Glencoe Physics Chapter 7Forces and Motion in

Two Dimensions

Objectives 7.1

- Determine the force that produces equilibrium

when three forces act on an object - Analyze the motion of an object on an inclined

plane with and without friction

Objectives 7.2

- Recognize that the vertical and horizontal

motions of a projectile are independent - Relate the height, time in air (hang time), and

initial vertical velocity of a projectile using

its vertical motion , then determine the range - Explain how the shape of the trajectory of a

moving object depends upon the frame of reference

from which it is observed

Objectives 7.3

- Explain the acceleration of an object moving in a

circle at a constant speed - Describe how centripetal acceleration depends

upon the objects speed and the radius of the

circle - Recognize the direction of the force that causes

centripetal acceleration - Explain how the rate of circular motion is

changed by exerting torque on it

7.1 Forces in Two Dimensions

- We have already learned about how to find the

resultant force of two forces acting on an

object, as well as 3 or more forces. - What we want to consider now are cases where Net

Force acting on an object is zero, and the object

is in Static equilibrium.

Static Equilibrium

- Condition of an object when net forces equal zero
- Object is motionless

Hanging sign f.b.d.

Free Body Diagram

Since the sign is not accelerating in any

direction, its in equilibrium. Since its not

moving either, we call it Static Equilibrium.

Thus, red green black 0.

Equilibrant

- If you remove one force from a static condition,

the system is no longer in static equilibrium - The removed force is called the EQUILIBRANT.
- The equilibrant is the force, that if added to

other forces, will bring the other forces into

static equilibrium. - Equilibrant is always equal to, but opposite, the

sum of all the other force vectors

What force represents the equilibrant of the

tensions of the strings?

Example problem

- What is the tension in the ropes holding the mass

in place?

22.5

T 22.5 m 168 N

168 N

Solution

- Fnet 0 (system is in equilibrium)
- Fax -Fbx
- Fay Fby mg 0
- FaSin 22.5 FbSin 22.5 168 N
- 2Fa Sin 22.5 168 N
- Fa 220 N

Weight of the Picture?

Weight is equal to F1y F2y So. 25N 25 N 50

N

What is the weight of this picture?

Components Scalar Equations

If in Equilibrium..the following would be true

Sample Problem

- A mother and daughter are outside playing on the

swings. The mother pulls the daughter and swing

(total mass 55.0 kg) back so that the swing makes

an angle of 40.0 with the vertical (50.0 from

horizontal) - A. What is the tension in each chain holding the

swing seat and the daughter? - B. How hard did the mother have to pull to hold

the daughter at that position?

A. 703N

B. 452N

How to budge a stubborn mule

It would be pretty difficult to budge this mule

by pulling directly on his collar. But it would

be relatively easy to budge him by tying the rope

to a tree and then pulling up (or pushing

sideways) in the middle. Why would this work?????

The tension in the rope has two components, one

horizontal and one vertical

FT

Little Force

FTx

Big Force

Sample Problem

- Is this a case of equilibrium?
- Calculate the magnitude of the net force

Equilibrium or Motion along an Inclined Plane

Is the sled on the inclined plane in

equilibrium? What are the forces acting on the

sled? Draw a Free Body Diagram of the forces

Inclined Plane

Inclined Plane )

Be careful what ? you use in your

calculation..remember you must use the angle, as

measured from (X) (East)

Fgx

Fgy

This calculation will look different than what is

in your book (p. 152)but is easier to use if you

remember the above caution!!!!!

The next 3 slides show the effect

Fn on block

of increasing the angle.

Fgx

Fn on block

F gx

Fg on block

CD 5.3 p-22

CD 5.3 p 21

Fn on block

F gx

Fg on block

CD 5.3 p-22

CD 5.3 p 21

net force greatest at

acceleration greatest at

Speed increases, acceleration

A

A

A

decreases

B

C

Fe on block

CD 5.3 p 22

Example p. 153 of your book.

- A 62 kg person on skis is going down a hill

sloped at 37. The coefficient of kinetic

friction between the skis and the snow is 0.15. - A) What is the Horizontal component of the

skiers weight? - B) What is the Vertical component of the skiers

weight? - C) What is the Normal force acting on the skier?
- D) What is the Frictional force acting on the

skier? - E) What is the Net Force acting on the skier?
- F) What is the acceleration of the skier?
- G) How fast is the skier traveling after 5.0s,

starting from

7.2 Projectile Motion in Two Dimensions

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Fountain at Explora! Science Museum,

Albuquerque, NM

What is Projectile Motion?

- Any object moving horizontally, with only the

force of gravity acting on it, will exhibit

projectile motion. (not moving straight up or

down) - An object displaying projectile motion will

follow a trajectory, or a curved,

parabolic-shaped path.

Projectile Motion

At a given location on the earth and in the

absence of air resistance, all objects fall with

the same uniform acceleration. Thus, two objects

of different masses, dropped from the same

height, will hit the ground at the same time.

An objects horizontal and vertical motion are

independent. Object projected horizontally will

reach the ground in the same time as an object

dropped vertically. No matter how large the

horizontal velocity is, the downward pull of

gravity is always the same.

Why does a projectile follow a parabolic path?

What happens to the horizontal displacement

during each successive time period? What happens

to the vertical displacement during each

successive time period?

Horizontal motion.

Vertical parabolic path

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Horizontal Vs. Vertical Velocities

- We can see from the diagrams that the horizontal

velocity of a projectile remains constant (with

no air resistance) - We can also see that that vertical velocity is

constantly changing, at a rate of 9.8 m/s2.

Horizontal Vs. Vertical Velocities

- The horizontal and vertical velocities are also

independent of each other, that is, they have no

effect on each other. - The only thing the horizontal and vertical

motions have in common is that each motion takes

exactly the same time to occur.

Equations of Projectile Motion

Working Projectile motion problems

- Separate the problem into two problemsvertical

motion and horizontal motion - Vertical motion is exactly that of an object

dropped or thrown straight up or down. Use

free-fall equations (or accelerated motion

equations) - Horizontal motion is constant velocity (Newtons

1st Law), work this part using constant velocity

equations. - Both Vertical and horizontal motion are tied

together by TIME. (TvTh)

Sample Problem.

- A rock is thrown horizontally off a tall cliff at

20.m/s. If the cliff is 70.m high - A) How far horizontally from the bottom of the

cliff does the rock land? - B) What is the magnitude of the rocks total

velocity at the instant before landing?

Another example in book on page 157

Monkey Hunter

- A young native hunter in the Amazon is hunting

monkeys with a bow and arrow. The hunter see a

monkey sitting on a branch, and since the bow

does not have a sight, the hunter pulls back the

bow and points the arrow directly at the monkey.

As the hunter releases the arrow the monkey

notices the movement and drops off the branch.

What happens next?

Throw at monkey with no gravity

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Throw above monkey with gravity

Throw at monkey fast with gravity

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Horizontal RangeWhat is the horizontal range of

the projectile?

Firstfind time using vertical motion..

Then, use that time to find range!

?Vy at

- dx vxt
- dx (v0 cos T)t

t ?vy a

Notice nice equation to use to find vertical time

in air!!!

- V0Sin25.0 -V0Sin25.0 -9.8m/s2

dx (v0Cos T) (V0 Sin T/4.9)

2.58s

dx (30.0 m/s Cos 25.0)(30.0 m/s Sin 25.0)/4.9

dx 70.4 m

Alternate Range equationsame thing.

Range (dx) V02 Sin 2 T g

Max height hang time depends only on initial

vertical velocity

Each initial velocity vector below has the a

different magnitude (speed) and the projectiles

will have different ranges (green the greatest),

but each object will spend the same time in the

air and reach the same max height. This is

because each vector has the same vertical

component.

Max Range at 45?

Over level ground at a constant launch speed,

what angle maximizes the range, R ? First

consider some extremes When ? 0, R 0,

since as soon as the object is launched its back

on the ground. When ? 90?, the object goes

straight up and lands right on the launch site.

The best angle is 45?, smack dab between the

extremes.

All launch speeds are the same only the angle

varies.

76?

45?

38?