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Kinematics in Two Dimensions Kinematics in Two Dimensions x = x 0 + v 0x t + 1/2 a x t 2 v x = v 0x + a x t v x 2 = v 0x 2 + 2a x Δx y = y 0 + v 0y t + 1/2 a y t 2 v y = v 0y + a y t v y 2 = v 0y 2 + 2a y Δy x and y motions are independent! They share a common time t Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 10/10/13 1 Kinematics for Projectile Kinematics for Projectile Motion Motion a a x = 0 a = 0 a y = -g = -g x = x 0 + v x t v x = v 0x y = y 0 + v 0y t - 1/2 gt 2 v y = v 0y - gt v y 2 = v 0y 2 - 2g Δy
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### Transcript of Kinematics in Two Dimensions l x = x 0 + v 0x t + 1/2 a x t 2 l v x = v 0x + a x t l v x 2 = v 0x 2... Kinematics in Two DimensionsKinematics in Two Dimensions

x = x0 + v0xt + 1/2 axt2

vx = v0x + axt

vx2 = v0x

2 + 2ax Δx

y = y0 + v0yt + 1/2 ayt2

vy = v0y + ayt

vy2 = v0y

2 + 2ay Δy

x and y motions are independent!They share a common time t

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 1

Kinematics for Projectile MotionKinematics for Projectile Motion aaxx = 0 a = 0 ayy = -g = -g

x = x0 + vxt

vx = v0x

y = y0 + v0yt - 1/2 gt2

vy = v0y - gt

vy2 = v0y

2 - 2g Δy Example 1Example 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?

3. They both hit the ground at the same time correct

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 y=0

• Therefore, t=sqrt(2y0/g)

• Result is independent of v0x

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 2 Example 2Example 2A 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 carcorrect

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.

The train and the ball have the same horizontal velocity and by throwing the ball straight up, the horizontal component is not changed.

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 3 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 monkey2 Below the monkey3 Above the monkey

Example 4Example 4

correct

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.

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 4 Example 4: Shooting the Monkey... (II)Example 4: Shooting the Monkey... (II)

x x = = xx00

yy = -1/2 gg t2

xx = = vv0 0 tt

yy = -1/2 gg t2

Dart hits the monkey!

No monkeys were harmed during the

making of this slide

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 5 Example 4: Shooting the Monkey... (III)Example 4: Shooting the Monkey... (III)

yy = vv0 t - 1/2 gg t2

At an angle, still aim at the monkey! yy = y0 - 1/2 gg t2

Dart hits the monkey!

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 6 Newton's LawsNewton's Laws

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 7

After this lecture, you should know about:Force, mass, inertia.Newton’s first and second law.Inertial and non-inertial reference frame.Gravitation.Action = reaction.Normal force.Free body diagram. Classical Mechanics and forcesClassical 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 (< atomic sizes)

» objects moving near the speed of light Force:

Usually think of a force as a push or pullVector quantityMay 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”

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 8 ForcesForces

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 9 Contact and Field ForcesContact and Field Forces

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 10 Fundamental ForcesFundamental Forces

TypesStrong nuclear forceElectromagnetic forceWeak nuclear forceGravity

CharacteristicsAll field forcesListed in order of decreasing strengthOnly gravity and electromagnetic in mechanics

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 11 NewtonNewton’’s First Laws 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 Newton’s 2nd Law…)

If v=0, 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

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 12 Example 7Example 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 totaltotal force on the plane:

1. is pointing upward2. is pointing downward 3. is pointing forward 4. is pointing backward5. is zero

lift

weight

drag thrust

correct

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.

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 13 Inertia and massInertia 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 quantitySI units are kg

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 14 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…

Newton’s 1st law is a way to define inertial frames

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 15

Inertial vs non-inertial reference frameInertial vs non-inertial reference frame NewtonNewton’’s Second Laws 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

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 16

Newton’s second law applies only in inertial reference frames

Applying it in non-inertial ones leads to pseudo-forces, e.g. centrifugal force Example 8Example 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)(1) 2m1 (2)(2) m1 (3)(3) m1/2

F a1

m1 F a2 = 2a1

m2

• F=ma • F= m1a1 = m2a2 = m2(2a1)• Therefore, m2 = m1/2

• Or in words…twice the acceleration means half the mass

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 17 Gravitational Force and WeightGravitational Force and Weight Mutual force of attraction between any two objects Expressed by Newton’s Law of Universal Gravitation:

The magnitude of the gravitational force acting on an object of mass m near the Earth’s surface is called the weight w of the objectw = m g is a special case of Newton’s 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

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 18 Weight and MassWeight and Mass

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 19 Example 9Example 9

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 20

What is the approximate weight force of a bar of Chocolate of 100g on sea level ?

Use g~10m/s2 : W~ 0.1kg x 10 m/s2 = 1 N

1 N NewtonNewton’’s Third Laws Third Law For every action, there is an equal and opposite reaction.

• Finger pushes on box • Ffingerbox = force exerted on box by finger

Ffingerbox

Fboxfinger• Box pushes on finger

• Fboxfinger = force exerted on finger by box

• Third Law: Fboxfinger = - Ffingerbox

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 21 Newton's Third Law...Newton's Third Law...

FFA ,B = - FFB ,A. is true for all types of forces

FFw,m FFm,w

FFf,m

FFm,f

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.

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 22 Since FFm,b = -FFb,m why isn’t FFnet = 0, and aa = 0 ?

a ??a ??FFb,m FFm,b

ice

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 23 Example of Good ThinkingExample of Good Thinking Consider only the boxonly the box!

FFon box = maabox = FFm,b

Free Body Diagram (more on this next time)

aaboxbox

FFb,m FFm,b

ice

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 24 Example 10AExample 10ASuppose 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 pushing:

1. FA = FS 2. FA > FS

3. FA < FS

correctThird Law!

Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University10/10/13 25 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 26

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 > aS

3. aA < aS

Example 10BExample 10B

correct

a=F/mF same, hence lower mass gives larger a 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 27

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

Example 11Example 11 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 28

The Normal ForceThe 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 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 29

Do You Feel The Normal Force?Do You Feel The Normal Force?Yes, you can feel an

F = gm = 9.8x100 = 980 Newtons!

That force is spread out over the area of you foot so it’s not

Pressure: P = F/Area = N/m2 Action-Reaction Pairs vs forces acting on Action-Reaction Pairs vs forces acting on an objectan 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

10/10/13 30Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University Forces Acting on an ObjectForces Acting on an Object

Newton’s Law uses the forces acting on an object

are acting on the object

are acting on other objects

10/10/13 31Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 32

• Newton’s First Law:The motion of an object does not change unless it is acted on by a net external force

• Newton’s Second Law:

• Newton’s Third Law:

Summary of NewtonSummary of Newton’’s lawss laws Applications of NewtonApplications of Newton’’s Lawss Laws

AssumptionsObjects behave as particles

» can ignore rotational motion (for now)Masses of strings or ropes are negligibleInterested only in the forces acting on the object or the “system of

interest”

» can neglect reaction forces

10/10/13 33Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 34

Example 12Example 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

mg

N

Free Body Diagram of the person: 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 35

Frictional ForceFrictional Force Friction

Opposes motion between systems in contactParallel to the contact surfaceDepends 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 frictionμK: coefficient of kinetic friction 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 nVariations of the

coefficient with speed will be ignored

Friction (II)Friction (II)

10/10/13 36Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 37

Example 13Example 13You 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. 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 38

Example 14AExample 14AYou 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

weight/gravity (W)normal (FN)

W

FN V

correct 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 39

Example 14BExample 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

W

FN V

W

FN ΣF = ma = 0

correct 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 40

Example 14CExample 14CYou are driving a car up a hill with constant acceleration.

How many forces are acting on the car? 12345

W

FN a

correct

weight/gravity (W)normal (FN) 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 41

Example 14DExample 14DYou are driving a car up a hill with constant acceleration.

The net force on the car is now:1. Zero 2. Pointing up the hill 3. Pointing down the hill 4. Pointing vertically downward 5. Pointing vertically upward

W

FN

W

FN ΣF = ma = up the hill

a

correct 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 42

Example 14- SummaryExample 14- Summary

W

FN V

W

FWpara

FWperp

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 < Froad on car

Accelerate up the hill

If Fw parallel > Froad on car

Accelerate down the hill

Fw parallel > fmax = FN μs= μsMgcosφ

Slide down the hill FWpara = Mg sinφFwperp=FN = Mg cosφ

φ

φ 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 43

TensionTensionTension is a force along the

length of a mediumTension can be transmitted around corners

If there is no friction in the pulleys, T remains the same 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 44

More on tensionMore 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. 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 45

What is the tension in the string?

A) T<W

B) T=W

C) W<T<2W

D) T=2W

Example 14: Pulley IExample 14: Pulley I

WW

W

T

W

Look at Free Body Diagram: T=W

Net Force = 0 = acceleration 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 46

Example 15: Pulley IIExample 15: Pulley II

What is the tension in the string?

A) T<W

B) T=W

C) W<T<2W

D) T=2WW

2Wa a

2Wa

T

2W

T<2W: Wa

T

W

W<T:

Look at FreeBody Diagrams: 10/10/13 Medical Physics, Winter 2013/14, Vita-Salute San Raffaele University 47

Example 15Example 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

Fh T T Fh

2Fh 2T