Table of Contents

## Definition of Derived Units

The derived units are unlimited as they are formed by different operations on the base units.

- For derived units, the dimensions are expressed in terms of the dimensions of the base units.
- The derived units might also be expressed with the combination of base and derived units.
- SI-derived units are units of measurement derived from the seven base units specified by the International System of Units (SI). They are either dimensionless or can be expressed as a product of one or more of the base units, possibly scaled by an appropriate power of exponentiation.
- The SI has special names for 22 of these derived units (for example, hertz, the SI unit of measurement of frequency), but the rest merely reflect their derivation: for example, the square meter (m2 ), the SI derived unit of area; and the kilogram per cubic meter (kg/m3 or kg⋅m-3 ), the SI derived unit of density.
- The International System of Units or SI system, by international agreement, has fixed measurement units for seven fundamental properties: length, mass, time, temperature, electric current, amount of substance, and luminosity. These are called the SI base units.
- Units of measurement derived from the mathematical combination of SI base units are called SI-derived units.

## Table of Derived Units

Derived Units are created by mathematical relationships between other Base Units and are expressed in a combination of fundamental and base quantities.

Quantity | Symbol | Unit | Unit Abbreviation | Derivation |

Area | A | square meter | m^{2} | length x width |

Volume | V | cubic meter | m^{3} | length × width × height |

Density | D | kilograms/cubic meter | kg/m^{3} | mass / volume |

Concentration | c | moles/litre | mol/L | amount / volume |

Speed(velocity) | v | meters/second | m/s | length / time |

Acceleration | a | meters/second/second | m/s^{2} | speed / time |

Force | F | newton | N | mass × acceleration |

Energy | E | joule | J | force × length |

**These derived units are explained below**

### 1. Area

The area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object.

The area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape or the amount of paint required to cover the surface with a single coat.

It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square meter (written as m2 ), which is the area of a square whose sides are one meter long.

A shape with an area of three square meters would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles.

Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundaries, calculus is usually required to compute the area.

Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.

#### Conversion of Area

Although there are 10 mm in 1 cm, there are 100 mm^{2} in 1cm^{2}.

Calculation of the area of a square whose length and width are 1 meter would be:

1 meter × 1 meter = 1 m^{2}

and so, a rectangle with different sides (say length of 3 meters and width of 2 meters) would have an area in square units that can be calculated as:

3 meters × 2 meters = 6 m^{2}

This is equivalent to 6 million square millimeters. Other useful conversions are :

- one square kilometer = 1,000,000 square meters.
- 1 square meter = 10,000 square centimeters = 1,000,000 square millimeters.
- 1 square centimeter = 100 square millimeters.

### 2. Volume

Volume is the quantity of three-dimensional space occupied by a liquid, solid, or gas.

- Common units used to express volume include liters, cubic meters, gallons, milliliters, teaspoons, and ounces, though many other units exist.
- Capacity is the measure of the content of a vessel that holds liquids, grains, or other materials that take the shape of the container. Capacity is not necessarily the same as volume. It is always the interior volume of the vessel. Units of capacity include the liter, pint, and gallon, while the unit of volume (SI) is derived from a unit of length.
- In thermodynamics, the volume of a system is an important extensive parameter for describing its thermodynamic state. The specific volume, an intensive property, is the system’s volume per unit of mass. Volume is a function of state and is interdependent with other thermodynamic properties such as pressure and temperature.

**For example**, volume is related to the pressure and temperature of an ideal gas by the ideal gas law .From the base unit of length, we can define volume, and from the base units of length, mass, and time, we can define energy. Since volume is length cubed, its SI derived unit is m^{3}.

### 3. Density

Densityi is the mass of a unit volume of a material substance. The formula for density is d = M/V, where d is density, M is mass, and V is volume.

- Density is commonly expressed in units of grams per cubic centimeter. For example, the density of water is 1 gram per cubic centimeter, and Earth’s density is 5.51 grams per cubic centimeter.
- Density can also be expressed as kilograms per cubic meter (in meter-kilogram-second or SI units). For example, the density of air is 1.2 kilograms per cubic meter.
- The densities of common solids, liquids, and gases are listed in textbooks and handbooks. Density offers a convenient means of obtaining the mass of a body from its volume or vice versa; the mass is equal to the volume multiplied by the density (M = Vd), while the volume is equal to the mass divided by the density (V = M/d).
- The weight of a body, which is usually of more practical interest than its mass, can be obtained by multiplying the mass by the acceleration of gravity. Tables that list the weight per unit volume of substances are also available; this quantity has various titles, such as weight density, specific weight, or unit weight. See also specific gravity. The expression particle density refers to the number of particles per unit volume, not to the density of a single particle, and it is usually expressed as n.

### 4. Concentration

There are several ways to express the amount of solute present in a solution.

- The concentration of a solution is a measure of the amount of solute that has been dissolved in a given amount of solvent or solution. A concentrated solution has a relatively large amount of dissolved solute.
- A dilute solution has a relatively small amount of dissolved solute. However, these terms are relative, and we need to express concentration in a more exact, quantitative manner.
- Still, concentrated and dilute are useful terms to compare one solution to another. Also, be aware that “concentrate” and “dilute” can be used as verbs. If you were to heat a solution, causing the solvent to evaporate, you would be concentrating it because the ratio of solute to solvent would be increasing. If you were to add more water to an aqueous solution, you would be diluting it because the ratio of solute to solvent would be decreasing.

**There are four quantities that describe concentration :**

#### Mass Concentration

The mass concentration pi is defined as the mass of a constituent mi divided by the volume of the mixture V:

P_{i} = m_{i} / V

The SI unit is kg/m^{3} (equal to g/L).

#### Molar Concentration

The molar concentration ci is defined as the amount of a constituent ni (in moles) divided by the volume of the mixture V:

C_{i} = n_{i} / V

The SI unit is mol/m^{3}. However, more commonly the unit mol/L (= mol/dm^{3}) is used.

#### Number Concentration

The number concentration ci is defined as the number of entities of a constituent Ni in a mixture divided by the volume of the mixture V:

C_{i} = N_{i} / V

The SI unit is 1/m^{3} .

#### Volume Concentration

The volume concentration sigmai is defined as the volume of a constituent Vi divided by the volume of the mixture V:

Sigma_{i} = V_{i} / V

Being dimensionless, it is expressed as a number, e.g., 0.18 or 18%; its unit is 1.

### 5. Speed

The speed (commonly referred to as v) of an object is the magnitude of the rate of change of its position with time or the magnitude of the change of its position per unit of time; it is thus a scalar quantity.

- The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as the duration of the time interval approaches zero.
- Speed has the dimensions of distance divided by time. The SI unit of speed is the meter per second (m/s), but the most common unit of speed in everyday usage is the kilometer per hour (km/h) or, in the US and the UK, miles per hour (mph). For air and marine travel, the knot is commonly used.

**Units of speed include :**

- metres per second (symbol ms
^{-1}or m/s), the SI derived unit. - kilometres per hour (symbol km/h);
- miles per hour (symbol mi/h or mph);
- . knots (nautical miles per hour, symbol kn or kt);
- feet per second (symbol fps or ft/s);
- Mach number (dimensionless), speed divided by the speed of sound;
- in natural units (dimensionless), speed divided by the speed of light in vacuum (symbol c = 299792458 m/s).

### 6. Acceleration

Acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction).

The orientation of an object’s acceleration is given by the orientation of the net force acting on that object.

The magnitude of an object’s acceleration, as described by Newton’s Second Law, is the combined effect of two causes:

- the net balance of all external forces acting onto that object — magnitude is directly proportional to this net resulting force;
- that object’s mass, depending on the materials out of which it is made — magnitude is inversely proportional to the object’s mass.
- The SI unit for acceleration is meter per second squared (ms
^{-2}, m/s^{2}). - Acceleration has the dimensions of velocity (L/T) divided by time, i.e., L T
^{-2}. The SI unit of acceleration is the meter per second squared (ms^{-2}); or “meter per second squared,” as the velocity in meters per second changes by the acceleration value, every second.

### 7. Force

Force equals mass multiplied by acceleration. This derived unit of force is called a newton and has the symbol N.

So, one newton is one kilogram-meter divided by seconds squared.

The newton (symbol: N) is the International System of Units (SI) derived unit of force. It is named after Isaac Newton in recognition of his work on classical mechanics, specifically Newton’s second law of motion.

One newton is the force needed to accelerate one kilogram of mass at the rate of one meter per second squared in the direction of the applied force.

The units “meter per second squared” can be understood as change in velocity per time, i.e. an increase of velocity by 1 meter per second every second.

### 8. Joule

The joule is a derived unit of energy in the International System of Units.

In 1843, James Prescott Joule independently discovered the mechanical equivalent in a series of experiments. The most famous used the “Joule apparatus”: a descending weight, attached to a string, caused rotation of a paddle immersed in water, practically insulated from heat transfer.

It showed that the gravitational potential energy lost by the weight in descending was equal to the internal energy gained by the water through friction with the paddle.

In the International System of Units (SI), the unit of energy is the Joule, named after Joule. It is a derived unit. It is equal to the energy expended (or work done) in applying a force of one newton through a distance of one meter.

However, energy is also expressed in many other units not part of the SI, such as ergs, calories, British Thermal Units, kilowatt-hours, and kilocalories, which require a conversion factor when expressed in SI units.

The SI unit of energy rate (energy per unit time) is the watt, a joule per second.

Thus, one joule is one watt-second, and 3600 joules equal one watt-hour. The CGS energy unit is the erg, and the imperial and US customary unit is the foot-pound.

Other energy units such as the electronvolt, food calorie, or thermodynamic kcal (based on the temperature change of water in a heating process), and BTU are used in specific areas of science and commerce.

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