Chapter 1

1. UNIT, PHYSICAL QUANTITIES AND VECTOR PHYSICS MECHANICS DAU 10103

Learning Outcomes After completing this chapter, students should be able to: • Identify the units of measurement based on SI base units. • Adapt the measurement accurately with its units, scientific notation and uncertainty depending on the measurement tool used. • Draw the resultant vector and resolve into x and y components. • Solve any problem related to vector.

MEASUREMENT ◦ Definition: the estimation of the magnitude of some attributes of an object, such as length and weight, relative to a unit of measurement. ◦ Usually involves using a measuring instrument, which is calibrated to compare to the object to some standard, such as a meter or kilogram respectively. ◦ Physical quantity: a quantity that can be measured by a scientific instrument. ◦ Example of scientific instruments: please type in chat box

BASE & DERIVED Three most QUANTITIES used base quantities ◦ Base quantities: ◦ Quantities that cannot be derived. ◦ Also known as fundamental quantities Base quantity Units Time Seconds (s) Mass Kilogram (kg) Length Meter (m) Temperature Kelvin (K) Amount of substance Mole (Mol) Ampere (A) Electric current Candela (cd) Light intensity

◦ Derived Quantities: Basic derived ◦ A combination of two or more base quantities formulae ◦ Obtained from the derivation of the base quantity ������ Derived quantity Name Symbol ������ = ������ Area Square meter m2 Cubic meter m3 ������ Volume Meter per second ms-1 ������ = ������ Speed, velocity Meter per second square ms-2 Acceleration Kilogram per cubic meter ������ Mass density Ampere per square meter kgm-3 ������ = ������ Current density Ampere per meter Am-2 ������ = ������������ Magnetic field strength Am-1 ������ ������ = ������ = ������������ Amount of substance Mole per cubic meter Molm-3 concentration Candela per square meter cdm-2 Luminance

Example of special names and symbols: Derived SI Derived Unit Quantity Name Symbol Expression Expression Frequency (other SI (in SI Unit) Force Hertz Unit) Newton Pressure Pascal Hz - s-1 Energy, work, heat Joule N - kgms-1 Power Watt Electric charge Coulomb Pa Nm-2 kgm-1s-2 Potential difference Volt J Nm kgm2s-2 W Js-1 kgm2s-3 C- sA V WA-1 kgm2s-3A-1

UNITS AND STANDARDS ◦ SI Units ◦ Système International / International System of Units ◦ A set of convenient units that have been widely accepted in both everyday commerce and in science ◦ The modern, revised form of the metric system ◦ Was developed in Paris, 1960 at the 11th General Conference on Weights and Measures. ◦ Two base unit systems: ◦ Metric system ◦ Imperial system

UNITS AND STANDARDS ◦ Metric system comprises of two systems: ◦ Centimetre-gram-second (cgs) system ◦ Meter-kilogram-second (mks) system ◦ Imperial system: ◦ Also known as British Engineering System ◦ Used the units of feet-pound-second (British Units) ◦ Is no longer implemented in most measurement since the use of SI units.

DEFINITION OF BASE UNITS Base Quantity Units (Symbol) Definition Time Seconds (s) the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom Mass Kilogram (kg) 1kg equals to the mass of the international Length Meter (m) prototype of the kilogram Temperature Kelvin (K) the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second the fraction 1/273.16 of the thermodynamic temperature of the triple point of water

Base quantity Units (symbol) Definition Amount of Mole (Mol) substance Ampere (A) the amount of substance of a system which contains as many elementary entities as Electric current Candela (cd) there are atoms in 0.012 kilogram of carbon-12 Light intensity that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross- section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2x107 newton per meter of length the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian

SCALAR AND VECTOR ◦ Scalar Quantities: ◦ A quantity that has a magnitude only, no directional component ◦ Example: time, speed, temperature, volume ◦ Vector Quantities: ◦ A quantity that has both magnitude and direction ◦ Example: velocity, force, displacement

DIMENSIONAL ANALYSIS ◦ Dimensions of a physical quantity is associated with combinations of mass, length, time, electric charge and temperature which represented by symbols M, L, T, Q and θ respectively, each rose to rational powers. ◦ A useful way to determine an appropriate unit of a quantity. Quantity Unit Dimension Mass kg M Length m L Time s T C Q Electric charge K θ Temperature

Example 1.1: State the dimension and unit in SI for speed, v. Solution:

Example 1.2: State the dimension and unit in SI for force, F. Solution:

Example 1.3: Determine whether or not the following equation is dimensionally correct. a) v = at b) t = 2x a c) v3 = 2ax2 d ) F = mvx

Derived Quantity Dimension Volume Energy Power Acceleration Pressure Momentum

SCIENTIFIC NOTATION AND PREFIXES ◦ Scientific notation: ◦ A shorter method to express very large or very small numbers ◦ Based on power of the base number 10 ◦ Example: 123 000 000 000 = 1.23 x 1011 *The number 1.23 = coefficient 1 ≤ coefficient < 10 *The number 1011 = scientific notation raised to a power of 11.

◦ Prefix modifiers of the metric system that are multiples of 10. Prefix Symbol Factor Numer Factor Word Factor Power Tera T 1 000 000 000 000 Trillion 1012 Giga G 1 000 000 000 Billion 109 Mega M 1 000 000 Million 106 Kilo k 1 000 Thousand 103 Hecto h 100 Hundred 102 Deca da 10 101 Deci d 0.1 Ten 10-1 Centi c 0.01 Tenth 10-2 Milli M 0.001 Hundredth 10-3 Micro μ 0.000001 Thousandth 10-6 Nano n 0.000000001 Millionth 10-9 Pico p 0.000000000001 Billionth 10-12 Trillionth

Example 1.4: Write these values into the scientific notation: 7 040 000 000 000 000 000 000 000 kg 0.000 000 000 000 635 m 456 852 569 534 ms-2 Solution:

APPROXIMATION & COMPARISON VALUE ◦ Absolute error: The magnitude of the difference between the exact value and the approximation ◦ Relative error: ◦ The absolute error divided by the magnitude of the exact value (ratio) Example 1.5: Given; Exact value = 50 and approximation value = 49.9 Absolute error = 50 – 49.9 = 0.1 Relative error

UNIT CONVERSION ◦ Need to be done in some circumstances e.g. to perform an operation, or to compare the value of a quantity. ◦ The original measurement must be multiplied with a conversion factor. ◦ Conversion factor: a ratio of units that is equal to unity (1). ◦ Example: 1000g = 1 1kg 1kg = 1 1000g ◦ These ratio can be used as conversion factor.

Example 1.6: Write down these quantities in SI units: 40 mg 35 cm2 56 mm3 Solution:

Example 1.7: If a car is travelling at a speed of 30.0 m/s, is it exceeding the speed limit of 58 mi/h? Given 1 mile = 1609m. Solution:

Example 1.8: Someone is 2.00 yard tall. Using the fact that 1 inch is exactly 2.54 cm, how tall is the person in centimeters? Given 1 yard = 3 feet and 1 feet = 12 inch. Solution:

SIGNIFICANT FIGURES ◦ Significant figures: digits that are meaningful to the accuracy of a number. ◦ Example: ◦ 4.68 cm → 3 s.f. ◦ 0.002cm → 1 s.f. ◦ 0.000122300 → 6 s.f. ◦ For addition and subtraction operation, the number of s.f. is based on the less precise quantity.

Significant Figures Rules: i. All nonzero digits are significant (e.g. 177 is 3 s.f) ii. Zeros between significant digits are significant (e.g. 4003 is 4 s.f) iii.Zeros to the left of nonzero digits are not significant (e.g. 120 is 2 s.f) iv.Zeroes at the end of a number are significant only if they are to the right of the decimal point (e.g. 2.030 is 4 s.f)

Number s.f Number s.f 70.2 3 100 0.045 2 706 70.0 3 400.0 4.7 2 2 0.002 0.0020 0.002047 2 1.0 3 104,020 1.20 x 103 2.00 x 10-3 (illustrating 1200 with 3 s.f)

Example 1.9: There are three measurements with different precision instrument and the results obtained are 3.76 cm, 46.855 cm and 0.2 cm. What are the total measurement? Solution:

Example 1.10: The measurement of the length of two different sticks are recorded as 2.32 cm and 4.562 cm. How much is the difference between both sticks? Solution:

Rounding Rules: i. If the number ends in something greater than 5, then you round up. ii. If the number ends in something less than 5, then you round down. iii.The Arcane Rounding Rule: If the number ends in a perfect 5 (in other words all or no zeros after the five), you round to the even number.

Original Number of Rounded Original Number of Rounded Number s.f wanted Number Number s.f wanted Number 36.0501 3 36.1 26.5 2 0.01249 2 0.012 275 2 26.50000000 2 26 275.0000000 2 0000 0000 2.0105 4 4.00130 3 0.0032 1 7.35 2

Introduction To Vector ◦ Vector is known as the shortest distance between two points that consists of both magnitude and directions. ◦ The direction of vector is reversible by contrary in positive and negative sign. ◦ A notation of vector A; ���Ԧ���, ������ or A.

◦ Length of an arrow = magnitude of a vector ◦ Direction of an arrow = direction of a vector

Physical Quantities That Applied Vector ◦Force ���റ��� ◦Velocity ���റ��� ◦Acceleration ◦Weight ������ ◦Displacement ◦Momentum ���റ���

Addition Vector Geometrically ���റ��� ���റ��� ���റ��� −������ ������ ���റ��� ���റ��� + ������ = ���റ��� ���റ��� − ������ =? NOTE: Vector must be added from head to tail ���റ��� ≠ ���റ��� different direction

◦Adding vector in different way ���റ��� ������ ������������������������������ ���റ��� ������������������������������ ���റ��� + ������ = ������ + ���റ��� ������

◦ Note: ◦ Negative sign of vector gives the same magnitude but in reverse direction. ◦ Adding the same vector that oppose each other and same direction will result in zero. ������ + (−������) = 0 ������ −������

Scalar Multiplication With Vector ◦The multiplication of a vector A by a scalar  ◦ Will result in vector B ������ = ������������ ◦ Whereby magnitude is changed but not direction If  is positive value, the vector A is in the same direction of vector A, and vice versa ◦If  = 0, B= ������������=0 result in zero vector ������ ������������ = ������������������ = ������ ������������ ������ + ������ ������ = ������������ + ������������

The direction of a vector can be stated is terms of Angle, Bearing, and Poles (1) Angle (2) Bearing (3) Poles

Components of Vector • Component of vector is the projection of vector on axis (x-y) y ���റ��� x ���റ���������  ���റ���������

◦ The process of finding the components of a vector is called resolving vector. ���റ��� ���റ��������� = ������ sin ������  (vertical component) ���റ��������� = a cos  (horizontal component)

◦ The MAGNITUDE OF VECTOR also found by solving Pythagorean theory ������ = ������������2 + ������������2 ◦ The DIRECTION of vector can be calculated using ������ = tan−1 ������������ ������������

Example 1.11: A small airplane leaves an airport on an overcast day and is later sighted 215 km away, in a direction making an angle of 22º east due north. How far east and north the airplane from the airport when sighted? Solution:

Resolving Vector IMPORTANT! Please consider ◦ If there is more than one vector ±x and ±y axes ◦ The sum of vector is by connecting all vector from head to tail (geometrically). ◦ To calculate the resultant vector, ���റ��� each component of resultant vector ���റ��� must be sum of all component of corresponding vector. ෍ ���റ��������������������� = ���റ���1 + ���റ���2 + ���റ���3 + ���റ���4

Resolving Vector ◦ Resolve the x and y component separately ෍ ���റ���������������������, ������ = ���റ���1, ������ + ���റ���2, ������ + ���റ���3, ������ + ���റ���4, ������ ෍ ���റ���������������������, ������ = ���റ���1, ������ + ���റ���2, ������ + ���റ���3, ������ + ���റ���4, ������ ◦ To find the magnitude of ���റ���������������������, from its component, use Pythagoras theory ������������������������ = ������������������������, 2 + ������������������������, 2 ������ ������ ◦ And find angle by using trigonometry function ������ = tan−1 ������������������������, ������ ������������������������, ������

Example 1.12: Two vectors ������ റ and ������ റ in figure above have equal magnitude of 10.0 m and the angles are θ1= 30º and θ2=105º. Find (a) the x and y component of their vector sum ������ റ (b) the magnitude of ������ റ and y (c) the angle ������ റ makes with the positive direction of the x axis ������ 2 ���റ��� x 1 0

Example 1.13: A car moves at a velocity of 50 m/s in a direction north 30° east. Calculate the component of the velocity, due north and due east respectively.

Example 1.14: The figure shows three forces F1, F2 and F3 acted on a particle O. Calculate the magnitude and direction of the resultant force on particle O.