General Physics I (Mechanics and Properties of Matter) Lecture 1 PDF

COURSE TITLE: GENERAL PHYSICS I (MECHANICS AND PROPERTIES OF MATTER) COURSE CODE: PHY111 (2 UNITS) Introduction Physics is the science that deals mainly with the description of matter, energy, space and times as they affect the universe. Physics attempt to find the most basic laws that govern t...

COURSE TITLE: GENERAL PHYSICS I (MECHANICS AND PROPERTIES OF MATTER) COURSE CODE: PHY111 (2 UNITS) Introduction Physics is the science that deals mainly with the description of matter, energy, space and times as they affect the universe. Physics attempt to find the most basic laws that govern these occurrences and express these laws in the most precise and simplified way possible. Objectives: To help the students understand the concept of (a) space and time; (b) fundamental/ basic units, and derived units and (c) dimension as algebraic quantities.  SPACE AND TIME Space and time are fundamental aspects of our universe. They form the stage on which all physical events take place, and understanding their nature is crucial for studying physics and other natural sciences. In this lecture, we will explore the concepts of space and time, their measurement, and how they relate to each other. Space is three-dimensional, which means it has three independent directions: length, width, and height (or depth). These three dimensions allow us to specify the position of any object using a coordinate system. 1D space: Only one coordinate is needed to specify the position of a point (e.g., a straight line). 2D space: Two coordinates are needed (e.g., x and y on a flat surface or plane).  3D space: Three coordinates are needed (e.g., x, y, and z in our everyday world). In physics, we use a Cartesian coordinate system to describe space, where positions are given as (x, y, z) in three dimensions. The Nature of Space Absolute Space: According to Isaac Newton, space is an unchanging, rigid structure that provides the backdrop for physical processes. Relative Space: As per philosopher Gottfried Leibniz, space is relative, meaning it only exists as the distance between objects, and there is no "space" without matter. Non-Euclidean Geometry: In general relativity, space is not perfectly flat (Euclidean) but can be curved by the presence of mass and energy. Time refers to the ongoing sequence of events from the past, through the present, and into the future. It is used to quantify the duration of events and the intervals between them. Time is often considered as the fourth dimension that, along with the three spatial dimensions, forms a four-dimensional continuum known as spacetime. Time allows us to describe when events occur and provides a framework for understanding motion and change. The Nature of Time Absolute Time: According to Newton, time flows at a constant rate and is independent of the observer. Relative Time: In contrast, Einstein’s theory of relativity suggests that time is relative and can vary depending on the observer’s velocity and gravitational field. Time can dilate, meaning that for a fast-moving object, time passes slower compared to a stationary observer (time dilation). Space and Time in Classical Physics In classical mechanics, space and time are treated as independent and absolute. Galilean Transformation: In classical mechanics, we assume that time flows at the same rate for all observers, and space is the same regardless of the motion of objects. The transformation of coordinates between different observers moving at constant velocity is straightforward. However, classical physics only works well for everyday, low-speed phenomena. At very high speeds (near the speed of light) or in strong gravitational fields, Newtonian ideas about space and time break down. Special Relativity and Spacetime Albert Einstein revolutionized our understanding of space and time with his theory of special relativity in 1905. The Two Postulates of Special Relativity 1. The laws of physics are the same in all inertial frames: An inertial frame is a non-accelerating frame of reference. 2. The speed of light in a vacuum is constant: It is the same for all observers, regardless of their relative motion. In special relativity, space and time are unified into a single four- dimensional construct called spacetime. Event: A point in spacetime, specified by four coordinates (x, y, z, t), where t represents time. Spacetime Interval: The distance between two events in spacetime is called a spacetime interval, and it combines both spatial distance and time differences. Time Dilation and Length Contraction Time Dilation: Time moves slower for objects moving at high speeds relative to a stationary observer. This effect has been confirmed by experiments involving fast- moving particles and atomic clocks. Example: A clock on a fast-moving spaceship will tick more slowly than a clock on Earth. Length Contraction: Objects moving at high speeds appear shorter in the direction of motion relative to a stationary observer. Example: A fast-moving train would appear shorter to an outside observer compared to its length measured at rest. Examples 1. Consider two points in three-dimensional space: Find the distance between these two points. Solution: The distance d between two points in 3D space is given by the formula: 2. An astronaut is traveling in a spaceship at a speed of v = 0.8c (where c is the speed of light) relative to Earth. The astronaut measures the time interval for a certain event to be 2 hours in her frame of reference (proper time). How much time will pass on Earth for the same event? Solution: The time dilation formula is given by: Δto is the proper time (time interval in the astronaut's frame), Δt is the time interval in the Earth's frame, v is the velocity of the moving observer, c is the speed of light. 3. A spaceship is traveling past Earth at a speed of v = 0.9c. The spaceship measures its own length to be 100 meters. How long does the spaceship appear to an observer on Earth? Solution: The length contraction formula is given by: L0​ is the proper length (length in the spaceship's frame), L is the contracted length (length in Earth's frame), v is the velocity of the moving object, UNITS Unit indicates the standard used in the measurement of a parameter. The Metric system is the most acceptable international form of unit. It is based on powers of ten and referred to as the SI units (Systeme International d’Unites in French). We have two types of unit; fundamental basic units and the derived units. Fundamental Units are those units taken as independent, that do not depend on one another. The SI fundamental units used in mechanics are meter (m) for length, kilogram (kg) for mass and second (s) for time. For electrical current the base unit is ampere (A), for Temperature Kelvin (K), for amounts of matters (mol), for luminous intensity candela (cd). Derived units are formulated from combinations of the basic units. Example: Unit of velocity = ms-1 Unit of volume = (Unit of length) 3 = m3 Rules on Applications of SI Units 1. When spelling out a unit in full small letters are used. The symbols are also small letters except when a unit is named after a scientist. 2. The letter “s” is not attached to a symbol when the numerical value of a measurement is plural; but it can be attached if the unit is spelt out fully. Classwork 1. Derive the units for the following (a) force (b) work (c) power  The equation for the velocity V, in a gas states that V= γkbT m v is velocity, ɣ is a constant, T is temperature in Kelvin (K), m is mass. What is the unit of the Boltzmann constant kb.? Solution γ k bT v = m Square both sides and make kb subject of formula γ k bT v2 = m 2 mv kb = γ T = kg(ms-1)2K-1 [m:kg, v2:(ms-1)2, T-1:K-1] = kgm2s-2K-1 SI Prefixes Prefix(abbre Multiplying Example viation) Factor tera (T) 1012 Freq. of mid. infra red waves 10THz (tera Hertz) giga (G) 109 Memory of computer 37.2GB (giga bytes) mega (M) 106 Capacity of a diskette 1.44MB (mega byte) Kilo (k) 103 Power consumed by an electric iron (kilo watt) deci (d) 10-1 Length of a ruler 3dm (decimeter) centi (c) 10-2 Length of a ruler 30cm (centimeter) milli (m) 10-3 Period of a sound wav 1ms (milli second) micro (μ) 10-6 Leakage current in diode 3.5 μm (micro amp) nano (n) 10-9 Wavelength of UV-B radiation 220nm (nanometer) pico (p) 10-12 Capacitance of a capacitor 4.7pf (picofarad) femto (f) 10-15 Energy of an X-ray photon 1fJ (femto joule) Classwork 1. A chain of micro-organisms each of 12 μm in length formed on a river, if the length of the chain is 0.60 km, what is the maximum number of organisms that can be found on the chain. 2. A task was accomplished within 3.2 x10-6 year. Express the time taken in minutes and seconds. Solution 0.60 10 3 60 −6 = 10 = 5 10 7 7 12 10 12 3.2 x10-6year = 0.0000032 x 365days x 24hours x 60min = 1.68 min = 0.0000032 x 365days x 24hours x 60mins x 60mins = 100.9 s DIMENSIONAL ANALYSIS Dimensional analysis is a powerful tool in physics used to check equations and derive relationships between physical quantities. It involves expressing physical quantities in terms of their fundamental dimensions (such as mass [M], length [L], time [T], etc.) Example: Dimension for area is L2, volume is L3, density is ML-3, and speed is LT-1 Classwork 1. Derive the dimensions for the following (a) acceleration (b) force (c) work (d) power. 2. Determine the values of the indices x, y and z in the relation T = kax ρy γz , where T is the period of vibration of a string, a is the radius, ρ is the density and γ is the surface tension which is force per unit length. work Power = time Work = Force x distance = mass x acceleration x distance length Work = Force x distance = mass x time 2 x length Power = mass x (length)2. (time)-3 = M(L2)T-3 Solution to 2: T is the period has dimension of time = T k is a constant and has no dimension a is the radius has dimension of length = L dimension of density, ρ = M.L-3 γ the surface tension has dimension of force per unit length = MLT-2L-1 = M.T-2 Equate the indices of M, L, and T on both sides, both M and L must cancel out on the right side leaving only T. T = kax ρy γz Equating the dimensions, we have T = (L)x.(ML-3)y.(MT-2)z = Lx. My. L-3y. Mz. T-2z Equating the indices of M , L, and T on both sides 0=y+z... (1) 0 = x – 3y... (2) 1 = -2z... (3) From equation (3), z= -1/2 Plug z = -1/2 into equation (1), y = 1/2 Plug y = 1/2 into equation (2), x = 3/2 Thus, aρ3 T = k a3/2 ρ1/2 γ-1/2 = k(a3 ρ γ-1)1/2 = k γ 3. The centripetal force (F) acting on a particle (moving uniformly in a circle) depends on the mass (m) of the particle, its velocity (v) and radius (r) of the circle. Derive dimensionally formula for force (F). Solution: Given, F ∝ ma.vb.rc ∴ F = kma.vb.rc (where k is constant) Putting dimensions of each quantity in the equation, [M1L1T-2] = [M1L0T0]a. [M0L1T-1]b. [M0L1T0]c = [MaLb+c+T-b] ⇒ a =1, b +c = 1, -b = -2 ⇒ a= 1, b = 2, c = -1 ∴ F = km1.v2.r-1 = kmv2/r ASSIGNMENT 1 1. Compute the dimensional formula of electrical resistance (R) 2. Hooke’s law states that the force, F, in a spring extended by a length x is given by F = −kx. Calculate the dimension of the spring constant k. 3. The Force between two wires 1, 2, length 1 metres separated a distance d metres and carrying currents I1, I2 amperes is given by kI 1 I 2 F= d Find (i) the units of the constant k (ii) dimension of k 4. Using unit and dimension, determine x, y, z in the expression v =kηxry(P/L)z where v is the volume of liquid passing per second, η (in Pa. s) is the viscosity of liquid, r is the radius of the pipe, P is pressure difference and L is the length of the pipe. CLASSWORK The formula for the gravitational force between two masses is given by Newton’s Law of Gravitation with all parameters having their usual meaning. Find the dimensions of the gravitational constant G.