Physics Grade 11 Mechanics Unit-1 (Unit and Dimension)

Unit and Dimension 

Physical Quantity

The quantities which can be measured are called physical quantities. The physical quantities when measured, are expressed by magnitude (numerical value) and unit.

The physical quantities can be divided into two categories; fundamental quantity and derived quantity.

Fundamental Quantity: The basic physical quantity which can be taken as a standard to measure other physical quantities is known as a fundamental quantity. In general, seven physical quantities are considered fundamental quantities.

1. What are fundamental Quantities?

They are Mass, Length, Time, Temperature, Luminous Intensity, electric current, and Amount of Substance. In addition, there are other two sub-fundamental quantities, which are plane angle and solid angle.

Derived Quantity: A quantity obtained from fundamental quantities is called derived quantity. Area, volume, density, work, power, etc. are examples of derived physical quantities. Derived physical quantities can be expressed in terms of fundamental quantities.

Measurement: The process of comparison of an unknown physical quantity with a known physical quantity is called measurement.

Unit: The known quantity used as the standard for the measurement is called unit. Unit is of two types: Fundamental unit and derived unit.

Fundamental Units: The units of fundamental quantities are called fundamental units. Derived Units: The units of derived quantities are called derived units.

2. What are the different systems of units in measurement? 

System of units:

FPS System: Length is measured in the foot; mass is measured in pounds and time is measured in seconds.

CGS System: Length is measured in centimeters; mass is measured in grams and time is measured in seconds.

MKS System: Length is measured in meters; mass is measured in kilogram and time is measured in the second

SI System: The International System of Units (SI, abbreviated from the French Système International (d'unités)) is the modern form of the metric system. It is the only system of measurement with official status in nearly every country in the world.

It comprises a coherent system of units of measurement starting with seven base units, which are the second (time= s), meter (length= m), kilogram (mass= kg), ampere (electric current= A), kelvin (thermodynamic temperature= K), mole (amount of substance, mol), and candela (luminous intensity, cd).

The system allows for an unlimited number of additional units, called derived units, which can always be represented as products of the powers of the base units.

Besides this, there are other two units called supplementary units. They are radian (unit of plane angle) and steradian (unit of solid angle)

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Advantages of SI system

ØSI system has advantages over other system of units

•SI system is a metric system. Like the CGS system and MKS system, the multiples and sub-multiples of the unit can be expressed as powers of 10. For example: multiples and sub-multiples of meter are shown in the picture below

SI system is a coherent system. In the SI system, dividing and multiplying the base or supplementary unit without introducing numerical factors, all the derived units can be obtained. For example: 1J work = 1kgm2s^-2.

• (a system of units is said to be coherent if all the units are either base units or derived from base units without introducing any numerical factors other than 1.)

•It is a rational system. SI system uses only one unit for one physical quantity. For example: SI unit of Energy (all types of energy) is joule.

Dimensions of a physical quantity

•The dimensions of a physical quantity are defined as the powers of the fundamental quantities which are involved in the physical quantity.

The dimension of mass is [M], that of length is [L] and that of time is [T]. Similarly, dimensions of temperature are [K], dimension of electric current is [A], dimension of luminous intensity is [J] and the dimension of amount of substance(mole) is [N].

For example, acceleration = velocity time=displacementtime2

Therefore, dimensional equation of acceleration is [a]=[L][T2] =[LT−2]

Hence the dimensions of acceleration are 1 in length and -2 in time.

Dimensional formula

ØThe dimensional formula of a physical quantity is defined as the expression showing how and which basic quantities are involved in the derived quantity. It is generally written in square bracket [ ]. Here, [LT^-2] is the dimensional formula of acceleration.

Example: The dimensional equation for velocity is [v] = [s][t]

=[L][T] =[LT^-1]

The hence dimensional formula for velocity is [LT^-1]

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Sl. No	Physical Quantity	Formula	Dimensional Formula	S.I Unit
1	Area (A)	Length x Breadth	[M0L2T0]	m2
2	Volume (V)	Length x Breadth x Height	[M0L3T0]	m3
3	Density (d)	Mass / Volume	[M1L-3T0]	kgm-3
4	Speed (s)	Distance / Time	[M0L1T-1]	ms-1
5	Velocity (v)	Displacement / Time	[M0L1T-1]	ms-1
6	Acceleration (a)	Change in velocity / Time	[M0L1T-2]	ms-2
7	Acceleration due to gravity (g)	Change in velocity / Time	[M0L1T-2]	ms-2
8	Specific gravity	The density of body/density of water at 4oC	No dimensions [M0L0T-0]	No units
9	Linear momentum (p)	Mass x Velocity	[M1L1T-1]	kgms-1
10	Force (F)	Mass x Acceleration	[M1L1T-2]	N
11	Work (W)	Force x Distance	[M1L2T-2]	J (Joule)
12	Energy (E)	Work	[M1L2T-2]	J
13	Impulse (I)	Force x Time	[M1L1T-1]	Ns
14	Pressure (P)	Force / Area	[M1L-1T-2]	Nm-2
15	Power (P)	Work / Time	[M1L2T-3]	W
16	The universal constant of gravitation (G)		[M-1L3T-2]	Nm2kg-2
17	Moment of inertia (I)	Mass x (distance)2	[M1L2T0]	kgm2
18	Moment of a force, a moment of the couple	Force x distance	[M1L2T-2]	Nm
19	Surface tension (T)	Force / Length	[M1L0T-2]	Nm-1
20	Surface energy (E)	Energy/unit area	[M1L0T-2]	Nm-1
21	Force constant (x)	Force / Displacement	[M1L0T-2]	Nm-1
22	Coefficient of viscosity ( η )		[M1L-1T-1]	Nsm-2
23	Thrust (F)	Force	[M1L1T-2]	N
24	Tension (T)	Force	[M1L1T-2]	N
25	Stress	Force / Area	[M1L-1T-2]	Nm-2
26	Strain	Change in dimension / Original dimension	No dimensions [M0L0T-0]	No unit
27	Modulus of Elasticity (E)	Stress / strain	[M1L-1T-2]	Nm-2
28	The radius of gyration (k)	Distance	[M0L1T0]	m
29	Angle ( θ), Angular displacement	Arc length / Radius	No dimensions [M0L0T-0]	rad
30	Trigonometric ratio ( sin θ, cos θ, tan θ, etc)	Length / length	No dimensions [M0L0T-0	No unit
31	Angular velocity( ω )	Angle / Time	[M0L0T-1]	rad s-1
32	Angular acceleration( α )	Angular velocity / Time	[M0L0T-2]	rad s-2
33	Angular momentum (J)	Moment of inertia x Angular velocity	[M1L2T-1]	kgm2s-1
34	Torque (𝞽)	Moment of inertia x Angular acceleration	[M1L2T-2]	Nm
35	Velocity gradient	Velocity / Distance	[M0L0T-1]	s-1
36	Rate flow	Volume / Time	[M0L3T-1]	m3s-1
37	Wavelength( 𝛌 )	Length of a wavelet	[M0L1T0]	m
38	Frequency()	Number of vibrations/second or 1/time period	[M0L0T-1]	Hz or s-1
39	Angular frequency (ω)	2π x frequency	[M0L0T-1]
40	Planck’s constant (h)	Energy / Frequency	[M1L2T-1]	Js
41	Buoyant force	Force	[M1L1T-2]	N
42	Relative density	Density of substance / density of water at 4oC	No dimensions [M0L0T-0]	No unit
43	Pressure gradient	Pressure / Dstance	[M1L-2T-2]	Nm-3
44	Pressure energy	Pressure x Volume	[M1L2T-2]	J
45	Temperature	——	[M0L0T0K1]	K
46	Heat (Q)	Energy	[M1L2T-2]	J
47	Latent heat (L)	Heat / Mass	[M0L2T-2]	Jkg-1
48	Specific heat (S)		[M0L2T-2K-1]	Jkg-1K-1
49	Thermal expansion coefficient or thermal expansivity		[M0L0T0K-1]	K-1
50	Thermal conductivity		[M1L1T-3K-1]	Wm-1K-1
51	Bulk modulus or (compressibility)-1		[M1L-1T-2]	Nm-2 or Pascals
52	Centripetal acceleration		[M0L1T-2]
53	Stefan constant (σ)		[M1L0T-3K-4]	Wm−2K−4
54	Wien constant	Wavelength X temperature	[M0L1T0K1]	mK
55	Gas constant (R)		[M1L2T-2K-1]	JK-1
56	Boltzmann constant (K)	Energy / temperature	[M1L2T-2K-1]	JK-1
57	Charge (q)	Current x time	[M0L0T1A1]	C
58	Current density	Current / area	[M0L-2T0A1]	A m−2
59	Electric potential (V), voltage, electromotive force	Work / Charge	[M1L2T–3A-1]	V
60	Resistance (R)	Potential difference / Current	[M1L2T–3A-2]	ohms (Ω)
61	Capacitance	Charge / potential difference	[M–1L–2T4A2]	F (Farad)
62	Electrical resistivity or (electrical conductivity)-1		[M1L3T-3A–2]	Ωm ( resistivity)
63	Electric field (E)	Force / Charge	[M1L1T-3A-1]	NC-1
64	Electric flux	Electric field X area	[M1L3T–3A-1]	Nm2C-1
65	Electric dipole moment	Torque / electric field	[M0L1T1A1]	C m
66	Electric field strength or electric intensity	Potential difference / distance	[M1L1T-3A-1]	NC-1
67	Magnetic field (B), magnetic flux density, magnetic induction		[M1L0T-2A-1]	T (Tesla)
68	Magnetic flux (Φ)	Magnetic field X area	[M1L2T-2A-1]	Wb (Weber)
69	Inductance	Magnetic flux / current	[M1L2T-2A-2]	H (Henry)
70	Magnetic dipole moment	Torque /field or current X area	[M0L2T0A1]	Am2
71	Magnetic field strength (H), magnetic intensity or magnetic moment density	Magnetic moment / volume	[M0L-1T0A1]	Am-1
72	Hubble constant	Recession speed / distance	[M0L0T-1]	s-1
73	Intensity of wave	(Energy/time)/area	[M1L0T-3]	Wm-2
74	Radiation pressure	Intensity of wave / speed of light	[M1L–1T-2]
75	Energy density	Energy / volume	[M1L-1T-2]	Jm-3
76	Critical velocity		[M0L1T-1]	ms-1
77	Escape velocity		[M0L1T-1]	ms-1
78	Heat energy, internal energy	Work ( = Force X distance)	[M1L2T-2]	J
79	Kinetic energy		[M1L2T-2]	J
80	Potential energy	Mass X acceleration due to gravity X height	[M1L2T-2]	J
81	Rotational kinetic energy		[M1L2T-2]	J
82	Efficiency		No dimensions [M0L0T0]	No unit
83	Angular impulse	Torque X time	[M1L2T-1]	Js (Joule second)
84	Permitivity constant (of free space)		[M-1L-3T4A2]	F m-1
85	Permeability constant (of free space)		[M1L1T-2A-2]	NA-2
86	Refractive index		No  dimensions [M0L0T0]	No unit
87	Faraday constant (F)	Avogadro constant X elementary charge	[M0L0T1A1 mol-1]	C mol-1
88	Wave number		[M0L-1T0]
89	Radiant flux, Radiant power	Energy emitted / time	[M1L2T-3]	W(Watt)
90	Luminosity of radiant flux or radiant intensity		[M1L2T-3]	W sr-1 (Watt/steradian)
91	Luminous power or luminous flux of source		[M1L2T-3]	lm (lumen)
92	Luminous intensity or illuminating power of source	Luminous flux / Solid angle	[M1L2T-3]	cd (candela)
93	Intensity of illumination or luminance (Lv)		[M1L0T-3]	cd m-2
94	Relative luminosity	Luminous flux of a source of given wavelength / luminous flux of peak sensitivity wavelength(555 nm) source of the same power	No dimensions [M0L0T0]	No unit
95	Luminous efficiency	Total luminous flux / Total radiant flux	No dimensions [M0L0T0]	No unit
96	Illuminance or illumination	Luminous flux incident / Area	[M1L0T-3]	lx (lux)
97	Mass defect	(Sum of masses of nucleons) – (mass of the nucleus)	[M1L0T0]
98	Binding energy of nucleus		[M1L2T-2]
99	Decay constant	0.693 / half-life	[M0L0T-1]
100	Resonant frequency		[M0L0T-1A0]
101	Quality factor or Q-factor of coil		No dimensions [M0L0T0]	No unit
102	Power of lens		[M0L-1T0]	D (dioptre)
103	Magnification	Image distance / Object distance	No dimensions [M0L0T0]	No unit
104	Fluid flow rate		[M0L3T-1]	m3s-1
105	Capacitive reactance (Xc)	(Angular frequency X capacitance)-1	[M1L2T-3A-2]	ohms (Ω)
106	Inductive reactance (XL)	(Angular frequency X inductance)	[M1L2T-3A-2]	ohms (Ω)

 

 

Dimensional formula of physical quantities

 Dimensional equation

The dimensional formula and SI units for more than 100 physical quantities are given in the table below.

ØAn equation obtained by equating a physical quantity with dimensional formula is called dimensional equation.

Example: dimensional equation of equation is [v] = [LT^-1]

 

Principle of homogeneity of dimensions

Principle of homogeneity of dimensions’ states that, “for the physical relation to be correct, the dimensions of the fundamental quantities on the left-hand side of the equation is equal to the dimensions of the fundamental quantities on the right-hand side of the equation.”

For illustration, we consider an equation A = B + C. For this equation to be correct, dimensions of A must be equal to dimensions of B and C.

Categories of physical quantities in terms of dimensional analysis

•Dimensional variable, example: distance, speed, acceleration, force, work, energy, power etc.

•Dimensional constant, example: Plank’s constant(h), Stefan’s constant(σ), universal gravitational constant(G) etc.

•Dimensionless variable, example: angle, refractive index, strain, relative density (specific gravity) etc.

•Dimensionless constant, example: pie(π), counting numbers etc.

Uses of dimensional analysis

•To check the correctness of physical relation

•To derive the relation between various physical quantities.

•To convert the value of physical quantity from one physical quantity from one system of unit into another system of units

•To find the dimensions of constants in the given equation.

Limitations of dimensional analysis

•It does not give any information about dimensionless constants.

•If the quantity depends on more than three other physical quantities having dimensions, the formula cannot be derived.

•We cannot derive the formula containing trigonometric functions, logarithmic functions, exponential functions etc.

•The exact form of relation cannot be formed when there is more than one part in any relation.

•It gives no information about the physical quantity, whether it is vector or scalar.

If ‘m’ is the mass, ‘c’ is the velocity of light and x = mc2, then dimensions of ‘x’ will be:

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How to master paragraph writing? Writing a perfect paragraph on any topic.

 Unit-2 English Book

Freewriting ( Writing Section )

What is paragraph writing?

Paragraph Writing

A paragraph can be defined as a piece of writing in which one's thought as the central idea or controlling idea is developed using supporting details which can be examples, reasons, comparative sentences, or other related facts. Every good paragraph should have a controlling idea, to which sentences are related. 

A paragraph is regarded as a good one only if the readers can completely understand the unit of information it contains, and if its central or controlling idea is completely developed using appropriate supporting ideas.

While producing paragraphs for our purpose, we have to arrange the sentences clarifying the idea so that we can meet our aim of expressing our thoughts. Coherently organized sentences overtly clarify the central idea of the paragraph. The sentences that do not relate to the central idea must not be included in the paragraph.

What are the types of paragraphs?

Classification of Paragraphs

Paragraphs are simply categorized as descriptive, narrative, reflective, and expository or argumentative. The classification depends on the nature of the topics and the purpose of writing. They are discussed as below:

1. Descriptive Paragraph

A descriptive paragraph describes places, objects/things such as countries, rivers, towns, buildings, and so on and so forth. A descriptive paragraph is an accurate account of the subject matter which imparts factual information. For example:

Portsmouth is an industrial city on the southern coast of England Although the largest proportion of the working population is employed in the naval dockyards, there are many other industries in the area. For example, there are clothing factories, manufactures of electrical components, light machine industries, and chemical industries. The naval dockyards have been in existence for five centuries, but most of the other industries have been established during the past fifteen years (Chaplen, 1976).

The above paragraph is a simple and clear description of an industrial city in England. The paragraph explains the changes taking place during the past fifteen years and why they have taken place as well.

2. Narrative Paragraph

A narrative paragraph describes the historical events, stories biographies, and so on. The writer of the paragraph narrates the subject matter. It is, to some extent, reflective, too. 

For example:

Only last year I witnessed what might have been a fatal accident. I was out for a stroll on the Mall At the crossroads near the General Post Office, I saw a car coming down, which swept round the corner of the road and crashed into another coming up. Happily, no one was killed but several people sustained injuries. The driver was knocked senseless. The two cars have completely wrecked The policeman on point duty had raised his hand to stop one of those cars But coming as it was at the speed of about eighty miles an hour the driver could not apply brakes in time (Dave et al. 1988)

The aforementioned paragraph narrates a motorcar accident that was witnessed by the author of the paragraph.

3. Reflective Paragraph

A reflective paragraph consists of reflections upon some topics, which are generally of an abstract nature; such as habits, politics, discipline, and so on. Reflective paragraphs include the experience of the writer on a particular topic. For example.

Perseverance is the very hinge of all virtues. We fail in our undertakings in the world not because we lack talents, but because we have no strength of will and tenacity of spirit to carry out schemes We sometimes waste our force on imaginary difficulties. Difficulties that melt into the nothingness of a dream before dogged perseverance. It has been well said "Patience and perseverance will overcome mountains The smallest brook on the earth, by continuing to run, has hollowed out for itself a considerable valley to flow in Let, therefore, extol the virtue of perseverance Without it all the rest are little better than fancy-gold which glitters in your purse, but when taken to market proves to be slate or cinders (Dave et al., 1988).

The paragraphs as such reflect the opinion of the writer upon some topics.

4. Argumentative/Expository Paragraph

Argumentative or expository paragraphs explain the saying or exposition. The writer of the paragraphs should try to give his/her thought to the readers to establish the truth of the saying. The writer tries to make readers believe his saying. The sayings or quotations are suitable topics for these types of paragraphs. For example:

Ruskin called books Kings Treasuries Books are indeed the richest possession of mankind The world would be barren without books Life would love half of its charm if books were banished from the world. Books are the friends of the friendless. Perhaps no other thing has such power to lift the poor out of his poverty, the wretched out of his misery to make the burden-bearer forget his burden, the sick his suffering, the sorrower his grief, the down-trodden his degradation as books. They are friends to the lonely, companions to the deserted, joy to the joyless, good cheer to the disheartened, a helper to the helpless, and a consoler to the afflicted. They bring light into darkness and sunshine into shadow (Dave et al, 1988)

The paragraph above exposits or argues why and to what extent books are friends of everyone. In this kind of writing, the writer tries to clarify his opinion by mentioning his arguments.

Some Features of a Good Paragraph

How to write a perfect paragraph? 

1. Unity

The connectedness of the supporting details with the central idea is called unity. A paragraph is a piece of writing in which a central or controlling idea is developed using supporting details. Every sentence in the paragraph must be closely connected with the topic of the paragraph The central idea should be fully developed, and should not be left half-developed. All the supporting details should directly be related to the central idea. Unnecessary details should not be introduced in the paragraph. A paragraph is said to be badly constructed if it lacks unity.

 2. Placement of a Topic Sentence

The sentence having the main idea of a paragraph is called the topic sentence. The terms main idea, central idea, and controlling idea are synonymously used. The topic sentence can be written as below: 

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1. The Topic Sentence at the Beginning

The topic sentence written at the beginning of the paragraph is clarified using supporting details. It is regarded as the most usual system of writing a paragraph. There are certainly some advantages in placing the topic sentence or pilot sentence at the beginning of the paragraph. The writer will case himself to plan the paragraph properly. On the other hand, the reader will be facilitated to grasp the information imparted by the paragraph For example:

Geographers have found that there are some important differences between the northern and southern polar regions of the earth The Arctic region, in the north, is mostly sea, surrounded by masses of land The south pole, on the other hand, is situated in a landmass surrounded by the oceans, in fact, Antarctica is regarded as a continent by itself. Both the regions have a very cold climate, but the winter in the Arctic is not so severe or so long as that in the Antarctic. Antarctica is snowbound almost throughout the year, the snow in the Arctic melts in summer. There is some vegetation in the Arctic, but Antarctica is a desert with virtually no plant or animal life (Narayanaswami, 2000)

2. The Topic Sentence in Middle

The topic sentence also appears in the middle of the paragraph with the supporting details after and before it. In such a paragraph, the reader will have to face the problem of getting the central idea of the paragraph. For example:

When the motorcar was first introduced it was a clumsy ty machine that labored along the street at a pace no farter than that of a trotting horse People looked at it with suspicion and fear. Since then the motor car has come a long way and is today a combination of beauty, luxury, and efficiency The modern automobile, sleek and streamlined with its brilliant hues, is an aesthetically appealing object, almost a work of art. You can travel in it in absolute comfort at an incredible speed and yet feel no fatigue at the end of the day's journey The engine is a masterpiece of mechanical ingenuity, and seldom lets you down if you maintain it in good condition (Narayanswami, 2000).

3. The Topic Sentence at the End

The topic sentence can also be placed at the end of the paragraph Facts related or the supporting details are presented one after another leading to a summing up in the topic sentence at the end. While writing such a paragraph, the last sentence is presented, to sum up, the central idea of the whole writing. For example

Recently there were reports in the newspapers on violent incidents in Bangladesh during a football match The fans supporting the two teams clashed openly in the streets, and stones and acid bombs were hurled at the spectators and at the police who were trying to control the crowds In Latin American countries, too, passions run high during football matches, and large-scale violence erupts during the matches English football fans have indulged in violence during the European Cup matches Such incidents lead us to conclude that our belief in supports as a means of establishing friendship among nations is misplaced (Narayanaswami, 2000)

1.2.3 Coherence

The logical sense relation between lexical items in a text is simply referred to as coherence. The thought in a paragraph should be connected and developed in a logical order to maintain coherence. The paragraph should be so constructed that one sentence leads on naturally to the next, and there should be a regular logical progression of thought. The material of a paragraph can be ordered in many ways, but the appropriate methods can be noted as chronological order, spatial order, logical order, climax, comparison and contrast, and so on.

1.2.4 Cohesion

It refers to the intra-textual relation of the grammatical and lexical kind that makes the parts of the text together as a whole to convey the complete meaning of it.

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Some Examples of Paragraphs

Some Guided Paragraphs

Here, the paragraphs given along with basic points are called guided paragraphs. Some guided paragraphs are given below:

A. An Accident I Witnessed

Last week, I witnessed a fearful accident while I was going to Pokhara to meet my relatives Early in the morning. I started my journey from Kathmandu. My bus left on time. Every seat of the bus was occupied. While we were traveling, it was raining heavily. As we reached Naubise, our bus driver accidentally jammed on the brakes to stop. I watched through the window and saw that there was a long queue of vehicles. As we got off the bus, we were informed that two vehicles collided with each other. As I heard it, I was extremely excited to know what really happened. I rushed to the spot where the accident happened 1 knew that the bus was going to Pokhara carrying 54 passengers. It met with the accident at 6 a.m. some unharmed passengers told me. Both of the vehicles were badly damaged as they fell 70 feet below the road. Unfortunately, nine of the bus passengers and the driver of the truck died on the spot. Fifteen of them were badly injured. They were immediately taken to hospitals Fortunately, the rest of the passengers were unharmed We saw that there was a pool of blood which made my eyes water. We were blocked there for an hour and moved ahead to our destination safely.

2. My Aim in Life

Education functions as the most important factor in the upliftment of the society, country, and the whole human civilization as well Public awareness can be fostered through education. To provide education to the people, I want to be a teacher in my life. After I pass graduation, I want to go to a remote place being a secondary level teacher. I will try to teach most of the children about the place. If I find any children not going to school, I will try to find the prime causes of this situation. I will try to convince their parents to send them to school. Furthermore, I will run many adult literacy classes for illiterate adults. I hope they will be co-operative with me. I will try to uplift their consciousness. I do hope that they will be literate and conscious after some years. If they are conscious, they can understand what is right and what is wrong. I will try to provide them health education, too because I have seen illiterate people are very careless about their health I will try to convince them not to follow bad and dead traditions because they are the hindrances that stop the human civilization from getting uplifted Because of the aforementioned reasons, I want to be a teacher in my life in the days to come.

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