CPO Science Foundations of Physics - tnpec.org and Forces in 2 and 3 dimensions.pdf · Unit 3:...

Unit 3, Chapter 7

CPO Science

Foundations of Physics

Chapter 9

Unit 3: Motion and Forces in 2 and 3

Dimensions

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

Chapter 7 Objectives

1. Add and subtract displacement vectors to describe

changes in position.

2. Calculate the x and y components of a displacement,

velocity, and force vector.

3. Write a velocity vector in polar and x-y coordinates.

4. Calculate the range of a projectile given the initial

velocity vector.

5. Use force vectors to solve two-dimensional

equilibrium problems with up to three forces.

6. Calculate the acceleration on an inclined plane when

given the angle of incline.

Chapter 7 Vocabulary Terms

vector

scalar

magnitude

x-component

y-component

cosine

parabola

Pythagorean

theorem

displacement

resultant

position

resolution

right triangle

sine

dynamics

tangent

normal force

projectile

trajectory

Cartesian

coordinates

range

velocity vector

equilibrium

inclined plane

polar coordinates

scale component


Key Question:

How do we accurately

communicate length

and distance?

*Students read Section 7.1 AFTER Investigation 7.1


A scalar is a quantity that

can be completely

described by one value:

the magnitude.

You can think of

magnitude as size or

amount, including units.


A vector is a quantity that

includes both magnitude

and direction.

Vectors require more than

one number.

— The information “1

kilometer, 40 degrees east

of north” is an example of a

vector.


In drawing a vector as

an arrow you must

choose a scale.

If you walk five meters

east, your displacement

can be represented by a

5 cm arrow pointing to

the east.


Suppose you walk 5 meters

east, turn, go 8 meters north,

then turn and go 3 meters

west.

Your position is now 8 meters

north and 2 meters east of

where you started.

The diagonal vector that

connects the starting position

with the final position is

called the resultant.


The resultant is the sum of

two or more vectors added

together.

You could have walked a

shorter distance by going 2 m

east and 8 m north, and still

ended up in the same place.

The resultant shows the most

direct line between the

starting position and the final

position.

7.1 Calculate a resultant vector

An ant walks 2 meters West, 3 meters

North, and 6 meters East.

What is the displacement of the ant?

7.1 Finding Vector Components

Graphically Draw a

displacement vector

as an arrow of

appropriate length

at the specified

angle.

Mark the angle and

use a ruler to draw

the arrow.

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 the

magnitude using the Pythagorean theorem.

7.1 Adding Vectors

Writing vectors in components make it easy to add

them.

7.1 Subtracting Vectors

7.1 Calculate vector magnitude

A mail-delivery robot

needs to get from where

it is to the mail bin on

the map.

Find a sequence of two

displacement vectors

that will allow the robot

to avoid hitting the desk

in the middle.


Vector

Any object that is

moving through the air

affected only by gravity

is called a projectile.

The path a projectile

follows is called its

trajectory.


Vector

The trajectory of a

thrown basketball

follows a special type

of arch-shaped curve

called a parabola.

The distance a

projectile travels

horizontally is called

its range.


Vector

The velocity vector (v) is a

way to precisely describe

the speed and direction of

motion.

There are two ways to

represent velocity.

Both tell how fast and in

what direction the ball

travels.

7.2 Calculate magnitude

Draw the velocity vector

v = (5, 5) m/sec and

calculate the magnitude

of the velocity (the

speed), using the

Pythagorean theorem.

7.2 Components of the Velocity Vector

Suppose a car is driving

20 meters per second.

The direction of the

vector is 127 degrees.

The polar representation

of the velocity is v = (20

m/sec, 127°).

7.2 Calculate velocity

A soccer ball is kicked at a speed of 10 m/s and an

angle of 30 degrees.

Find the horizontal and vertical components of the

ball’s initial velocity.

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 the

water.

The water is also moving with another velocity relative to

the land.

7.2 Adding Velocity Components

7.2 Calculate velocity components

An airplane is moving at a velocity of 100 m/s in a

direction 30 degrees NE relative to the air.

The wind is blowing 40 m/s in a direction 45 degrees SE

relative to the ground.

Find the resultant velocity of the airplane relative to the

ground.

7.2 Projectile Motion

Vx

Vy

x

y

When we drop a ball

from a height we know

that its speed

increases as it falls.

The increase in speed

is due to the

acceleration gravity, g

= 9.8 m/sec2.

7.2 Horizontal Speed

The ball’s horizontal velocity remains constant while it falls because gravity does not exert any horizontal force.

Since there is no force, the horizontal acceleration is zero (ax = 0).

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

7.2 Horizontal Speed

The horizontal distance a projectile moves can

be calculated according to the formula:

7.2 Vertical Speed

The vertical speed (vy) of the

ball will increase by 9.8

m/sec after each second.

After one second has

passed, vy of the ball will be

9.8 m/sec.

After the 2nd second has

passed, vy will be 19.6 m/sec

and so on.

7.2 Calculate using projectile motion

A stunt driver steers a car

off a cliff at a speed of 20

meters per second.

He lands in the lake below

two seconds later.

Find the height of the cliff

and the horizontal

distance the car travels.

7.2 Projectiles Launched at an Angle

A soccer ball kicked

off the ground is

also a projectile, but

it starts with an

initial velocity that

has both vertical

and horizontal

components.

*The launch angle determines how the initial velocity

divides between vertical (y) and horizontal (x) directions.

7.2 Steep Angle

A ball launched

at a steep angle

will have a large

vertical velocity

component and a

small horizontal

velocity.

7.2 Shallow Angle

A ball launched at

a low angle will

have a large

horizontal velocity

component and a

small vertical one.

7.2 Projectiles Launched at an Angle

The initial velocity components of an object launched at a

velocity vo and angle θ are found by breaking the

velocity into x and y components.

7.2 Range of a Projectile

The range, or horizontal distance, traveled by a

projectile depends on the launch speed and the

launch angle.


The range of a projectile is calculated from the

horizontal velocity and the time of flight.


A projectile travels farthest when launched at

45 degrees.


The vertical velocity is responsible for giving

the projectile its "hang" time.

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 you

reach.

Measure this distance with a meter stick.

The vertical distance formula can be rearranged to

solve for time:


Vector

Key Question:

Can you predict the landing spot of a projectile?

*Students read Section 7.2 BEFORE Investigation 7.2

Marble’s Path

Vy

x = ?

y

Vx

t = ?

In order to solve “x” we must know “t”

Y = vot – ½ g t2

2y = g t2

vot = 0 (zero)

Y = ½ g t2

t2 = 2y

g

t = 2y

g


Force is also represented in x-y components.

7.3 Force Vectors

If an object is in

equilibrium, all of the

forces acting on it are

balanced and the net

force is zero.

If the forces act in two

dimensions, then all of

the forces in the x-

direction and y-direction

balance separately.

7.3 Equilibrium and Forces

It is much more difficult

for a gymnast to hold

his arms out at a 45-

degree angle.

To see why, consider

that each arm must still

support 350 newtons

vertically to balance the

force of gravity.


Use the y-component to find the total force in the

gymnast’s left arm.


The force in the right arm must also be 495 newtons

because it also has a vertical component of 350 N.


When the gymnast’s arms

are at an angle, only part

of the force from each

arm is vertical.

The total force must be

larger because the

vertical component of

force in each arm must

still equal half his weight.

7.3 Forces and Inclined Planes

An inclined plane is a straight surface, usually

with a slope.

Consider a block sliding

down a ramp.

There are three forces

that act on the block:

— gravity (weight).

— friction

— the reaction force

acting on the block.


When discussing forces, the word “normal”

means “perpendicular to.”

The normal force

acting on the block is

the reaction force

from the weight of the

block pressing

against the ramp.


The normal force

on the block is

equal and

opposite to the

component of the

block’s weight

perpendicular to

the ramp (Fy).


The force parallel

to the surface (Fx)

is given by

Fx = mg sinθ.

7.3 Acceleration on a Ramp

Newton’s second law can be used to calculate the

acceleration once you know the components of all the

forces on an incline.

According to the second law:

a = F

m

Force (kg . m/sec2)

Mass (kg)

Acceleration

(m/sec2)


Since the block can only accelerate along the ramp, the

force that matters is the net force in the x direction,

parallel to the ramp.

If we ignore friction, and substitute Newtons' 2nd Law,

the net force is:

Fx =

a =

m sin θ g

F m


To account for friction, the horizontal component of

acceleration is reduced by combining equations:

Fx = mg sin θ - m mg cos θ


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

usually in the range 0.1 - 0.3.

The resulting equation for acceleration is:

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 skier’s acceleration?


Key Question:

How do forces balance

in two dimensions?

*Students read Section 7.3 BEFORE Investigation 7.3

Application: Robot Navigation

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