## PHYSICAL QUANTITIES AND UNITS

**Physical quantity:**

All the measurable quantities are physical quantity. The measurement is associated with numerical value. The quantities can be measured directly or indirectly. E.g. measuring the length of road, weight of rice, etc are the direct measurement. Measuring the mass of earth, electron, etc are indirect measurement.

**Aphysical quantity:**

The quantities that cannot be measured are aphysical quantity. They do not have any scale to measure. They are also called the abstract quantity and are considered not to be present in this physical world.

**Unit:**

They are the standard reference used to express the numerical value (magnitude) of physical quantity.

- A physical quantity must have numerical value and unit i.e. Physical Quantity = Magnitude + Unit
- In lack of any of this, it can’t be called a physical quantity.

**SI Unit:**

Till 1960, there were many measurement system used all over the world. They were local and not standard or commonly practiced all over the world. It was also creating problem in global communication. So, in 1960, people of different part of the world agreed to start a common unit i.e. SI system of measurement.

SI is a unit that is recognized universally for universal measurement.

- Ø It is the system of units accepted worldwide to bring uniformity in measurement.
- Ø SI units are founded on the basis of seven base (fundamental) units. They are as follows:

S.No. |
Physical quantity | Symbol | Units | Symbol |

1. | Mass | m | Kilogram | Kg |

2. | Length | l | Metre | m |

3. | Time | t | Second | s |

4. | Temperature | T or Ө | Kelvin | k |

5. | Current | I | Ampere | A |

6. | Amount of substance | (N) | Mole | Mol |

7. | Luminous Intensity | (J) | Candela |
cd |

Here, (N) and (J) are general symbols.

**Derived units:**

All the remaining units apart from seven SI-base units are derived units.

Multiplier System in SI-unit:

Multiplier | Prefix | Symbol |

10^{3} |
Kilo | K |

10^{6} |
Mega | M |

10^{9} |
Giga | G |

10^{12} |
Tera | T |

10^{15} |
Peta | P |

10^{18} |
Exa | E |

Sub-Multiplier System:

Sub-multiplier | Prefix | Symbol |

10^{-3} |
Mili | m |

10^{-6} |
Micro | μ |

10^{-9} |
Nano | n |

10^{-12} |
Pico | p |

10^{-15} |
Femto/Fermi | f |

10^{-18} |
Atto | a |

**Derived units:**

The unit obtained by the process of division or multiplication of two or more than two physical quantity is known as derived unit.

Physical quantity | Formula | Derived unit |

Speed/velocity | Displacement
Time |
m/s = ms^{-1} |

Acceleration | Velocity
Time |
Ms^{-1}/s = ms^{-2} |

Force | Mass X acceleration | Kgms^{-2} |

Work done/Energy | Force X distance | Kgm^{2}s^{-2} |

Power | Work done
Time |
Kgm^{2}s^{-3} |

**Use of SI-Base Unit:**

We use SI base unit to

- Find out the unknown unit of given physical quantity.
- To check homogeneity of given equation.

**To find out unknown unit:**

Process:

- Remove all unit less constant.
- Write all the units of known physical quantity in terms of SI-base unit.
- Work out mathematical operation.

Example:

You are given a equation,

F = 6πηrv

F = resistive force

r = radius of ball

v = speed of ball

Find unit of η in terms of SI-base unit.

Solution:

F = 6πηrv

F = ηrv

Kgms^{-1} = ηm.ms^{-1}

η = kgms^{-1}

m^{2}s^{-1}

= kgm^{-1}s^{-1}

Therefore, units of η = kgm^{-1}s^{-1}

**Homogeneity:**

An equation is said to be homogeneous if each term associated with it has same unit.

Example: s = ut +½gt^{2}

Units of I^{st} term = m

Units of II^{nd} term = m/s X s = m

Units of III^{rd} term = m/s^{2} X s^{2} = m

Hence, the equation is homogeneous.

Whereas, s = ut + gt^{2}, this equation is homogeneous but incorrect.

*Conclusion:*

A homogeneous equation may not be correct equation is always homogeneous.

**Plotting the graph in A/AS Level Physics:**

General Rule:

- Use maximum area of the graph.
- Write the variable along with their unit to be plotted (separated by a ‘/’).
- Use more than 60% of the axis to plot the data.
- Draw line of best fit.
- Use ‘x’ or ‘’ to represent a point.

Calibration Curve:

It is standard line of best fit (curve or straight) obtained when two interrelated quantities are plotted.

**Significant Figures (digits):**

These are truly known digits about measurement. You can depend on these digits to make scientific comment. Digital informations obtained from direct measurement all are significant.

Rules to find out significant figures:

- All the non-zero digits are significant. E.g. 15612131 → 8 s.f.
- Starting zeros and ending zeros are not significant. E.g. 0.0135 → 3s.f. and 15000 → 2s.f.
- Zeros between two non-zero digits are significant. E.g. 2008 → 4s.f. and 0.030501 → 5s.f.
- Ending zeros after decimal are significant. E.g. 1500 → 2s.f., 0.1500 → 4s.f. and 15.00 → 4s.f.
- All the natural number must have infinite significant figure.

Addition and Subtraction of numbers:

E.g. 1. 12.501500 8s.f.

01.56 . 3s.f.

14.06 4s.f.

2. 1.23456700

1.30 .

2.53

- If an addition or subtraction of given two number, the answer must be written in such a way that the number of significant figures (s.f.) is equal to the least number of s.f. in either of the numbers.

Division and Multiplication:

E.g. 1. 2.510/2.1

= 1.195238

= 1.2 or 1.19

**Uncertainty:**

It is the range of measured value in which the true value lies. It is also termed as error in physics. Some time, it is known as absolute error as well.

**Digital Instruments:**

In digital instrument, manufacturing company itself states the error.

Example:

A digital ammeter records a current of 5.21 mA. Its uncertainty as specified by the company is (±2% ±3). Find the

a. absolute uncertainty in the reading

b. fractional uncertainty in the reading

c. % uncertainty in the reading

Solution:

a) Measured Value = 5.21mA

So, Absolute Uncertainty = 2% of 5.21 ± 0.03

= 2/100 X 5.21 + 0.03

= 10.42/100 + 0.03

= 0.104 + 0.03

= 0.134

= 0.13mA

b) fractional uncertainty = absolute uncertainty

measured value

= 0.13

5.21

= 0.0249

= 0.025

c) % uncertainty = f.u. X 100%

= 0.025 X 100

= 2.5%

Hence, measured value = (5.21 + 0.13)mA

**Combining uncertainty (Error):**

Rule 1

In case of addition or subtraction, the error (absolute uncertainty) is obtained by adding individual absolute uncertainty.

Rule 2

In case of division or multiplication, the fractional uncertainty in the answer is given by the individual fractional uncertainty.

**Precision:**

It concerns with the closeness of observed data. If the measured values are highly scattered, then the measurement cannot be precise.

A sensitive instrument increases the precision of the measurement and hence there is increase in significant figures/digits.

**Accuracy:**

It concerns how close the measured value with its true value is. An accurate measurement is possible even if it is imprecise. For this our measuring scale should be error free.

**Types of Error:**

There are two types of errors associated with our measurement:

- Random error
- Systematic error

*1. Random Error:*

The fluctuation of observed value on either side of the true value is called random error. The fluctuation of points on both sides of line of best fit represent random error in your experiment. Due to presence of large random error we cannot obtain precise measurement though it is accurate.

Source of Random Error:

- Parallax error (It is due to different angles of observation).
- Measurement without fixing reference point.
- Fluctuation in controlled system.

Ways to Remove Errors:

- Observe normally while measuring with analog meters. Place mirror behind the pointer and take reading in such a way that the image is completely blocked by the pointer itself.
- Fix standard reference point in measurement.
- Get average values and draw line of best fit.

*2. Systematic Error:*

It is the constant deviation of observed value from its true value in single direction only. Due to this error, our measurement can be precise but it cannot be accurate.

Causes of Systematic Error:

- Zero error in measurement instrument.
- Wrongly calibrated scale
- Use of less sensitive instrument
- Human reaction time

Ways to Remove the Error:

i) Correct the zero error of your measuring instrument i.e. positive, negative error.

ii) Re-calibrate by using standard scale.

iii) Use more sensitive instrument.

iv) Measure the time interval in such a way that the human reaction time itself become negligible.

**Notes:**

- When there is no random error (i.e. systematic error), the data is precise only.
- When there s no systematic error, the data is accurate only.
- When there is no error, data is precise and accurate.
- When there are both errors, data is neither precise nor accurate.

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