball is about 160 g. Considering the two balls together as a single
system, calculate:

a) the initial momentum pi of the system

b) the final momentum p f of the system
c) Is momentum conserved? Why or why not?
4.
Bonus! The kinetic energy of an object is defined as K = 12 mv 2 . Is kinetic energy
conserved for this system of two balls?
10
Physical Sciences 2: Lecture 1a
September 3, 2015
Activity 3: Measuring Mass
•
A cannon rolls freely on wheels on a flat
horizontal surface. When the cannon fires, the
cannonball moves to the right while the cannon
itself recoils to the left.
If the mass of the cannonball is 10 kg, the final
speed of the cannonball is 100 m/s, and the final
speed of the cannon is 1 m/s, what is the mass of
the cannon?
•
Bonus! You repeat the cannonball experiment on the moon, where the strength of gravity
is only 1/6 of that here on Earth. Would anything be different?
•
How can you use momentum conservation to measure the mass of a person?
11
Physical Sciences 2: Lecture 1a Pre-Reading
September 3, 2015
Additional Reading on Momentum
•
In physics, the concept of motion is linked with the concept of momentum. In everyday
speech, we say that something has “momentum” if it has a tendency to continue along its
current trajectory. Here are several recent headlines from the New York Times that
include the term “momentum”:
Services Sector Growth Suggests Strong Economic Momentum
Iran Nuclear Deal Gains Momentum With Endorsements of Four House Democrats
Momentum Slipping Away in Iowa, Walker Adopts a Trump-Like Stance
•



In physics, momentum is a vector quantity p defined by: p = mv


⎛  dr ⎞
where m is the mass and v is the velocity ⎜ v = ⎟
⎝
dt ⎠
Momentum is important in physics because: Momentum is Conserved.
To define “conserved,” we usually divide the universe into two parts: one part, called the
system, which contains the object(s) of interest, and another part, called the surroundings,
which is simply the rest of the universe. You should envision an imaginary boundary
around the system that separates it from the surroundings:
system
boundary
•
surroundings
In many instances, we can consider a system to be approximately isolated from the
surroundings: such a system does not interact in any way with the rest of the universe.
Although there are no truly isolated systems, an object in deep space (far away from
gravitational interactions with any planets or stars) would be isolated to a very high
degree of approximation. Given that definition of an isolated system, the Law of
Conservation of Momentum is:
The momentum of an isolated system is constant.
Or, expressed mathematically,

dp
For an isolated system,
=0
dt
According to this formulation, conservation of momentum means that the momentum of
an isolated system does not change with time.
3
Physical Sciences 2: Lecture 1a Pre-Reading
September 3, 2015
•
For an isolated system containing only one object, the mass of that object will be
constant, and so using our definition of momentum, we know:


d
dv
For a single isolated object,
(mv ) = 0 which implies that
=0
dt
dt
•
So momentum conservation means that the velocity of an isolated object is constant.
This is Newton’s First Law: An isolated object at rest will remain at rest; an isolated
object in motion will remain in motion in a straight line at constant speed.
•
Since objects are never truly isolated, we often study situations in which objects act as if
they are isolated. We can define an object as functionally isolated if it will keep going
in a straight line at constant speed without any intervention. Examples include objects
sliding on ice, rolling without friction, floating on water, or gliding on a cushion of air.
•
If a system contains two or more objects, the total momentum is the vector sum of the
momenta of the individual objects:

 




ptotal = p1 + p2 + p3 +… = m1v1 + m2 v2 + m3v3 +…
•
If an isolated system contains two or more objects, the total momentum is constant, but
momentum can be transferred from one object to another within the system.
•
Momentum is transferred from one object to another by interactions. If a system is not
isolated, that means it must be interacting in some way with its surroundings.
•
There are two basic kinds of macroscopic interactions that we will discuss in this course:
- Contact interactions require direct contact between the objects. Pushing and pulling
are examples of contact interactions. Friction is a more subtle example of a contact
interaction, as is drag (friction in a fluid). The important idea is: any time one object is
in contact with another, there will be some kind of contact interaction.
- Long-range interactions do not require contact. The most important example for this
course is gravity, which is a universal attractive interaction between all objects. Other
examples include electric and magnetic forces, which can act without direct contact.
•
All physical interactions actually arise from one of the four “fundamental interactions,”
which are known to physicists as:
- The electromagnetic force (electricity and magnetism)
- The weak nuclear force (responsible for some kinds of nuclear decay)*
- The strong nuclear force (holds protons and neutrons together inside nuclei)
- The gravitational force (gravity!)
Essentially all macroscopic interactions—pushing, pulling, friction, drag—as well as
microscopic interactions down to the level of atoms and molecules are electromagnetic
interactions. It’s amazing that electromagnetism and gravity describe almost all of the
interactions responsible for everyday physics, chemistry, and biology. *(Indeed,
electromagnetism and the weak force are part of the same “electroweak” interaction.)
4
Physical Sciences 2: Lecture 1a
September 3, 2015
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Physical Sciences 2: Lecture 1a
September 3, 2015
One-Minute Paper
Your name: _______________________________
•
What was the single most important thing you learned today?
•
What single topic left you most confused after today’s class?
•
Any other comments or reflections on today’s class?
13