ChapteChapter 5r 5

Forces in Two Dimensions

Vector vs. Scalar ReviewVector vs. Scalar Review

Vector vs. Scalar ReviewVector vs. Scalar ReviewAll physical quantities encountered in

this text will be either a _____ or a ______.

A vector quantity has both ________ (size) and ________.◦Add/subtract vectors require special rules

A scalar is completely specified by only a _________ (size).◦Add/subtract vectors require no special

rules

Vector NotationVector NotationWhen handwritten, use an arrow:When printed, will be in bold print

with an arrow: When dealing with just the

magnitude of a vector in print, an italic letter will be used: A◦Italics will also be used to represent

scalars

A

A

Properties of VectorsProperties of Vectors

Equality of Two Vectors◦Two vectors are ________if they have

the same magnitude and the same direction.

◦These are parallel rays.Movement of vectors in a diagram

◦Any vector can be moved _________ to itself without being affected.

More Properties of VectorsMore Properties of VectorsNegative Vectors

◦Two vectors are negative if they have the same _______ but are _________(opposite directions).

◦ Resultant Vector

◦The ________ vector is the sum of a given set of vectors.

◦

; 0 A B A A

R A B

Parallel VectorsParallel Vectors

A

B

These two vectors are equal.

A

B

These two vectors are not equal.

Adding VectorsAdding VectorsWhen adding vectors, their ________

must be taken into account.Units must be the same. Geometric Methods

◦Use scale drawingsAlgebraic Methods

◦More convenient◦More on this later

Adding Vectors Geometrically Adding Vectors Geometrically (Triangle Method)(Triangle Method)

Choose a scale Draw the first vector with the appropriate

length and in the direction specified, with respect to a coordinate system

Draw the next vector using the same scale with the appropriate length and in the direction specified, with respect to a coordinate system whose origin is the end of vector and parallel to the coordinate system used for .

AA

Graphically Adding VectorsGraphically Adding VectorsContinue drawing the vectors “________”The resultant is drawn from the origin of

to the end of the last vectorMeasure the length of and its angle

◦ Use the scale factor to convert length to actual magnitude.

A

R

Graphically Adding VectorsGraphically Adding Vectors

When you have many vectors, just keep repeating the process until all are included.

The resultant is still drawn from the origin of the first vector to the end of the last vector.

Sample ProblemSample Problem

Find the magnitude of the sum of a 15 km displacement and a 5 km displacement when the angle between them is 90o and when the angle between them is 135o. Use the graphical method of addition.

Notes about Vector Notes about Vector AdditionAdditionVectors obey the _______________ Law

of Addition◦ The order in which the vectors are added

doesn’t affect the result◦

A B B A

Sample ProblemSample Problem

A car travels 20 km due north and then 35 km in a direction 60o west of north. Find the magnitude and direction of a single vector that gives the net effect of the car’s trip. This vector is called the car’s resultant displacement.

Vector SubtractionVector Subtraction

A B A B

Special case of vector addition:◦Add the negative

of the subtracted vector

Continue with

standard vector addition procedure

Components of a VectorComponents of a Vector

A component is a _______.

It is useful to use rectangular components◦ These are the

projections of the vector along the x- and y-axes.

Components of a Vector Components of a Vector

The x-component of a vector is the projection along the ________:

The y-component of a vector is the projection along the _________:

Then,

cosA A x

sinyA A

x y A A A

More About Components of a More About Components of a VectorVector

The previous equations are valid only if ____ is measured with respect to the _______.

The components can be positive or negative and will have the same units as the original vector.

We can use various trig. functions to solve for the length and angle of the resultant.

More About ComponentsMore About ComponentsThe components are __________________

whose hypotenuse is

◦ May still have to find θ with respect to the positive x-axis.

◦ The value will be correct only if the angle lies in the first or fourth quadrant.

2 2 1tan yx y

x

AA A A and

A

A

Trig. and Vectors Trig. and Vectors Trig. Ratios:

Law of Cosines: ◦R2 = A2 + B2 – 2ABcosθ◦R =resultant vector◦A/B = two vectors added together ◦Θ = angle between A and B

Law of Sines:

Adding Vectors Adding Vectors AlgebraicallyAlgebraicallyChoose a coordinate system and

sketch the vectors.Find the x- and y-components of

all the vectors.Add all the x-components

◦This gives Rx: xx vR

Adding Vectors Adding Vectors AlgebraicallyAlgebraically

Add all the y-components◦This gives Ry:

Use the _____________ to find the magnitude of the resultant:

Use the ________________to find the direction of R:

yy vR

2y

2x RRR

x

y1

R

Rtan

Sample Problem Sample Problem A GPS receiver indicates that your home is 15.0 km and 40.0o north of west, but the only path through the woods leads directly north. If you follow the path 5.0 km before it opens into a field, how far, and in what direction, would you walk to reach your home?

Forces of FrictionForces of FrictionWhen an object is in motion on a

_______ or through a _____________, there will be a resistance to the motion.◦This is due to the interactions between the

object and its environment.◦The “viscous” mediums include water or

air.This is resistance is called __________.

◦Static friction◦Kinetic friction

Friction plays an important role in our daily lives.◦Friction allows us to run and walk and

allows the wheels on our cars to move the car.

More About FrictionMore About FrictionFriction is ______________ to the

_____________.The force of static friction is

generally ________ than the force of kinetic friction.

The direction of the frictional force is __________ the direction of motion.

The _______________(µ) depends on the surfaces in contact.

The coefficients of friction are nearly __________ of the _____________.

Static Friction, ƒStatic Friction, ƒss

Static friction acts to keep the object from moving. ◦ As long as F = ƒs no

motion occurs.◦ When F >ƒs motion

occurs.ƒs arises from the nature

of the two surfaces in contact with each other.◦ µs accounts for this

ƒs µs n◦ This gives us the magnitude

of the force of static friction.◦ Use = sign for impending

motion only.

Static Friction, Static Friction, ƒƒssIf F increases,

so does ƒs

If F decreases, so does ƒs

ƒs, max is the maximum limit to the force of static friction. ◦When an object

is in motion the friction force is less than the ƒs,

max

Kinetic Friction, Kinetic Friction, ƒƒkk

The force of kinetic friction acts when the object is in motion.

Fnet = F – ƒk At constant velocity (a =

0): ◦ F = ƒk

If F = 0◦ ƒk will provide an

acceleration in the direction opposite of the movement and eventually bring the object to rest.

ƒk = µk n◦ Variations of the coefficient

with speed will be ignored

Sample ProblemSample Problem

You push a 25.0 kg wood box across a wooden floor at a constant speed of 1.0 m/s. How much force do you exert on the box? Use Table 5.1 (p.129) to help you solve this problem.

Some Coefficients of FrictionSome Coefficients of Friction

Friction and the Motion of a Friction and the Motion of a CarCar

Friction plays an important role in the movement of a car.

Friction between the moving car’s wheels and the road is static friction.◦ This moves the car

forward.◦ Unless the car is skidding.

Also have the air resistance,◦ This slows the car down

R

EquilibriumEquilibriumAn object either at rest or moving

with a constant velocity is said to be in ____________.

The ________acting on the object is zero (since the acceleration is zero.)

0F

Equilibrium Equilibrium Easier to work with the above equation in terms of its components:

This could be extended to three dimensions.◦We would also have to consider ΣFz = 0

A zero net force does not mean the object is not moving, but that it is not accelerating.

0 0x yF and F

Inclined Planes Inclined Planes ProblemsProblemsThese are

important and common types of problems here and elsewhere.

Choose the coordinate system with x along the incline and y perpendicular to the incline

Replace the force of gravity with its components.◦ Fg itself must

remain perpendicular to the ground.

Sample Problem Sample Problem

A crate weighing 562 N is resting on a plane inclined 30.0o above the horizontal. Find the component of the weight forces that are parallel and perpendicular to the plane.

ChapteChapter 5r 5

Forces in Two Dimensions

THE END