© 2001-2005 shannon w. helzer. all rights reserved. 1 unit 6 analytical vector addition

© 2001-2005 Shannon W. Helzer. All Rights Reserved. 1

Unit 6Analytical Vector Addition


Multiplying a Vector by a Scalar

A

A

C = -1/2 A

B = 2A

C

BA

½ A


Adding “-” Vectors

C = A + B

D = A - B

D = A + (- B)

A

B

C -B

D

Add “negative” vectors by keeping the same magnitude but adding 180 degrees to the direction of the original vector.


Components of Vectors

A = Ax + Ay

Ay

Ax

Recall: Vectors are always added “head to tail.”

A



A = Ax + Ay

Finding the components when you know A.

coscos AAA

Ax

x

sin sinyy

AA A

A

Recall: is measured from the positive x axis.


If A = 3.0 m and = 45, find Ax + Ay.

Ay

Ax

A

?)cos(0.3cos anglemAAx

)45cos(0.3cos mAAx

mAx 1.2

)45sin(0.3sin mAAy

mAx 1.2

cosAAx sinAAy

From the + x Axis



Finding the vector magnitude and direction when you know the components.

Recall: is measured from the positive x axis.

22yx AAA

x

y

x

y

A

A

A

Aarctantan

Caution: Beware of the tangent function.

Always consider in which quadrant the vector lies when dealing with the tangent function.


If Ax = 2.0 m and Ay. = 2.0 m, then Find A and .

Ay

Ax

A

822 2222 yx AAA

452

2arctanarctan

x

y

A

A

315360


22yx AAA

x

y

A

Aarctan

cosAAx sinAAy

Adding Vectors With Components.

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CBAD

A

B

C

D


Follow this type of methodology when doing these problems.

Magnitude Angle Rx Ry

A=72.4m 58 38.37 m 61.4 m

B = 57.3 m 216 -43.36 m -33.68 m

C = 17.8 m 270 0.0 m -17.80 m

Rx = -7.99 m Ry = 9.92m

R = 12.7 m Angle = 129 degrees

Component Template


Hand Glider Trip


Analytical Vector Addition – Hand Glider

Vector Magnitude Angle x component y component Quad

-------------- -------------- -------------- -------------- -------------- --------------

Find the Final displacement of the hand glider using analytical vector addition.

This problem is similar to the following problems: WS 21, 1; WS 22, 1; and WS 23, 4.


Analytical Vector Addition


-------------- -------------- -------------- -------------- -------------- --------------

Three Physics track robots pull on a book as shown.

They pull with the following forces: 7.0 N @ 45, 8.0 N @ 180, and 5.0 N @ 270 .

Find the net force applied to this most valuable book.

This problem is similar to WS 21, 2.


Statics – Hanging Sign Draw the Static FBD for the sign below What do we need to do with the tension (T)? Resolve it into its components (Tx & Ty).


Statics – Hanging Sign

Draw the Static FBD for the sign below What do we need to do with the tension (T)? Resolve it into its components (Tx & Ty).


Inclined Plane Problems

Draw the FBD for the piano on the inclined plane. What will we have to do with the Normal Force (N) and the force of

friction (Ff)? Resolve them into their x and y components.


Inclined Plane Problems

Would you like to do less work? How could we do this problem by resolving only one force? Try rotating the FBD so that the N is in the y plane and the Ff is in the x

plane.




F1

F2

F3

F4

-------------- -------------- -------------- -------------- -------------- --------------

F

Use the table below when performing analytical vector addition. Do WS 23 numbers 1 & 2.


Vr

Relative Velocity


Relative Velocity




F1

F2

F3

F4

-------------- -------------- -------------- -------------- -------------- --------------

F

Do WS 23 number 3. This problem is similar to WS

24, 3 and WS 22 , 2.


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Where we are committed to Excellence In Mathematics And Science

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F1

F2

F3

F4

-------------- -------------- -------------- -------------- -------------- --------------

F

A


Setting the Standard When we do problems involving

kinematics, it is important that we stick to a standard when imputing data into the know-want table.

This standard enables us to take into account the vector nature of acceleration, velocity, displacement, etc.

Here is a diagram we will use in order to help us correctly input data into the table.

This standard is based upon the Cartesian Coordinate system.

If a body travels West, then what sign would you give its velocity?

If a body travels at an angle of 90 degrees, then what sign would you give its velocity?

3-3


Dot Products

Write each of the three vectors given in their unit vector notation.

A = 15.0 m @ 30 B = 22.0 m @ 225 C = 9.0 m @ 267 Calculate the Dot Products below.

3-3

A B

B C

B B

C A

A C


Dot Products – Finding the angle

Given the vectors below, find the angles between the following vectors. A and C. B and A. C and E. D and E.

3-3

ˆˆ ˆ3 2 4A i j k

ˆˆ ˆ3.5 4 2B i j k

ˆˆ ˆ1.5 4 2C i j k

ˆ5D j

ˆˆ1.8 2.2E i k


3-D Cartisian Coordinate System

+x

+z

+y

zyx AAAA

222zyx AAAA

1-17


Unit Vectors

-i

-j

j

k i

-k

z

y

x

A = Axi + Ayj + Azk

Note: Remember to put the “^” over the hand written vector when writing unit vectors.

1-18


Scalar or “Dot” Product

B

A

B

A

The Dot product gives the projection of one vector onto another.

You can also use the dot product to find the angle between the vectors.

BA = Projection of B onto A.

BA

AB = Projection of A onto B.

AB

1-19



B

A

The Dot product results in a Scalar quantity.

ABBA coscos BAAB

1-20


Scalar or “Dot” Product & Unit Vectors

2222 614462124324 yxyxyyxxyxyxyx

You “multiply” the dot product in a similar way as below.

A = Axi + Ayj B = Bxi + Byj

A•B = (Axi + Ayj) • (Bxi + Byj)

A•B = Ax i • Bxi + Ax i • Byj+ Ayj • Bxi + Ayj • ByjHowever,

i • i = j • j = k • k = 1

i • j = i • k = j • k = 0

A•B = Ax Bx + Ay By1-21



B

A

One use for the dot product is to determine the angle between two

vectors.

AB

BABA

AByyxx

BA

arccos1-22


ABR 2BAR 1

Vector or “Cross” Product

Right hand rule: Place the fingers of your right hand in the direction of the first vector in the cross product. Rotate your fingers towards the second vector. Your thumb tells you the direction of the resultant vector.

B

A

B

A

2R

1R

The Cross product results in a VECTOR quantity.

1-23



BAR 1

B

A

1R

The Cross product results in a VECTOR quantity.

The magnitude of the vector is given by

sin1 ABR

WARNING: AB sin DOES NOT EQUAL BA sin

A x B DOES NOT EQUAL B x A

However, A x B = - B x A 1-24



AxB = (Axi + Ayj) x (Bxi + Byj)

AxB = Ax i x Bxi + Ax i x Byj+ Ayj x Bxi + Ayj x Byj

However,i x i = j x j = k x k = 0

i x j = -j x i = k j x k = -k x j = iK x i = -i x k = j

AxB = Ax i x Byj+ Ayj x Bxi

AxB = (AxBy )i x j + (AyBx)j x i

AxB = (AxBy )k - (AyBx)k

1-25



A x B = i j k i jAx Ay Az Ax AyBx By Bz Bx By

A = Axi + Ayj + Azk B = Bxi + Byj + Bzk

A x B = (Axi + Ayj + Azk) x (Bxi + Byj + Bzk)

A x B = AyBzi - AzByi+ AzBxj - AxBzj+ AxByk- AyBxk

A x B = (AyBz –AzBy)i + (AzBx-AxBz)j + (AxBy-AyBx)k

Determinant method of solving for the cross product.

A x B = Rx i + Ry j + Rz k1-26


y

x

z

r

z

x

y

r

y

r x y

siny r

sinr cosx r

cosz

sin siny

sin cosx

ˆˆ ˆxi yj zk

Spherical Coordinates


Advanced Physics Unit 5Applications of Newton’s Laws


Newton’s Laws – A Review Newton’s First Law - An object remains at rest, or in uniform motion in a

straight line, unless it is compelled to change by an externally imposed force.

Newton’s first law describes an Equilibrium Situation. An Equilibrium Situation is one in which the acceleration of a body is

equal to zero. Newton’s Second Law – If there is a non-zero net force on a body, then it

will accelerate. Newton’s Second Law describes a Non-equilibrium Situation. A Non-equilibrium Situation is one in which the acceleration of a body is

not equal to zero.

Newton’s Third Law - for every action force there is an equal, but opposite, reaction force.


Free Body Diagrams – A Review

When solving problems involving forces, we must draw FBDs of all bodies involved in the force interactions.

Since torque is related to force, we must modify the FBD concept to apply to bodies upon which a torque acts.

Before we carry out this modification, lets review problems involving force using FBDs.

If the crate started from rest, then which way did it accelerate? Draw the FBD for the crate. What type of a situation is depicted below?

DigDug


Free Body Diagrams – Inclined Plane (WS 14 # 8) A block slides down an inclined plane as

shown. Draw the FBD for the block as it slides down

the ramp at a constant speed. Write the Newton’s laws in vector form for

the block in both the horizontal and vertical directions.

Now convert from vector form to math form.

x Fx x xF F N ma

y Fy y yF F N W ma

x Fx x xF F N ma

y Fy y yF F N W ma


WS 14 Problems 1-4 A mass rest on an inclined plane as shown. What type of friction is acting on the mass? Now suppose the mass begins to slide down the plane. What type of friction is acting on the mass as it slides? Draw the FBD for the mass while at rest and while

sliding down the plane.


Static Friction v. Kinetic Friction Static friction exists when an object wants to move but is held in

place by the force of friction. This force of friction is greater than the component of the weight

acting down the plane. If we continue to rotate the plane, the component of the weight

acting down the plane will eventually become larger than the normal force.

When this happens, the object will begin to slide changing from static friction to kinetic friction.

Force of Friction v. Time

-7

-6

-5

-4

-3

-2

-1

0

1

0 2 4 6 8 10

Time (s)

Fri

ctio

n (

N)


Like WS 14 Problem 5 MeanyBot and PhysicsBot are moving a crate as shown. MeanyBot is pulling with a force F2 = 10,000 N and the PhysicsBot

is pushing with a force of F1 = 6,000 N.

Additionally, the coefficient of kinetic friction, k, is 0.459.

The mass of the crate is 1000 kg. Determine the net force and the acceleration of the crate.


WS 14 Problem 9

Derive the equations needed to determine the tension and the acceleration of the weights (m1<m2) on the Atwood’s machine shown to the right..

What type of a situation do we have when the masses first begin to move?

Define this situation with its two predominate characteristics.


WS 14 Problem 10 – Elevator Problems

An elevator (m = 675.0 kg) ascending at a rate of 8.5 m/s comes to a stop in a distance of 22.0 m.

Find the Tension in the three cables supporting the weight of the elevator and the acceleration experienced by the elevator.

What type of a situation is the elevator in while coming to a stop?

Now suppose Dr. Physics (m = 62.5 kg) is standing on a scale inside the elevator.

After three seconds of descending, the elevator begins traveling at a constant speed of 9.0 m/s.

What does the scale say that Dr. Physics weighs while he descends?

What situation is the elevator in once it begins traveling at a constant velocity?


WS 15 Problem 1

Block A below weighs 90.0 N. The coefficient of static friction

between the block and the table is s = 0.30.

Block B weighs 15.0 N. The system is in equilibrium. Draw and label the FBDs for Body A

& Body B. What is meant by the term

“equilibrium” above? Find the friction force acting on

Block A.

B

45°A

AW BW

N

AT

BT


WS 15 Problem 2

A wooden block (m1) rests on a plane inclined at an angle of .

This block is attached to mass m2 held at a height of y above the ground.

The coefficient of friction between the block and the incline is K.

Derive the equations (in terms of m1, m2, K, y, , and g) needed to calculate the tension in the string, the acceleration of the system, and the time needed for to hit the ground.

1W 2W

N

1T

2T


WS 15 Problem 3

Derive the equations needed to determine the tension in each chain given the angle , that the angle between chain 2 and the post is 90, and the fact that the weight of the bug zapper is W.

1T

2T


WS 15 Problem 4

Derive the equations needed to determine m2 and the tensions in the strings given angles and that the mass of weight one is m1.

Suppose m1 = 10.0 kg. What are the tensions and what is the value of m2?



1m

2m

3T 1T 2T


222222 A wrapped box (m1) rests on

a table and is attached to a hanging weight (m2) as shown.

The coefficient of friction between the box and the table is K.

The weight is released pulling the box to the right as shown.

Derive the equations (in terms of m1, m2, K, and g) needed to calculate the tension in the string and the acceleration of the system.

1W 2W

N

1T

2T


Relative Velocity


Relative Velocity

1T

2T


Torque a F l

1 1 1W l


Static Friction v. Kinetic Friction Static friction exist when an object wants to move but is held in

place by the force of friction. This force of friction is greater than the component of the weight

acting down the plane. If we continue to rotate the plane, the component of the weight

acting down the plane will eventually become larger than the normal force.

When this happens, the object will begin to slide changing from static friction to kinetic friction.


-7

-6

-5

-4

-3

-2

-1

0

1

0 2 4 6 8 10

Time (s)

Fri

ctio

n (

N)


Torque a F l

1 1 1W l


Torque a


Torque a F l

1 1 1W l


Advanced Physics Unit 5 Exam

QUESTION 1 - 3

QUESTION 4

QUESTION 5

QUESTION 6


Unit 5 Exam Problems 1-3 The graph below shows the force of friction verses the pull time. What type of friction is represented in the portion of the graph that is blue in

color? What type of friction is represented in the portion of the graph that is red in

color? What is physically happening at the point where the graph changes from blue

to red?


-7

-6

-5

-4

-3

-2

-1

0

1

0 2 4 6 8 10

Time (s)

Fri

ctio

n (

N)

RETURN


Unit 5 Exam Problem 4

Under what conditions would a Elevator passenger appear to weigh more than his or her actual weight: while accelerating upwards, while accelerating downwards, or while riding at a constant speed?

Justify your answer using verbal explanations or equations as needed.

RETURN

RIDE


Unit 5 Exam Problem 5 A wrapped box (m1) rests on

a table and is attached to a hanging weight (m2) as shown.

The coefficient of friction between the box and the table is K.

The weight is released pulling the box to the right as shown.

Derive the equations (in terms of m1, m2, K, and g) needed to calculate the tension in the string and the acceleration of the system.

RETURN


Unit 5 Exam Problem 6

Derive the equations needed to determine the tension in each chain given angles and and the fact that the weight of the bug zapper is W.

1T 2T

RETURN

© 2001-2005 shannon w. helzer. all rights reserved. 1 unit 6 analytical vector addition

Documents

x axis slide

components of vectors

vectors c

b c b d

components t x t y

hand glider trip slide

vector magnitude

positive x axis