PhysicsChpt7

8/3/2019 PhysicsChpt7

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Unit 3, Chapter 7

CPO Science

Foundations of Physics

Chapter 9


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Unit 3: Motion and Forces in 2 and 3Dimensions

7.1 Vectors and Direction

7.2 Projectile Motion and the Velocity

Vector 7.3 Forces in Two Dimensions

Chapter 7 Using Vectors: Forces and Motion


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Chapter 7 Objectives1. Add and subtract displacement vectors to describe

changes in position.

2. Calculate the xand ycomponents of a displacement,velocity, and force vector.

3. Write a velocity vector in polar and x-ycoordinates.

4. Calculate the range of a projectile given the initialvelocity vector.

5. Use force vectors to solve two-dimensionalequilibrium problems with up to three forces.

6. Calculate the acceleration on an inclined plane when

given the angle of incline.


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Chapter 7 Vocabulary Terms vector

scalar

magnitude

x-component

y-component

cosine

parabola

Pythagoreantheorem

displacement

resultant

position

resolution

right triangle

sine

dynamics

tangent

normal force

projectile

trajectory

Cartesiancoordinates

range

velocity vector

equilibrium

inclined plane

polar coordinates

scale component


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7.1 Vectors and DirectionKey Question:

How do we accurately

communicate lengthand distance?

*Students read Section 7.1 AFTER Investigation 7.1


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A scalar is a quantity thatcan be completely

described by one value:the magnitude.

You can think of

magnitude as size oramount, including units.


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A vector is a quantity thatincludes both magnitude

and direction.

Vectors require more thanone number.

The information 1kilometer, 40 degrees eastof north is an example of a

vector.


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7.1 Vectors and Direction In drawing a vector as

an arrow you mustchoose a scale.

If you walk five meterseast, your displacementcan be represented by a5 cm arrow pointing tothe east.


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7.1 Vectors and Direction Suppose you walk 5 meters

east, turn, go 8 meters north,then turn and go 3 meterswest.

Your position is now 8 metersnorth and 2 meters east of

where you started.

The diagonal vector thatconnects the starting positionwith the final position is

called the resultant.


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7.1 Vectors and Direction The resultant is the sum of

two or more vectors addedtogether.

You could have walked ashorter distance by going 2 meast and 8 m north, and still

ended up in the same place.

The resultant shows the mostdirect line between thestarting position and the final

position.


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7.1 Calculate a resultant vector

An ant walks 2 meters West, 3 metersNorth, and 6 meters East.

What is the displacement of the ant?


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7.1 Finding Vector ComponentsGraphically

Draw adisplacement vector

as an arrow ofappropriate lengthat the specifiedangle.

Mark the angle anduse a ruler to drawthe arrow.


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7.1 Finding the Magnitude of a Vector

When you know the x-and y-components of a vector,and the vectors form a right triangle, you can find themagnitude using the Pythagorean theorem.


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7.1 Adding Vectors Writing vectors in components make it easy to add

them.


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7.1 Subtracting Vectors


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7.1 Calculate vector magnitude A mail-delivery robot

needs to get from whereit is to the mail bin onthe map.

Find a sequence of twodisplacement vectors

that will allow the robotto avoid hitting the deskin the middle.


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7.2 Projectile Motion and the VelocityVector

Any object that is

moving through the airaffected only by gravityis called a projectile.

The path a projectilefollows is called itstrajectory.


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The trajectory of a

thrown basketballfollows a special typeof arch-shaped curvecalled a parabola.

The distance aprojectile travelshorizontally is called

its range.


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The velocity vector (v) is a

way to precisely describethe speed and direction ofmotion.

There are two ways torepresent velocity.

Both tell how fast and inwhat direction the ball

travels.


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7.2 Calculate magnitude

Draw the velocity vector

v = (5, 5) m/sec andcalculate the magnitudeof the velocity (thespeed), using the

Pythagorean theorem.


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7.2 Components of the Velocity Vector Suppose a car is driving

20 meters per second.

The direction of thevector is 127 degrees.

The polar representationof the velocity is v = (20m/sec, 127).


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7.2 Calculate velocity

A soccer ball is kicked at a speed of 10 m/s and anangle of 30 degrees.

Find the horizontal and vertical components of the

balls initial velocity.


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7.2 Adding Velocity Components Sometimes the total velocity of an object is a

combination of velocities.

One example is the motion of a boat on a river.

The boat moves with a certain velocity relative to thewater.

The water is also moving with another velocity relative to

the land.


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7.2 Adding Velocity Components


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7.2 Calculate velocity components

An airplane is moving at a velocity of 100 m/s in adirection 30 degrees NE relative to the air.

The wind is blowing 40 m/s in a direction 45 degrees SErelative to the ground.

Find the resultant velocity of the airplane relative to theground.


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7.2 Projectile MotionVx

Vy

x

y

When we drop a ballfrom a height we knowthat its speedincreases as it falls.

The increase in speedis due to theacceleration gravity, g= 9.8 m/sec2.


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7.2 Horizontal Speed The balls horizontal

velocity remains constantwhile it falls because

gravity does not exert anyhorizontal force.

Since there is no force, the

horizontal acceleration iszero (ax = 0).

The ball will keep movingto the right at 5 m/sec.


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7.2 Horizontal Speed The horizontal distance a projectile moves can

be calculated according to the formula:


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7.2 Vertical Speed The vertical speed (vy) of the

ball will increase by 9.8

m/sec after each second.

After one second haspassed, vyof the ball will be

9.8 m/sec.

After the 2nd second haspassed, vywill be 19.6 m/sec

and so on.


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7.2 Calculate using projectile motion

A stunt driver steers a caroff a cliff at a speed of 20meters per second.

He lands in the lake below

two seconds later.

Find the height of the cliffand the horizontal

distance the car travels.


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7.2 Projectiles Launched at an Angle A soccer ball kicked

off the ground isalso a projectile, butit starts with aninitial velocity thathas both verticaland horizontal

components.

*The launch angle determines how the initial velocitydivides between vertical (y) and horizontal (x) directions.


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7.2 Steep Angle

A ball launched

at a steep anglewill have a largevertical velocitycomponent and a

small horizontalvelocity.


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7.2 Shallow Angle

A ball launched at

a low angle willhave a largehorizontal velocitycomponent and a

small vertical one.


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7.2 Projectiles Launched at an AngleThe initial velocity components of an object launched at a

velocity voand angle are found by breaking thevelocity into xand ycomponents.


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7.2 Range of a Projectile The range, or horizontal distance, traveled by a

projectile depends on the launch speed and thelaunch angle.


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7.2 Range of a Projectile The range of a projectile is calculated from the

horizontal velocity and the time of flight.


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7.2 Range of a Projectile A projectile travels farthest when launched at

45 degrees.


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7.2 Range of a Projectile The vertical velocity is responsible for giving

the projectile its "hang" time.


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7.2 "Hang Time" You can easily calculate your own hang time.

Run toward a doorway and jump as high as you can,touching the wall or door frame.

Have someone watch to see exactly how high youreach.

Measure this distance with a meter stick.

The vertical distance formula can be rearranged to

solve for time:


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7.2 Projectile Motion and the VelocityVectorKey Question:

Can you predict the landing spot of a projectile?

*Students read Section 7.2 BEFORE Investigation 7.2


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Marbles Path

Vy

x = ?

y

Vx

t = ?


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In order to solve x we must know t

Y = vot g t2

2y = g t2

vot = 0 (zero)

Y = g t2

t2 = 2y

gt = 2y

g


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7.3 Forces in Two Dimensions Force is also represented in x-y components.


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7.3 Force Vectors If an object is in

equilibrium, all of theforces acting on it arebalanced and the netforce is zero.

If the forces act in twodimensions, then all ofthe forces in the x-direction and y-direction

balance separately.


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7.3 Equilibrium and Forces It is much more difficult

for a gymnast to holdhis arms out at a 45-degree angle.

To see why, considerthat each arm must stillsupport 350 newtonsvertically to balance theforce of gravity.


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7.3 Forces in Two Dimensions Use the y-component to find the total force in the

gymnasts left arm.


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7.3 Forces in Two Dimensions The force in the right arm must also be 495 newtons

because it also has a vertical component of 350 N.


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7.3 Forces in Two Dimensions When the gymnasts arms

are at an angle, only partof the force from eacharm is vertical.

The total force must belarger because thevertical componentofforce in each arm muststill equal half his weight.


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7.3 Forces and Inclined Planes An inclined plane is a straight surface, usually

with a slope.

Consider a block slidingdown a ramp.

There are three forcesthat act on the block:

gravity (weight).

friction

the reaction forceactin on the block.


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7.3 Forces and Inclined Planes

When discussing forces, the word normalmeans perpendicular to.

The normal forceacting on the block isthe reaction forcefrom the weight of the

block pressingagainst the ramp.


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The normal forceon the block is

equal andopposite to thecomponent of theblocks weight

perpendicular tothe ramp (Fy).


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The force parallel

to the surface (Fx)is given by

Fx = mgsin.


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7.3 Acceleration on a Ramp

Newtons second law can be used to calculate the

acceleration once you know the components of all theforces on an incline.

According to the second law:

a = Fm

Force (kg. m/sec2)

Mass (kg)

Acceleration(m/sec2)


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Since the block can only accelerate along the ramp, theforce that matters is the net force in the xdirection,parallel to the ramp.

If we ignore friction, and substitute Newtons' 2nd Law,the net force is:

Fx =

a =

m sin g

Fm


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To account for friction, the horizontal component ofacceleration is reduced by combining equations:

Fx = mg sin - m mg cos


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For a smooth surface, the coefficient of friction () is

usually in the range 0.1 - 0.3.

The resulting equation for acceleration is:


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7.3 Calculate acceleration on a ramp

A skier with a mass of 50 kg is on a hill making an angle

of 20 degrees.

The friction force is 30 N.

What is the skiers acceleration?


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Key Question:

How do forces balancein two dimensions?

*Students read Section 7.3 BEFORE Investigation 7.3


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Application: Robot Navigation

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