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PPT – Kinematics in Two Dimensions PowerPoint presentation | free to download - id: 5a3ae8-MzY0Z

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Kinematics in Two Dimensions

- x x0 v0xt 1/2 axt2
- vx v0x axt
- vx2 v0x2 2ax ?x

- y y0 v0yt 1/2 ayt2
- vy v0y ayt
- vy2 v0y2 2ay ?y

Kinematics for Projectile Motion ax 0

ay -g

- y y0 v0yt - 1/2 gt2
- vy v0y - gt
- vy2 v0y2 - 2g ?y

- x x0 vxt
- vx v0x

x and y motions are independent! They share a

common time t

Example 1

- You and a friend are standing on level ground,

each holding identical baseballs. At exactly the

same time, and from the same height, you drop

your baseball without throwing it while your

friend throws her baseball horizontally as hard

as she can. Which ball hits the ground first? - 1. Your ball
- 2. Your friends ball
- 3. They both hit the ground at the same time

They both have the same initial vertical

component with the same acceleration due to

gravity, therefore they hit the ground at the

same time.

No matter how much horizontal velocity is put on

an object it still falls at the same rate as any

other dropped object.

- y y0 voyt - gt2/2
- v0y 0 and y0
- Therefore, tsqrt(2y0/g)
- Result is independent of v0x

Example 2

A flatbed railroad car is moving along a track at

constant velocity. A passenger at the center of

the car throws a ball straight up. Neglecting

air resistance, where will the ball land ? 1.

Forward of the center of the car 2. At the center

of the car 3. Backward of the center of the car

The train and the ball have the same horizontal

velocity and by throwing the ball straight up,

the horizontal component is not changed.

The ball has no acceleration in the horizontal

direction. Therefore, the balls remains directly

above the center of the train at all times during

the flight and would fall directly back toward

the center of the train.

Example 4

- You are a vet trying to shoot a tranquilizer dart

into a monkey hanging from a branch in a distant

tree. You know that the monkey is very nervous,

and will let go of the branch and start to fall

as soon as your gun goes off. On the other hand,

you also know that the dart will not travel in a

straight line, but rather in a parabolic path

like any other projectile. In order to hit the

monkey with the dart, where should you point the

gun before shooting? - 1 Right at the monkey
- 2 Below the monkey
- 3 Above the monkey

If the shot is fired at the monkey the same time

the monkey drops, both objects will fall at the

same rate causing the shot to hit the monkey.

Example 4 Shooting the Monkey... (II)

x v0 t y -1/2 g t2

x x0 y -1/2 g t2

No monkeys were harmed during the making of this

slide

Example 4 Shooting the Monkey... (III)

y y0 - 1/2 g t2

- At an angle, still aim at the monkey!

y v0 t - 1/2 g t2

Newton's Laws

After this lecture, you should know

about Force, mass, inertia. Newtons first and

second law. Inertial and non-inertial reference

frame. Gravitation. Action reaction. Normal

force. Free body diagram.

Classical Mechanics and forces

- Classical mechanics
- Describes the relationship between the motion of

objects in our everyday world and the forces

acting on them - Conditions when Classical Mechanics does not

apply - very tiny objects (lt atomic sizes)
- objects moving near the speed of light
- Force
- Usually think of a force as a push or pull
- Vector quantity
- May be a contact force or a field force
- Contact forces result from physical contact

between two objects - Field forces act between disconnected objects
- Also called action at a distance

Forces

Contact and Field Forces

Fundamental Forces

- Types
- Strong nuclear force
- Electromagnetic force
- Weak nuclear force
- Gravity
- Characteristics
- All field forces
- Listed in order of decreasing strength
- Only gravity and electromagnetic in mechanics

Newtons First Law

- The motion of an object does not change unless it

is acted upon by a net external force (for the

definition of force see Newtons 2nd Law) - If v0, it remains 0
- If v is some value, it stays at that value
- Hence
- If no net force
- velocity is constant in magnitude and direction
- acceleration is zero
- Hence An object traveling at a constant velocity

along a straight line will continue to do so as

long as there is no net external force acting on

it - External forces are forces that result from the

interaction between the object and its

environment

Example 7

- An airplane is flying from Madison to O'Hare.

Many forces act on the plane, including weight

(gravity), drag (air resistance), the thrust of

the engine, and the lift of the wings. At some

point during its trip the velocity of the plane

is measured to be constant (which means its

altitude is also constant). At this time, the

total force on the plane 1. is pointing

upward2. is pointing downward 3. is pointing

forward 4. is pointing backward5. is zero

When the velocity is constant the objects

acceleration is equal to zero. The only time

acceleration is equal to zero is when the sum of

the net force is equal to zero.

An object traveling at a constant velocity along

a straight line will continue to do so as long as

there is no net force acting on it (Newton's

First Law). The total force acting on the plane

is zero, because its motion is uniform in a

straight line.

Inertia and mass

- Inertia
- Is the tendency of an object to continue in its

original motion - (N.B. not a physics quantity in the strict

sense of the term) - Mass
- A measure of the resistance of an object to

changes in its motion due to a force - Scalar quantity
- SI units are kg

Inertial vs non-inertial reference frame

- Inertial reference frames are coordinate systems

which travel at constant velocity. - In such a frame, an object is observed to have no

acceleration when no forces are acting on it. - If a reference frame moves with constant velocity

relative to an inertial reference frame, it also

is an inertial reference frame. - There is no absolute inertial reference frame,

meaning that there is no state of velocity which

is special in the universe. All inertial

reference frames are equivalent. One can only

detect the relative motion of one inertial

reference frame to another. - (Approximate) example of inertial frame train

moving with constant velocity - Examples of non-inertial frames train

accelerating, bus braking, car on a curve - Newtons 1st law is a way to define inertial

frames

Newtons Second Law

- The acceleration of an object is directly

proportional to the net force acting on it and

inversely proportional to its mass. - Units
- F M a
- F kg-m/s2
- 1 Newton (N) 1 kg-m/s2
- A vector equation
- Fnet,x Max
- Fnet,y May

- Newtons second law applies only in inertial

reference frames - Applying it in non-inertial ones leads to

pseudo-forces, e.g. centrifugal force

Example 8

- A force F acting on a mass m1 results in an

acceleration a1.The same force acting on a

different mass m2 results in an acceleration a2

2a1. What is the mass m2?

(1) 2m1 (2) m1 (3) m1/2

- Fma
- F m1a1 m2a2 m2(2a1)
- Therefore, m2 m1/2
- Or in wordstwice the acceleration means half the

mass

Gravitational Force and Weight

- Mutual force of attraction between any two

objects - Expressed by Newtons Law of Universal

Gravitation - The magnitude of the gravitational force acting

on an object of mass m near the Earths surface

is called the weight w of the object - w m g is a special case of Newtons Second Law
- g is the acceleration due to gravity g 9.81

m/s2 - g can also be found from the Law of Universal

Gravitation - g GMearth/r2
- Weight is not an inherent property of an object
- mass is an inherent property
- Weight depends upon location

Weight and Mass

Example 9

What is the approximate weight force of a bar of

Chocolate of 100g on sea level ?

Use g10m/s2 W 0.1kg x 10 m/s2 1 N

1 N

Newtons Third Law

- For every action, there is an equal and opposite

reaction.

- Finger pushes on box
- Ffinger?box force exerted on box by finger

- Box pushes on finger
- Fbox?finger force exerted on finger by box

- Third Law
- Fbox?finger - Ffinger?box

Newton's Third Law...

- FA ,B - FB ,A. is true for all types of

forces

Whenever one body exerts a force on a second

body, the first body experiences a force that is

equal in magnitude and opposite in direction to

the one it exerts.

Example of Bad Thinking

- Since Fm,b -Fb,m why isnt Fnet 0, and a 0 ?

Fb,m

Fm,b

a ??

ice

Example of Good Thinking

- Consider only the box!
- Fon box mabox Fm,b
- Free Body Diagram (more on this next time)

What about forces on man?

Fb,m

Fm,b

abox

ice

Example 10A

- Suppose you are an astronaut in outer space

giving a brief push to a spacecraft whose mass is

bigger than your own. - 1) Compare the magnitude of the force you exert

on the spacecraft, FS, to the magnitude of the

force exerted by the spacecraft on you, FA, while

you are pushing1. FA FS 2. FA gt FS3. FA

lt FS

Third Law!

Example 10B

2) Compare the magnitudes of the acceleration

you experience, aA, to the magnitude of the

acceleration of the spacecraft, aS, while you

are pushing 1. aA aS 2. aA gt aS 3. aA lt aS

aF/m F same, hence lower mass gives larger a

Example 11

Consider a car at rest. We can conclude that the

downward gravitational pull of Earth on the car

and the upward contact force of Earth on it are

equal and opposite because 1. The two forces

form an action-reaction pair 2. The net force

on the car is zero 3. Neither of the above

The two forces cannot be an action-reaction pair

because they act on the same object (car). Car is

at rest - therefore, it has no net forces acting

on it. The forces acting on it add up to zero

The Normal Force

When person is holding the bag above the table he

must supply a force. When the bag is placed on

the table, the table supplies the force that

holds the bag on it That force is perpendicular

or normal to the surface of the table

Do You Feel The Normal Force?

Yes, you can feel an upward force on your feet. F

gm 9.8x100 980 Newtons! That force is

spread out over the area of you foot so its not

so bad. Pressure P F/Area N/m2

Action-Reaction Pairs vs forces acting on an

object

- is the normal force, the force the table

exerts on the TV - is perpendicular to the surface
- is the reaction force the TV exerts on the

table

- is force the Earth exerts on object
- is force object exerts on the earth

Forces Acting on an Object

- Newtons Law uses the forces acting on an object
- are acting on the object
- are acting on other objects

Summary of Newtons laws

- Newtons First Law
- The motion of an object does not change unless it

is acted on by a net external force - Newtons Second Law
- Newtons Third Law

Applications of Newtons Laws

- Assumptions
- Objects behave as particles
- can ignore rotational motion (for now)
- Masses of strings or ropes are negligible
- Interested only in the forces acting on the

object or the system of interest - can neglect reaction forces

Example 12

Consider a person standing in an elevator that is

accelerating upward. The upward normal force N

exerted by the elevator floor on the person

is a) larger than b) identical to c) less

than the downward weight W of the person.

Person is accelerating upwards - net upwards

force is non zero

Frictional Force

- Friction
- Opposes motion between systems in contact
- Parallel to the contact surface
- Depends on the force holding the surfaces

together - Normal force (N)
- Static friction
- Force required to move a stationary object
- fs is less than or equal to µs N
- Object remains stationary
- Kinetic friction
- Frictional force on an object in motion
- Is generally less than static friction
- Note Equation contains only magnitudes of forces

since friction and normal force have different

directions

µS coefficient of static friction µK

coefficient of kinetic friction

Friction (II)

- Static friction acts to keep the object from

moving - If F increases, so does ƒs
- If F decreases, so does ƒs
- ƒs µs n
- The force of kinetic friction acts when the

object is in motion - ƒk µk n
- Variations of the coefficient with speed will be

ignored

Example 13

You are pushing a wooden crate across the floor

at constant speed. You decide to turn the crate

on end, reducing by half the surface area in

contact with the floor. In the new orientation,

to push the same crate across the same floor with

the same speed, the force that you apply must be

about a) four times as great b) twice as

great c) equally as great d) half as

great e) one-fourth as great as the force

required before you changed the crate orientation.

Frictional force does not depend on the area of

contact. It depends only on the normal force and

the coefficient of friction for the contact.

Example 14A

- You are driving a car up a hill with constant

velocity. On a piece of paper, draw a Free Body

Diagram (FBD) for the car. How many forces are

acting on the car? 12345

V

weight/gravity (W)normal (FN)engine/motor

(Fcar_on_road(action) gt (reaction) Froad on car)

Example 14B

- The net force on the car is 1. Zero 2. Pointing

up the hill 3. Pointing down the hill 4.

Pointing vertically downward 5. Pointing

vertically upward

SF ma 0

Example 14C

- You are driving a car up a hill with constant

acceleration. - How many forces are acting on the car?

12345

a

weight/gravity (W)normal (FN)engine/motor

(Fcar_on_road(action) gt (reaction) Froad on car)

Example 14D

- You are driving a car up a hill with constant

acceleration. - The net force on the car is now1. Zero 2.

Pointing up the hill 3. Pointing down the hill

4. Pointing vertically downward 5. Pointing

vertically upward

a

FN

Froad on car

W

Example 14- Summary

- Often important to resolve the weight into

components parallel and perpendicular to the

hill. - Then if Fw parallel Froad on car
- Constant velocity
- if Fw parallel lt Froad on car
- Accelerate up the hill
- If Fw parallel gt Froad on car
- Accelerate down the hill
- Fw parallel gt fmax FN µs µsMgcosf
- Slide down the hill

f

FWpara

f

W

FWperp

Tension

Tension can be transmitted around corners If

there is no friction in the pulleys, T remains

the same

Tension is a force along the length of a medium

More on tension

For massless cords passing over frictionless

pulleys or surfaces the whole rope is

characterized by a single tension, which is

usually denoted as T. If a rope is in tension,

then at any cross section along its length, the

left part pulls on the right by a force T and the

right side pulls on the left by a force, T.

Hence 1. There is a single tension, T,

characterizing an ''ideal'' cord. 2. A rope can

only pull along its length. It never pushes and

it never exerts a force perpendicular to its

length. Rule 1) sets the magnitude of the forces

produced by a cord and rule 2) determines the

direction of the force produced on an object in

contact with the cord.

Example 14 Pulley I

- What is the tension in the string?
- A) TltW
- B) TW
- C) WltTlt2W
- D) T2W

Same answer

Example 15 Pulley II

- What is the tension in the string?
- A) TltW
- B) TW
- C) WltTlt2W
- D) T2W

Example 15

In the 17th century, Otto von Guricke, a

physicist in Magdeburg, fitted two hollow bronze

hemispheres together and removed the air from the

resulting sphere with a pump. Two eight-horse

teams could not pull the halves apart even though

the hemispheres fell apart when air was

readmitted! Suppose von Guricke had tied both

teams of horses to one side and bolted the other

side to a heavy tree trunk. In this case, the

tension on the hemisphere would be a) twice

what it was b) exactly what it was c) half

what it was