Physics 207: Lecture 4, Pg 1 Lecture 4 l Today: Ch. 3 (all) & Ch. 4 (start) Perform vector algebra...

Physics 207: Lecture 4, Pg 1

Lecture 4 Today: Ch. 3 (all) & Ch. 4 (start)

Perform vector algebra• (addition & subtraction) graphically or numerically • Interconvert between Cartesian and Polar coordinates

Begin working with 2D motion• Distinguish position-time graphs from particle trajectory plots• Trajectories

Obtain velocities Acceleration: Deduce components parallel and perpendicular to the trajectory path

Reading Assignment: For Monday read through Chapter 5.3 Reading Assignment: For Monday read through Chapter 5.3


Coordinate Systems and vectors

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 Coordinates (r,)

In 3 dimensions there are three commonly used systems,Cartesian Coordinates (x,y,z)Cylindrical Coordinates (r,,z)Spherical Coordinates (r,)


Vectors 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 [ in (r,) ] OR

Boston is 150 miles north and 150 miles east of New York [ in (x,y) ]

Boston

New York

rr


Vectors...

There are two common ways of indicating that something is a vector quantity:

Boldface notation: A

“Arrow” notation:

A or

A

A


Vectors

Vectors have both magnitude and a direction Examples: displacement, velocity, acceleration The magnitude of A ≡ |A| Usually vectors include units (m, m/s, m/s2)

For vector algebra ( +/ -)a vector has no particular position (Note: the position vector reflects displacement from the origin) Two vectors are equal if their directions, magnitudes & units

match.

A

B CA = C

A = B, B = C


Scalars

A scalar is an ordinary number. A magnitude ( can be negative ) without a direction Often it will have units (i.e. kg) but can be just a number Usually indicated by a regular letter, no bold face and no

arrow on top.Note: Lack of a specific designation 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


Exercise Vectors and Scalars

A. my velocity (3 m/s)B. my acceleration downhill (30 m/s2)C. my destination (the lab - 100,000 m east)D. my mass (150 kg)

Which of the following is cannot be a vector ?

While I conduct my daily run, several quantities describe my condition


Vectors and 2D vector addition

The sum of two vectors is another vector.

A = B + CB

C A

B

C

D = B + 2 C ???


2D Vector subtraction

Vector subtraction can be defined in terms of addition.

B - C

B

C

B

-C

B - C

= B + (-1)C

A Different direction and magnitude !A = B + C


Reference vectors: Unit Vectors

A 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

U = |U| U = |U| ûû

û û

x

y

z

ii

jj

kk

Useful examples are the cartesian unit vectors [ i, j, k ] or

Point in the direction of the x, y and z axes.

R = rx i + ry j + rz k

]ˆ,ˆ,ˆ[ zyx


Vector addition using components:

Consider, in 2D, C = A + B. (a) CC = (Ax ii + Ay jj ) + (Bx i i + By jj ) = (Ax + Bx )ii + (Ay + By )

(b) CC = (Cx ii + Cy jj )

Comparing components of (a) and (b):

Cx = Ax + Bx

Cy = Ay + By

|C| =[ (Cx)2+ (Cy)2 ]1/2

CC

BxAA

ByBB

Ax

Ay


ExampleVector Addition

A. {3,-4,2}

B. {4,-2,5}

C. {5,-2,4}

D. None of the above

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?


ExampleVector Addition

A.A. {{3,-4,2}

B. {4,-2,5}

C. {5,-2,4}

D. None of the above

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?


Converting Coordinate Systems

In polar coordinates the vector R = (r,)

In Cartesian the vector R = (rx,ry) = (x,y)

We can convert between the two as follows:

• In 3D cylindrical coordinates (r,z), r is the same as the magnitude of the vector in the x-y plane [sqrt(x2 +y2)]

tan-1 ( y / x )

22 yxr

y

x

(x,y)

rry

rx

j i

sin

cos

yxr

ryr

rxr

y

x


Resolving vectors into componentsA mass on a frictionless inclined plane

A block of mass m slides down a frictionless ramp that makes angle with respect to horizontal. What is its acceleration a ?

ma


Resolving vectors, little g & the inclined plane

g (bold face, vector) can be resolved into its x,y or x’,y’ components g = - g j g = - g cos j’ + g sin i’

The bigger the tilt the faster the acceleration…..

along the incline

x’

y’

x

yg


Dynamics II: Motion along a line but with a twist(2D dimensional motion, magnitude and directions)

Particle motions involve a path or trajectory

Recall instantaneous velocity and acceleration

These are vector expressions reflecting x, y & z motion

r = r(t) v = dr / dt a = d2r / dt2


Instantaneous Velocity But how we think about requires knowledge of the path. The direction of the instantaneous velocity is along a line that is

tangent to the path of the particle’s direction of motion.

v

The magnitude of the instantaneous velocity vector is the speed, s. (Knight uses v)

s = (vx2 + vy

2 + vz )1/2


Average Acceleration

The average acceleration of particle motion reflects changes in the instantaneous velocity vector (divided by the time interval during which that change occurs).

a The average

acceleration is a vector quantity directed along ∆v

( a vector! )


Instantaneous Acceleration

The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero

The instantaneous acceleration is a vector with components parallel (tangential) and/or perpendicular (radial) to the tangent of the path

Changes in a particle’s path may produce an acceleration The magnitude of the velocity vector may change The direction of the velocity vector may change

(Even if the magnitude remains constant) Both may change simultaneously (depends: path vs time)


Generalized motion with non-zero acceleration:

need both path & time

Two possible options:

Change in the magnitude of v

Change in the direction of v

a = 0

a = 0

aaa= +

Animation

22 with 0 tr aaaa

aar

||aat

a

v


Kinematics

The position, velocity, and acceleration of a particle in

3-dimensions can be expressed as:

r = x i + y j + z k

v = vx i + vy j + vz k (i , j , k unit vectors )

a = ax i + ay j + az k

All this complexity is hidden away in

r = r(t) v = dr / dt a = d2r / dt2

2

2

dtxd

ax 2

2

dtyd

ay 2

2

dtzd

az

dt

dxvx

dt

dyvy

dtdz

vz

)( txx )( tyy )( tzz

221

0 )( e.g. accel.,constant if with,0

tatvxtx xx


Special Case

Throwing an object with x along the horizontal and y along the vertical.

x and y motion both coexist and t is common to both

Let g act in the –y direction, v0x= v0 and v0y= 0

y

t0 4 x

t = 0

4y

t

x

0 4

x vs t y vs t x vs y


Another trajectory

x vs yt = 0

t =10

Can you identify the dynamics in this picture?

How many distinct regimes are there?

Are vx or vy = 0 ? Is vx >,< or = vy ?

y

x


Exercise 1 & 2Trajectories with acceleration

A rocket is drifting sideways (from left to right) in deep space, with its engine off, from A to B. It is not near any stars or planets or other outside forces.

Its “constant thrust” engine (i.e., acceleration is constant) is fired at point B and left on for 2 seconds in which time the rocket travels from point B to some point C Sketch the shape of the path

from B to C. At point C the engine is turned off.

Sketch the shape of the path

after point C (Note: a = 0)


Exercise 1Trajectories with acceleration

A. A

B. B

C. C

D. D

E. None of these

B

C

B

C

B

C

B

C

A

C

B

D

From B to C ?



A. A

B. B

C. C

D. D

E. None of these

B

C

B

C

B

C

B

C

A

C

B

D

From B to C ?



A. A

B. B

C. C

D. D

E. None of these

C

C

C

C

A

C

B

D

After C ?


Lecture 4

Assignment: Read all of Chapter 4, Ch. 5.1-5.3Assignment: Read all of Chapter 4, Ch. 5.1-5.3

Physics 207: Lecture 4, Pg 1 Lecture 4 l Today: Ch. 3 (all) & Ch. 4 (start) Perform vector algebra...

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Transcript of Physics 207: Lecture 4, Pg 1 Lecture 4 l Today: Ch. 3 (all) & Ch. 4 (start) Perform vector algebra...