Honors Physics, Pg 1 Honors Physics Vectors l Chapter Problems All problems prior to Mixed review l Vector addition graphically A vector can be represented

Honors Physics , Pg 1

Honors Physics Honors Physics VectorsVectors

Chapter Problems All problems prior to Mixed review

Vector addition graphically

A vector can be represented by an arrow tipped line segment


Vectors...Vectors...

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

Boldface notation: AA

“Arrow” notation:

A A =A

A

A“Hat”

Notation



The components of rr are it’s (x,y,z) coordinates. rr = (rx ,ry ,rz ) = (x,y,z)

Consider this in 2-D (since it’s easier to draw):rx = x = r cos ry = y = r sin

y

x

(x,y)

where r = |rr |

rr arctan( y / x )

NOTE: The book uses instead of . No big deal.



The magnitude (length) of rr is found using Pythagoras’ theorem:

r r x y2 2

rr y

x

The length of a vector clearly does notnot depend on it’s direction.


Angle Determination using the UNIT CircleAngle Determination using the UNIT Circle

90 degrees

Resultant Vector ‘C’

0 degrees

180 degrees

270 degrees

degrees


Angle Determination using the UNIT CircleAngle Determination using the UNIT Circle

90 degrees

Resultant Vector ‘C’

0 degrees

180 degrees

270 degrees

degrees

degrees


Vector addition:Vector addition:

Consider the vectors AA and BB. Find AA + BB.

AA

BB

AA BB AA BB

CC = AA + BB

Vectors can be added by placing the head of one vector at the head of another vector

» When adding two or more vectors, neither the length or the direction is changed

» The sum of the vectors is found by drawing a RESULTANT vector from the tail of the first vector to the head of the last (beginning to end)

We can arrange the vectors as we want, as long as we maintain their length and direction !!

See text: 3-2 and 3-3


Addition of several vectorsAddition of several vectors

Two students are pulling concurrently (same place, same time)5 N , 0 degrees and 7 N, 270 degrees on a 20kg oscilloscope (assume a frictionless surface)

a) determine the direction of motion

b) determine the net force and acceleration of the scope

5 N 0 degrees

7 N 270 degrees



5 N 0 degrees

7 N 270 degrees

Add vectors head to tail and draw resultant

With recognition that a right angle exists, the Pythagorean theorem and SOHCAHTOA can be used



5N,0 deg

7N 270 degr r 5 7 8 62 2 .

tan = 7/5

= 54.5 degrees but using unit circle as a reference the correct angle which describes the true direction of the resultant should be either 305.5 or -54.5 degrees

NOTE:



Two students are pulling concurrently (same place, same time)5 N , 0 degrees and 7 N, 270 degrees on a 20kg oscilloscope (assume a frictionless surface)



5 N 0 degrees

7 N 270 degrees Resultant: NET FORCE: 8.6N,305.5 degrees



The direction of the acceleration will be in the same direction of the NET FORCE


a=F(net)/m = 8.6N/20kg = .43 m/s2 ,305.5 degrees

5 N 0 degrees

7 N 270 degrees Resultant: NET FORCE: 8.6N,305.5 degrees


Addition of several vectors Addition of several vectors What if the angle between the vectors is not 90 degrees?What if the angle between the vectors is not 90 degrees?

Vector resolution and components verses trig laws of cosines and sines

Two students are pulling concurrently (same place, same time)5 N , 10 degrees and 7 N, 300 degrees on a 20kg oscilloscope such that it accelerates at .3 m/s2


b) determine the net force and coefficient of friction between the scope and the floor

5 N 10 degrees

7 N 300 degrees


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


Frictionless Inclined planeFrictionless Inclined plane

Vector view of FFTOT = maa

parallelparallel

mgN

ma

See text: 6-4

perpendicular


Inclined Plane without Friction:Inclined Plane without Friction:

Draw free-body diagram:

parallelparallel

perpendicularperpendicular

mgN

ma

See text: 6-4

See example p124.


Inclined plane...Inclined plane...

Consider xx and y y components of FFTOT = maa :

Parallel components

mg sin = ma

a / g = sin

See text: 6-4

parallelparallel

perpendicularperpendicular mg

N

ma

mg sin



Consider xx and y y components of FFTOT = maa :

Perpendicular components

perpen perpen N = mg cos

See text: 6-4

parallelparallel


mg cos

mgN

ma


Static Friction:Static Friction:

So far we have considered friction acting when something moves.We also know that it acts in un-moving “static” systems:

See text: 6-4

In these cases, the force provided by friction will depend on the configuration of the system (i.e. angle of the plane).




llll

mgN KN

ma

See text: 6-4


Inclined Plane with Friction:Inclined Plane with Friction:


parallelparallel


mgN

KNma

See text: 6-4

See example p124.



Consider ii and j j components of FFTOT = maa :

parallelparallel mg sin KN = ma


mg sin Kmg cos = ma

a / g = sin Kcos

mg

N

KN

ma

mg sin

mg cos

See text: 6-4

parallelparallel



Static Friction...Static Friction...

mg

N

ma = 0 (block is not moving)

The force provided by friction, fF , depends on as long as the block is “static”.

fF

mg sin ff

(Newtons 2nd law along x-axis)

See example p126: Static Friction

See text: 6-4


Static Friction...Static Friction... Definition: The maximum possible force that the friction between two objects can provide is fMAX =

sN, where s is the “static coefficient of friction”.

We can find s by increasing the ramp angle until the block slides:

M mg

N

SN In this case:

ma = mg sin MSmg cos M

Stan M

See text: 6-4

See example p126 Static Friction




llll

mgN KN

ma

See text: 6-4


Inclined Plane with Friction:Inclined Plane with Friction:


parallelparallel


mgN

KNma

See text: 6-4

See example p124.



Consider components of FFTOT = maa :

parallelparallel mg sin KN = ma


mg sin Kmg cos = ma

a / g = sin Kcos

mg

N

KN

ma

mg sin

mg cos

See text: 6-4

parallelparallel


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Honors Physics, Pg 1 Honors Physics Vectors l Chapter Problems All problems prior to Mixed review l Vector addition graphically A vector can be represented