Post on 29-Dec-2015
Physics 1501: Lecture 3, Pg 1
Physics 1501: Lecture 3Physics 1501: Lecture 3TodayToday’’s Agendas Agenda
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Physics 1501: Lecture 3, Pg 2
TodayToday’’s Topic :s Topic :
Finish Chapter 2Review of 1-D motionFree Fall
Strategy for Problem Solving
Begin Chapter 3Review of vectors
» Coordinate systems
» Math with vectors
» Unit vectors
Physics 1501: Lecture 3, Pg 3
Recap of 1-D motionRecap of 1-D motion
For constant acceleration:
From which we know:
Physics 1501: Lecture 3, Pg 4
Lecture 3, Lecture 3, Act 1Act 1Motion in One DimensionMotion in One Dimension
When throwing a ball straight up, which of the following is When throwing a ball straight up, which of the following is true about its velocity true about its velocity vv and its acceleration and its acceleration aa at the at the highest point in its path?highest point in its path?
(a)(a) BothBoth v = 0v = 0 andand a = 0a = 0..
(b)(b) v v 0 0, but , but a = 0a = 0..
(c) (c) v = 0v = 0, but , but a a 0 0..
y
Physics 1501: Lecture 3, Pg 5
Lecture 3, Lecture 3, Act 1Act 1SolutionSolution
x
tv
t
Going up the ball has positive velocity, while coming down Going up the ball has positive velocity, while coming down it has negative velocity. At the top the velocity is it has negative velocity. At the top the velocity is momentarily zero.momentarily zero.
Since the velocity is Since the velocity is
continually changing there mustcontinually changing there must
be some acceleration.be some acceleration. In fact the acceleration is caused In fact the acceleration is caused
by gravity ( by gravity (g = 9.81 g = 9.81 m/sm/s22).). (more on gravity in a few lectures)(more on gravity in a few lectures)
The answer is (c) The answer is (c) v = 0v = 0, but , but a a 0 0..
a
t
Physics 1501: Lecture 3, Pg 6
Free FallFree Fall
When any object is let go it falls toward the ground !! The force that causes the objects to fall is called gravity.
The acceleration caused by gravity is typically written as g
Any object, be it a baseball or an elephant, experiences the same acceleration (g) when it is dropped, thrown, spit, or hurled, i.e. g is a constant.
Physics 1501: Lecture 3, Pg 7
Gravity facts:Gravity facts:
g does not depend on the nature of the material!Galileo (1564-1642) figured this out without fancy clocks
& rulers!
demo - feather & penny in vacuum
Nominally, g = 9.81 m/s2 At the equator g = 9.78 m/s2
At the North pole g = 9.83 m/s2
More on gravity in a few lectures!
Physics 1501: Lecture 3, Pg 8
ProblemProblem:: On a bright sunny day you are walking around
the campus watching one of the many construction sites. To lift a bunch of bricks from a central area, they have brought in a helicopter. As the pilot is leaving, she accidentally releases the bricks when they are 1000 m above the ground. The worker below is getting ready to walk away in 10 seconds. Does he live?
Physics 1501: Lecture 3, Pg 9
Problem Solution Method:Problem Solution Method:Five Steps: (Similar to IDEA)
1) Focus the Problem- Draw a picture – what are we asking for?
2) Describe the physics- What physics ideas are applicable
- What are the relevant variables known and unknown
3) Plan the solution- What are the relevant physics equations
4) Execute the plan- Solve in terms of variables
- Solve in terms of numbers
5) Evaluate the answer- Are the dimensions and units correct?
- Do the numbers make sense?
Physics 1501: Lecture 3, Pg 10
Problem:Problem:
1. We need to find the time it takes for the brick to hit the ground.
1000 m
Physics 1501: Lecture 3, Pg 11
Problem:Problem:
2. Free Fall
Constant acceleration at value g
Choose coordinate system
Variables:h – height of helicopter = 1000 m
g – acceleration due to gravity = 9.81 m/s2
t – time to drop – goal
m – mass of brick – unknown
vo – initial velocity = 0 why?
1000 m
y = 0
y
Physics 1501: Lecture 3, Pg 12
Problem:Problem:
3. Next write position for constant acceleration:
1000 m
y = 0
y
Realize that v0y = 0
Solve for when y=0
• Why ??
Physics 1501: Lecture 3, Pg 13
Problem:Problem:
y0 = 1000 m
y
y = 0
Solve for t in symbols:
4. Substitute y=0 and vo=0
Solve for t in numbers:
Physics 1501: Lecture 3, Pg 14
Problem:Problem:
y0 = 1000 m
y
y = 0
5. Does it make sense?
Units worked out to be ‘s’
That’s correct for time
It takes 14.3 s.
That means that the man escapes !!!
Physics 1501: Lecture 3, Pg 15
Tips:Tips: Read !
Before you start work on a problem, read the problem statement thoroughly. Make sure you understand what information is given, what is asked for, and the meaning of all the terms used in stating the problem.
Watch your units !Always check the units of your answer, and carry the
units along with your numbers during the calculation.
Physics 1501: Lecture 3, Pg 16
Coordinate Systems / Chapter 3Coordinate Systems / Chapter 3
In 1 dimension, only 1 kind of system, Linear Coordinates (x) +/-
In 2 dimensions there are two commonly used systems,Cartesian Coordinates (x,y)Circular / polar Coordinates (r,)
In 3 dimensions there are three commonly used systems,Cartesian Coordinates (x,y,z)Cylindrical Coordinates (r,,z)Spherical Coordinates (r,)
Physics 1501: Lecture 3, Pg 17
Vectors (review):Vectors (review): In 1 dimension, we can specify direction with a + or - sign.
In 2 or 3 dimensions, we need more than a sign to specify the direction of something:
To illustrate this, consider the position vector rr in 2 dimensions.
ExampleExample: Where is Boston? Choose origin at New YorkNew York Choose coordinate system Boston is 212 miles northeast of
New York
or
Boston is 150 miles north and
150 miles east of New York
Boston
New York
r r
Physics 1501: Lecture 3, Pg 18
Vectors...Vectors... There are two common ways of indicating that
something is a vector quantity:
Boldface notation: AA
“Arrow” notation:
A A =
AA
AA
Physics 1501: Lecture 3, Pg 19
Vectors: definitionVectors: definition
A vector is composed of a magnitude and a directionexamples: displacement, velocity, accelerationmagnitude of A is designated |A|usually carries units
A vector has no particular position
Two vectors are equal if their directions and magnitudes match.
A
B CA = C
A = B, B = C
Physics 1501: Lecture 3, Pg 20
Vectors and scalars:Vectors and scalars:
A scalar is an ordinary number.a magnitude without a directionmay have units (kg) or be just a numberusually indicated by a regular letter, no bold face and no
arrow on top.
Note: the lack of specific designation of a scalar can lead to confusion
The product of a vector and a scalar is another vector in the same direction but with modified magnitude.
A BA = -0.75 B
Physics 1501: Lecture 3, Pg 21
Lecture 3, Lecture 3, ACT 2ACT 2Vectors and ScalarsVectors and Scalars
A) my velocity (3 m/s)
C) my destination (the pub - 100,000 m)
B) my acceleration downhill (30 m/s2)
D) my mass (150 kg)
Which of the following is not a vector ?(For bonus points, which answer has a reasonable
magnitude listed ?)
While I conduct my daily run, several quantities describe my condition
Physics 1501: Lecture 3, Pg 22
Lecture 3, Lecture 3, ACT 1AnswerAnswer
Which of the following is not a vector ?(For bonus points, which answer has a reasonable
magnitude listed ?)
While I conduct my daily run, severalquantities describe my condition
D) my mass (150 kg)
Answer: There is no way to assign a direction to a mass
Can only be a scalar
A) Velocity has direction (3 m/s east)B) Acceleration has directionC) Destination = position and has a direction ( pub is
100,000 m east of my house).
Physics 1501: Lecture 3, Pg 23
Lecture 3, Lecture 3, ACT 2ACT 2Bonus AnswerBonus Answer
A) my velocity(3 m/s)
C) my destination (the pub - 100,000 m)
B) my acceleration downhill (30 m/s2)
D) my mass (150 kg)
~ 6 mi/hr or 10 min miles
0 to 60 mi/hr in 1 seconds(3 times g)
100 km is about 60 mi
Pounds are a measure of forceOn the earth’s surface, 1 kg weighs
2.2 lbs150 kg weighs about 330 lbs
Answer = (A)
Physics 1501: Lecture 3, Pg 24
Converting Coordinate SystemsConverting Coordinate Systems In circular coordinates the vector R = (r,)
In Cartesian the vector R = (rx,ry) = (x,y)
We can convert between the two as follows:
• In cylindrical coordinates, r is the same as the magnitude of the vector
rx = x = r cos
ry = y = r sin
arctan( y / x )
y
x
(x,y)
rrry
rx
Physics 1501: Lecture 3, Pg 25
Vector addition:Vector addition:
The sum of two vectors is another vector.
A = B + C
B
C A
B
C
Physics 1501: Lecture 3, Pg 26
Vector subtraction:Vector subtraction:
Vector subtraction can be defined in terms of addition.
B - C
B
C
B
-C
B - C
= B + (-1)C
Physics 1501: Lecture 3, Pg 27
Unit Vectors:Unit Vectors:
A Unit Vector Unit Vector is a vector having length 1 and no units.
It is used to specify a direction. Unit vector uu points in the direction of UU.
Often denoted with a “hat”: uu = û
UU
û û
x
y
z
ii
jj
kk
Useful examples are the cartesian unit vectors [ i, j, ki, j, k ]
point in the direction of the x, y and z axes.
R = rxi + ryj + rzk
Physics 1501: Lecture 3, Pg 28
Vector addition using components:Vector addition using components:
Consider CC = AA + BB.
(a) CC = (Ax ii + Ay jj ) + (Bx i i + By jj ) = (Ax + Bx )ii + (Ay + By )jj
(b) CC = (Cx ii + Cy jj )
Comparing components of (a) and (b):
Cx = Ax + Bx
Cy = Ay + By
CC
BxAA
ByBB
Ax
Ay
Physics 1501: Lecture 3, Pg 29
Lecture 3, Lecture 3, ACT 3ACT 3Vector AdditionVector Addition
Vector A = {0,2,1} Vector B = {3,0,2} Vector C = {1,-4,2}
What is the resultant vector, D, from adding A+B+C?
(a)(a) {3,-4,2}{3,-4,2} (b)(b) {4,-2,5}{4,-2,5} (c)(c) {5,-2,4}{5,-2,4}
Physics 1501: Lecture 3, Pg 30
Lecture 3, Lecture 3, ACT 3ACT 3SolutionSolution
D = (AXi + AYj + AZk) + (BXi + BYj + BZk) + (CXi + CYj + CZk)
= (AX + BX + CX)i + (AY + BY+ CY)j + (AZ + BZ + CZ)k
= (0 + 3 + 1)i + (2 + 0 - 4)j + (1 + 2 + 2)k
= {4,-2,5}
Answer is (b)
Vector A = {0,2,1} Vector B = {3,0,2} Vector C = {1,-4,2}