# Measurement and Units

**Measurement and Units**

This chapter delves into the fundamental concepts of measurement and units in physics, equipping you with the skills to quantify the physical world accurately. We’ll explore the International System of Units (SI), conversion techniques, and the significance of significant figures in scientific notation.

**1.Understanding SI Units and Prefixes**

The International System of Units (SI) is the globally recognized system for measurement. It establishes a set of **fundamental base units** that serve as the building blocks for all other measurements. These base units are:

**Meter (m):**Length**Kilogram (kg):**Mass**Second (s):**Time**Ampere (A):**Electric current**Kelvin (K):**Temperature**Mole (mol):**Amount of substance**Candela (cd):**Luminous intensity

From these base units, we derive numerous **derived units** to quantify various physical quantities. For example, the derived unit for speed (meter per second) is written as m/s.

SI also employs a system of **prefixes** to denote multiples of ten. These prefixes range from **yotta (Y)** (10^24) to **yocto (y)** (10^-24), allowing for the expression of very large or very small quantities concisely.

Here’s a table summarizing the commonly used prefixes:

Prefix | Symbol | Multiplication Factor |

Yotta | Y | 10^24 |

Zetta | Z | 10^21 |

Exa | E | 10^18 |

Peta | P | 10^15 |

Tera | T | 10^12 |

Giga | G | 10^9 |

Mega | M | 10^6 |

Kilo | k | 10^3 |

Hecto |
h |
10^2 |

Deca |
da |
10^1 |

deci (d) |
d |
10^-1 |

centi (c) |
c |
10^-2 |

milli (m) |
m |
10^-3 |

micro (µ) |
µ |
10^-6 |

nano (n) |
n |
10^-9 |

pico (p) | p | 10^-12 |

femto (f) | f | 10^-15 |

atto (a) | a | 10^-18 |

yocto (y) | y | 10^-24 |

**Example:** The distance between the Earth and the Sun is approximately 149,600,000,000 meters. Using SI prefixes, we can express this as 149.6 Gm (gigameters).

**2. Converting Between Units**

Physics often involves measurements expressed in different units. The ability to convert between units is crucial. Here’s a two-step approach:

**Identify the conversion factor:**This is a ratio that equates two equivalent measurements. For instance, 1 inch = 2.54 centimeters. Here, the conversion factor is either 1 inch/2.54 cm or 2.54 cm/1 inch.**Set up a dimensional analysis:**Write the quantity you want to convert in terms of its original unit. Then, multiply by a series of conversion factors, strategically chosen to cancel out unwanted units and arrive at the desired unit.

**Example:** Convert 15 miles per hour (mph) to meters per second (m/s).

- Conversion factors: 1 mile = 1.609 kilometers (km) and 1 km = 1000 meters (m) ; 1 hour = 3600 seconds (s)
- Dimensional analysis:

15 mph * (1 km / 0.621 mi) * (1000 m / 1 km) * (1 h / 3600 s) = 6.71 m/s (approx.)

**Tips for Unit Conversion:**

- Pay close attention to the units in the conversion factors.
- Ensure the units cancel out appropriately to reach the desired unit.
- Round the final answer to an appropriate number of significant figures (covered in the next section).

**3. Significant Figures and Scientific Notation**

Scientific measurements are rarely exact.There’s always a degree of uncertainty associated with the measurement process. Significant figures (sig figs) convey this level of certainty.

**Significant figures:**These are all the digits in a measured value that are known with certainty, plus one estimated digit. The estimated digit reflects the limitation of the measuring instrument.

**Counting Significant Figures:**

- All non-zero digits are significant.
- Zeros between significant digits are significant. (e.g., 2.005 has 4 sig figs)
- Leading zeros (zeros to the left of the first non-zero digit) are not significant unless the number is written in scientific notation (explained later). (e.g., 0.00025 has 2 sig figs)
- Trailing zeros (zeros to the right of a decimal point and next to a significant digit) are significant. (e.g., 12.0 has 3 sig figs)

**Examples:**

- 36.52 (4 sig figs)
- 1.002 (4 sig figs) (leading zero becomes significant)
- 0.00508 (3 sig figs) (trailing zeros are significant)
- 200 (1 sig fig) (ambiguous; assume leading zero is not significant)

**Rules for Combining Significant Figures:**

- In multiplication and division, the final answer has the same number of significant figures as the measurement with the fewest sig figs. (e.g., 2.54 cm * 3.1 m = 7.8 (2 sig figs))
- In addition and subtraction, the answer has the same number of decimal places as the measurement with the least decimal places. (e.g., 12.5 + 3.002 = 15.502 (round to 4 decimal places))

**Scientific Notation:**

Scientific notation is a compact way to express very large or very small numbers. It represents a number in the form:

a x 10^b

where:

- a (coefficient) is a number between 1 and 10 (inclusive).
- b (exponent) is an integer that indicates the number of places the decimal is shifted to the right (positive exponent for large numbers) or left (negative exponent for small numbers).

**Examples:**

- 2.54 x 10^2 meters (same as 254 meters)
- 8.03 x 10^-4 grams (same as 0.000803 grams)

**Significant Figures in Scientific Notation:**

In scientific notation, only the coefficient (a) carries the significant figures. The exponent does not affect the number of sig figs.

**Example:**

- 3.1415 x 10^3 (5 sig figs; all digits in 3.1415 are significant)

**Using Significant Figures and Scientific Notation:**

- Rounding calculations: When performing calculations with sig figs, round the final answer to the same number of sig figs as the measurement with the least number of sig figs.
- Reporting measurements: Scientific measurements are reported with the appropriate number of significant figures to reflect the certainty of the measurement.

By understanding SI units, prefixes, unit conversion, significant figures, and scientific notation, you’ll be well-equipped to quantify the physical world accurately and effectively in your HESI A2 Physics endeavors.