physical quantities, vectors, fovce and newton's laws

Physics Assignment

Group “A”Usman Abrar, Kamran Sharif,

Muneeba Idrees, Hassan Amjad, Ali Asad

Measurement

Motion in one Dimension&

Force and Newton’s Laws

Chapter # 01, 02, 03

Physical Quantities:- “Quantities which can be measured are called Physical

Quantities.” These require Magnitude, Unit and Sometime Direction

for their complete description. Here we will discuss SI( System International) for

measurements…

Chap#01 Measurement

For example Height of the girl, l = 1.55 m

Symbol of the physical

quantity

Numerical value for the magnitude of the physical

quantity

Unit of the physical quantity

Types of physical Quantities:- Base/ Basic Quantities . Derived Quantities.Basic Quantities:-

“ Quantities which are chosen as a base and many other measuring quantities are derived from them are called basic quantities and their units are called base units…”

They are :

System International:-

Derived quantities:-“All quantities other then “7” basic quantities are called Derived quantities because they are derived from Base

quantities and their units are called derived units.”For Example :-

Physical Quantities:-

Units:-“Units are symbol associated with every physical quantity whether its basic or Derived quantity….”

Like, Meter(m) for length ; kilogram(kg) for Mass and

second(s) for time etc…Standards:-“Physicals quantities must have a standard value so that

every calculation is accurate and scientist in different place can calculate things without any ambiguity…”

Some standards are discussed here...

System international

The Standard of time:-Anything which repeats its motion periodically can be set as time standard like Oscillating Pendulum, A mass-Spring system, a Quartz crystal etc… But in SI time standard is defined as:“The second is the duration of 9,192,631,770 vibrations of

(specific) radiation emitted by a Specific isotope Of the Cesium atom”

Fig shows the current national frequancy standard, so-called Cesium foundation clock.

System international

The Standard of length:

SI unit of length is Meter (m)… defined as: “The meter is the length of the path traveled

by light in vacuum during a time interval of 1/299,792,458 of a second”

Length is also measured in cm, km, miles, feets etc…The Standard of Mass: SI unit Base unit for Mass is “Kg”

“The SI standard of mass is a platinum-iridium cylinder kept at IBWM”

System international

Precision:-“An indication of the scale on the measuring device

that was used.”In other words, the more correct a measurement is, the more accurate it is. On the other hand, the smaller the scale on the measuring instrument, the more precise the measurement.

Fig illustrates difference b/w Precision and accuracy.

Precision And Significant Figures

“The significant figures (also known as significant digits, and often shortened to sig figs) of a number are

those digits that carry meaning contributing to its precision. This includes all digits”

except: leading and trailing zeros which are merely place

holders to indicate the scale of the number. spurious digits introduced, for example, by

calculations carried out to greater precision than that of the original data, or measurements reported to a greater precision than the equipment supports.

Significant Figures

The rules for identifying significant figures when writing or interpreting numbers are as follows:  All non-zero digits are considered significant. For example, 91 has two significant

figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4 and 5).

Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has five significant figures: 1, 0, 1, 1 and 2.

Leading zeros are not significant. For example, 0.00052 has two significant figures: 5 and 2.

Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures since it has three trailing zeros. This convention clarifies the precision of such numbers; for example, if a measurement precise to four decimal places (0.0001) is given as 12.23 then it might be understood that only two decimal places of precision are available. Stating the result as 12.2300 makes clear that it is precise to four decimal places (in this case, six significant figures).

Identifying significant figures

The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is precise to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. Various conventions exist to address this issue:

A bar may be placed over the last significant figure; any trailing zeros following this are insignificant. For example, 1300 has three significant figures (and hence indicates that the number is precise to the nearest ten).

The last significant figure of a number may be underlined; for example, "2000" has two significant figures.

A decimal point may be placed after the number; for example "100." indicates specifically that three significant figures are meant.

In the combination of a number and a unit of measurement the ambiguity can be avoided by choosing a suitable unit prefix. For example, the number of significant figures in a mass specified as 1300 g is ambiguous, while in a mass of 13 h g or 1.3 kg it is not.

Identifying significant figures

Kinematics:-“Kinematics is the branch of mechanics that describes the

motion of objects without necessarily discussing what causes the motion.”

By specifying the velocity, position and acceleration f a object, we can describe how this object moves, including the direction of its motion. How that direction changes with time, whether the object speeds up or slows down and so forth…

Position, velocity, acceleration etc… can be found using vectors.

Chapter # 02 Motion in one dimension

Vector:-“Vector is that quantity that requires

magnitude, unit as well as Direction for its complete description…”

For example: Displacement, velocity (v) etc… Vector is represented by straight line with an

arrow head on its either side… And vector quantities are represented either

in Bold or making an arrow on is symbol… A.

Vectors

Representation of a Vector:-

Properties of Vectors

x

and

z

θ

A

Ax

Aan

d

Az

θsenAAx cos

θsenAsenAy θcosAAz

222zyx AAAAA

kAjAiAA zyx

Sum of Vectors

BA

R

BA C

C

Law of the polygon

Sum of Vectors

BA

R

BA C

C

Law of the polygon

The resulting vector is one that vector from the origin of the first vector to the end of the last

Vectors Properties

A

Opposite-A

Null 0 = A + ()-A

Unit vector AA

μ

ˆAA

Properties of the sum of vectors

Commutative Law

ABBAR Associative

Law

C)BA)CBAR

((

Difference

B-AR

)B(-AR

A

B A

-BR

Commutative (Method parallelogram)

The vectors A and B can be displaced parallel to find the vector sum

B

R = A

+ B

A

B R = B

+ A

BR =

A +

B

Multiplication of a vector by a scalar

Given two vectors ByA

Are said to be parallel if BA

BAsi

0

BAsi

0BAsi

1

Dot product of two vectors

θABBA cos

cosθAAB Projection of A on B

cosθBBA

Projection of B on A

1ˆˆ ii1ˆˆ jj

0ˆˆ ji

0ˆˆ kj0ˆˆ ki

xAiA

1ˆˆ kk

yAjA ˆ

zAkA ˆ

ZZYYXX BABABABA

Vector product of two vectors BAC

θABC sen

0ii

0ˆˆ

jj

0ˆˆ

kk

kji ˆˆˆ ikj ˆˆˆ

jik ˆˆˆ

)kBjBiB()kAjAiA(BAC zyxzyx

YZZYX BABAC

zxxzy BABAC

xyyxz BABAC

Demonstrate:

Distance vs Displacement

Distance ( d ) Total length of the path travelled Measured in meters scalar

Displacement ( ) Change in position (x) regardless of path x = xf – xi

Measured in meters vector

d

Distance

Displacement

A

B

Finding displacement

hbwlarea 2

1

velo

city

time

vo

t

v

v – vo = at

atttvd o 2

1

20 2

1attvd

The displacement divided by the elapsed time.

timeElapsed

ntDisplaceme velocityAverage

ttt o

o

xxx

v

Average Velocity

SI units for velocity: meters per second (m/s)

sm4s 2

m 8 Slope

t

x

Finding velocity

The instantaneous velocity indicates how fast the car moves and the direction of motion at each instant of time.

tt

xv

0lim

Instantaneous Velocity & Speed

The instantaneous speed is the magnitude of the instantaneous velocity

Instantaneous Velocity

The notion of acceleration emerges when a change in velocity is combined with the time during which the change occurs.

Acceleration

ttt o

o

vvv

a

The difference between the final and initial velocity divided by the elapsed time

SI units for acceleration: meters per second per second (m/s2)

2sm6s 2

sm 12 Slope

t

v

Finding acceleration

Determine the average acceleration of the plane.

sm0ov

s 0ot

hkm260v

s 29t

s

hkm0.9

s 0s 29

hkm0hkm260

a

Example Acceleration and Increasing Velocity

o

o

tt

vv

a

s 9s 12

sm28sm13

o

o

tt

vva

Example Acceleration and Decreasing Velocity

2sm0.5a

It is common to dispense with the use of boldface symbols overdrawn with arrows for the displacement, velocity, and acceleration vectors (AP does not show arrows on given equations nor expect them on open-ended problems). We will, however, continue to convey the directions with a plus or minus sign. (AP calls elapsed time “t” where t = t – to)

t

vva o

Equations of Kinematics for Constant Acceleration

o

o

tt

vv

a

t

vva o

ovvat

atvv o

Equations of Kinematics for Constant Acceleration

AP Equation#1

atvv o vvt

xx

0

0

2

1

221 attvxx oo

Equations of Kinematics for Constant Acceleration

vvv o 2

1

t

xxv 0

tvvxx 00 2

1

If, a is constant:

tatvvxx ooo 2

1

AP Equation#2

a

vvvvxx o

oo 21

t

vva o

a

vvt o

a

vvxx oo 2

22

Equations of Kinematics for Constant Acceleration

tvvxx 00 2

1

022 2 xxavv o

AP Equation#3

In the absence of air resistance, it is found that all bodiesat the same location above the Earth fall vertically with the same acceleration. If the distance of the fall is smallcompared to the radius of the Earth, then the accelerationremains essentially constant throughout the descent.

This idealized motion is called free-fall and the accelerationof a freely falling body is called the acceleration due to gravity.

22

22

sft30sm10

sft2.32or sm80.9

org

g

Free Fall

Freefalling bodies

I could give a boring lecture on this and work through some examples, but I’d rather make it more real…

Use same kinematic equations just substitute g for aChoose +/- carefully to make problem as easy as possible

Free fall problems

Two types of forces◦ Contact force

Force caused by physical contact◦ Field force

Force caused by gravitational attraction between two objects

Force

Born 1642

Went to University of Cambridge in England as a student and taught there as a professor after

Never married

Gave his attention mostly to physics and mathematics, but he also gave his attention to religion and alchemy

Newton was the first to solve three mysteries that intrigued the scientists

◦ Laws of Motion◦ Laws of Planetary Orbits◦ Calculus

Isaac Newton

Newton’s Laws of Motion are laws discovered by Physicist and mathematician, Isaac Newton, that explains the objects’ motions depending on forces acted on them

◦ Newton’s First Law: Law of Inertia◦ Newton’s Second Law: Law of Resultant Force◦ Newton’s Third Law: Law of Reciprocal Action

Three Laws of Motion

An Object at rest remains at rest, and an object in motion continues in motion with constant velocity (that is, constant speed in a straight line), unless it experiences a net external force.

The tendency to resist change in motion is called inertia

◦ People believed that all moving objects would eventually stop before Newton came up with his laws

Newton’s First Law

A force that causes resistance to motion Arises from contact between two surfaces◦ If the force applied is smaller than the

friction, then the object will not move If the object is not moving, then

ffriction=Fapplied

◦ The object eventually slips when the applied force is big enough

Friction

Friction

Friction was discovered by Galileo Galilee when he rolled a ball down a slope and observed that the ball rolls up the opposite slope to about the same height, and concluded that the difference between the initial height and the final height is caused by friction.

Galileo also noticed that the ball would roll almost forever on a flat surface so that the ball can elevate to the same height as where it started.

Static Friction◦ Friction that exists

while the object is stationary

◦ If the applied force on an object becomes greater than the maximum of static friction, then the object starts moving

◦ Fstatic ≤ μstatic n

Kinetic Friction◦ The friction that exists

when an object is in motion

◦ F – fkinetic produces acceleration to the direction the object is moving

◦ If F = fkinetic, then the object moves at constant speed with no acceleration

◦ fkinetic= μkineticn◦ Kinetic friction and the

coefficient of kinetic friction are smaller than static friction and the static coefficient

Two types of Friction

Newton’s First Law

When there is no force exerted on an object, the motion of the object remains the same like described in the diagram

◦ Because the equation of Force is F=ma, the acceleration is 0m/s². So the equation is 0N=m*0m/s²

◦ Therefore, force is not needed to keep the object in motion, when

◦ The object is in equilibrium when it does not change its state of motion

The car is traveling rightward and crashes into a brick wall. The brick wall acts as an unbalanced force and stops the car.

The truck stops when it crashes into the red car.

But the ladder falls in front of the truck because the ladder was in motion with the truck but there is nothing stopping the ladder when the truck stops.

Inertial Frames

Any reference frame that moves with constant velocity relative to an inertial frame is itself an inertial frame

A reference frame that moves with constant velocity relative to the distant stars is the best approximation of an inertial frame

We can consider the Earth to be such an inertial frame, although it has a small centripetal acceleration associated with its motion

Newton’s First Law – Alternative Statement

In the absence of external forces, when viewed from an inertial reference frame, an object at rest remains at rest and an object in motion continues in motion with a constant velocity

Newton’s First Law describes what happens in the absence of a force Does not describe zero net force

Also tells us that when no force acts on an object, the acceleration of the object is zero

Inertia and Mass

“The tendency of an object to resist any attempt to change its velocity is called inertia”

“Mass is that property of an object that specifies how much resistance an object exhibits to changes in its velocity”

Masses can be defined in terms of the accelerations produced by a given force acting on them:

The magnitude of the acceleration acting on an object is inversely proportional to its mass

1 2

2 1

m am a

Mass vs. Weight

Mass and weight are two different quantities

Weight is equal to the magnitude of the gravitational force exerted on the object Weight will vary with location

Example: wearth = 180 lb; wmoon ~ 30 lb mearth = 2 kg; mmoon = 2 kg

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass

Newton’s Second Law

Fnet

Acceleration

Unbalanced Force and Acceleration

Force is equal to acceleration multiplied by mass

◦ When an unbalanced force acts on an object, there is always an acceleration

Acceleration differs depending on the net force

The acceleration is inversely related to the mass of the object

Force is a vector◦ Because it is a vector, the net force can be

determined by subtracting the force that resists motion from the force applied to the object.

◦ If the force is applied at an angle, then trigonometry is used to find the force

Net Force

Fnet

θR

R*cos θR*sin θ

θR

The force that exerts all objects toward the earth’s surface is called a gravitational force.

◦ The magnitude of the gravitational force is called weight

The acceleration due to gravity is different in each location, but 9.80m/s² is most commonly used

Calculated with formula w=mg

Gravitational Force

If two objects interact, the force exerted on object 1 by object 2 is equal in magnitude but opposite in direction to the force exerted on object 2 by object 1

Forces always come in pair when two objects interact

◦ The forces are equal, but opposite in direction

Newton’s Third Law

Fg

Fn

As the man jumps off the boat, he exerts the force on the boat and the boat exerts the reaction force on the man.

The man leaps forward onto the pier, while the boat moves away from the pier.

Newton’s Third Law

Newton’s Third Law

Flow backward

Foil deflected down

Foil deflected up

Engine pushed forward

Flow pushed backward

Foil deflected down

Applications of Newton’s LawAssumptions

Objects can be modeled as particles Interested only in the external forces acting

on the object can neglect reaction forces

Initially dealing with frictionless surfaces Masses of strings or ropes are negligible

When a rope attached to an object is pulling it, the magnitude of that force is the tension in the rope.