General Physics I (Mechanics, Thermal Physics, and Waves) PDF

PHY 101: GENERAL PHYSICS I (Mechanics, Thermal Physics and Waves) By Dr. Eli DANLADI NCE (KSCOE), B.Tech Hons. (FUTMINNA), M.Sc., Ph.D (NDA), FHNR, MNIP, MNIM, MTRCN, MMSN, MNSPS, MSESN, MRAESON SPACE: is the property of the universe that enables physical pheno...

PHY 101: GENERAL PHYSICS I (Mechanics, Thermal Physics and Waves) By Dr. Eli DANLADI NCE (KSCOE), B.Tech Hons. (FUTMINNA), M.Sc., Ph.D (NDA), FHNR, MNIP, MNIM, MTRCN, MMSN, MNSPS, MSESN, MRAESON SPACE: is the property of the universe that enables physical phenomena to be extended into three mutually perpendicular directions for complete description of a physical phenomenon in space, time coordinate is required such that the event location and time occurrence can be specified as (x, y, z, t). A point on a line can be described with one coordinate. A point in a plane is located with two coordinates. Three coordinate are required to locate a point in space. A coordinate system is used to specify location in space and consists of; 1. A fixed reference point says O and is called the origin. 2. A set of specified axes or directions. 3. Instruction that tells us how to label a point in space relative to the origin and the axes. Coordinate systems are commonly Cartesian (rectangular) coordinate system (x, y, z) and polar coordinate system (r, ) as shown in the figures below. Cartesian (rectangular) coordinate Polar coordinate Using standard trigonometry, we can find conversions from Cartesian to polar coordinates; And, from polar to Cartesian coordinates; These relations are true if only is defined as above. Example: The Cartesian coordinate of a point are given by (x, y) = (-5,-7). Find the polar coordinate of this point. You will later learn in relativity (relativity simply means the physics theory which describes particles moving at very high speed) that space–time continuum simply is the geometry which includes the three dimensions of space x, y, z and a fourth dimension of time, t which according Newtonian physics, space and time and considered as separate entities whether or not events are simultaneous is a matter that is regarded as obvious to any competent observer. However, according to Einstein’s concept of the physical universe, space and time are entwined so that two observers in relative motion could disagree with the simultaneity of distant events. FRAME OF REFERENCE Physical observations are usually made with respect to a given reference point which we can refer to as the frame of reference and may be defined as a rigid frame work relative to which positions and movement could be measured e.g. latitude and longitude define positions on the Earth’s surface, the earth being used as a frame of reference. UNIT: The unit of measurement of a physical quantity is the primary identification key of the quantity; it is used to distinguish between two physical quantities having the same numerical values. For instance, 40 kg and 40 km, from their units of measurements, we can say, though they are of the same magnitude but the former is the measurement of the mass of some quantity while the latter is the distance or displacement of some physical quantity. Generally, without the unit of measurement the measurement of a physical quantity makes no sense. There are different systems of measurements; this includes cgs and the mks. In the cgs system of measurement, centimeter is the unit of measurement for Length, grams is the unit of measurement for mass and seconds is the unit of measurement for time. However, the universally agreed system of unit, also called the International System Of Units is the mks system, in which meter is the unit of measurement for Length, Kilograms is the unit of measurement for mass and seconds is the unit of measurement for time. The mks system of units is also called the. (systeme International) unit Fundamental quantities The three basic fundamental quantities are the Mass, Length and Time, however, there are other fundamental quantities, such as Temperature, Luminous intensity, Amount of Substance, etc. Below is a table showing the fundamental quantities, their unit of measurements and their symbols. Basic Quantities Basic Units S. I unit Length Metre M Mass Kilogram Kg Time Second S Electric Current Ampere A Temperature Kelvin K Amount of Substance Mole Mol Luminous Intensity Candela Cd Derived quantities on the other hand are obtained from the combination of two or more of the basic quantities. For instance, velocity is obtained by the combination of length (distance) and time. Derived quantities Formula S. I unit Area Length x breath m2 Volume Length x breath x width m3 Velocity Displacement/time m/s Acceleration Velocity/time m/s2 Pressure Force/Area N/m2 Density Mass/volume Kg/m3 Force Mass x acceleration Kgm/s2 Work Force x distance Nm Multiples and Submultiple Units When a measurement is too large, then it will be convenient to express this measurement in other units aside the S.I unit, these are called multiples while for very small measurement (Less than one), we use submultiples. Below are tables for multiples, submultiples and their respective values.  Multiples Submultiple Factor Prefix Symbol Factor Prefix symbol 1024 yotta Y 10-1 deci d 1021 zetta Z 10-2 centi c 1018 exa E 10-3 milli m 1015 peta P 10-6 micro  1012 tera T 10-9 nano n 109 giga G 10-12 pico p 106 mega M 10-15 femto f 103 kilo K 10-18 atto a 102 hecto H 10-21 zepto z 101 deka Da 10-24 yocto y DIMENSION: This is the analysis of how other physical quantities relates to the fundamental quantities in terms of the symbols T for time, L for length and M for mass. To obtain the dimension of a physical quantity, express the quantity in terms of the fundamental quantities, then replace each fundamental quantity by its dimension. Quantity Dimension Length L Mass M Time T Electric current A Temperature  Amount of substance N Usually square brackets denote the dimension or units of a physical quantity. Example 1: 1. [Area]= [length x breadth] =LxL = L2 Unit for area = m2 2. [velocity] = [displacement/time] =L/T =LT-1 Unit for velocity =ms-1  3. [Acceleration]= [change in velocity/time] =LT-1/T =LT-2 Unit for acceleration = ms-2 4. [Force]= [mass x Acceleration] =MLT-2 Unit for acceleration = ms-2 Assignment 1 Derive the dimension, hence the S.I units of the following quantities. a. Density b. Young’s modulus c. Universal gravitational constant. d. Electric Charge e. Frequency f. Strain Uses of units and dimensions 1. To check the homogeneity of physical equations 2. To derive the unit of a physical quantity 3. To derive the exact form of a physical equation Example 2: The position of a particle along the x-axis depends on the time according to the equation x = at2 – bt3, where x is in meters and t in seconds. What dimensions and units must a and b have? Example 3: For what value of x, y and z is the expression MxLyTz (i) velocity (ii) acceleration (iii) force (iv) pressure (v) power Example 4: From experiments, it was observed that the period ( ) of oscillation of a simple pendulum depends on the mass ( ) of the bob, the length ( ) of the string and acceleration due to gravity. Use the method of dimension analysis to obtain the correct relation. T  km x l y g z , where k is a constant. Example 5: The viscous drag force F between two layers of liquid with surface area of contact A dv dv in a region of velocity gradient is given by F  A dx dx Where  is the coefficient of viscousity of the liquid. What is the dimension of  ? Hence, write the unit for  in terms of the base units in S.I Example 6: The speed of sound v in a medium depends on its wavelength  , the young modulus E, and the density  of the medium. Use the method of base units to derive a fomula for the speed of sound v, in a medium. [unit for young modulus E: kgm-1s-2]. Assignment 2 1. Using dimension Analysis, verify whether or not the following physical quantities are correct, Where is distance, is seconds, is acceleration, is work, is mass and is momentum. 2. The viscous force ( ) of a ball of radius ( ) falling through a liquid of viscosity at the rate are related by the relation F  k c v b r a , where is a dimensionless constant, obtain the physical relation between these quantities. 3. The resonance frequency of a closed air column is known to depend on the pressure , the density of air and the length of air column , how are these quantities related? let f  kp x  y l z. 4. Poiseuille assumed that the rate of flow of a liquid through a horizontal tube under streamline flow depends on: (a) a, radius of the tube (b)  , viscousity of the liquid and p (c) , the pressure gradient along the tube l Where p  pressure difference across the length of the tube l  length of the tube Using poiseuille’s assumption, derive an expression for the rate of flow of a liquid through a horizontal tube in terms of a, l , p and . Limitation of dimensional analysis i. The value of dimensionless constants cannot be determined. ii. It cannot be applied to an equation involving exponential and trigonometric functions. iii. It cannot be applied to an equation involving more than three physical quantities. iv. A dimensionally correct equation may not always be the correct relation. v. It does not tell us whether a given quantity is a scalar or vector. Introduction to Vectors Scalar Quantities These are physical quantities having only magnitude. Examples of scalar quantities are mass, temperature, energy, area, distance, speed etc. Scalar quantities can be summed as normal to obtain a single value, for instance a distance of 49 km can be added to another of 69 km to obtain 118 km (49 km + 69 km = 118 km). Vector Quantities These are quantities having magnitude, direction and orientation in space. They are expressed geometrically by directed line segments, where the length of the line is proportional to the magnitude of the vector. Examples of vector quantities are velocity, displacement, acceleration, momentum, force etc. Representation of vectors We usually represent a vector quantity such as displacement by a single letter, such as A , in boldface italic type with an arrow above them. We do this to remind you that vector quantities have different properties from scalar quantities; the arrow is a reminder that vectors have direction. In handwriting, vector symbols are usually underlined or written with an arrow above them. When you write a symbol for a vector, always write it with an arrow on top. We always draw a vector as a line with an arrowhead at its tip. The length of the line shows the vector’s magnitude, and the direction of the line shows the vector’s direction. If two vectors have the same direction, they are parallel. If they have the same magnitude and the same direction, they are equal, no matter where they are located in space. Displacements A and A’ are equal because they have the same length and direction. These two displacements are equal, even though they start at different points. Two vector quantities are equal only when they have the same magnitude and the same direction. Displacement B has the same magnitude as A but opposite direction; B is the negative of A.  The vector B above, however, is not equal to A because its direction is opposite to that of A. We define the negative of a vector as a vector having the same magnitude as the original vector but the opposite direction. The negative of vector quantity A is denoted as - A and we use a boldface minus sign to emphasise the vector nature of the quantities. We usually represent the magnitude of a vector quantity (in the case of a displacement vector, its length) by the same letter used for the vector, but in light italic type with no arrow on top, rather than boldface italic with an arrow (which is reserved for vectors). An alternative notation is the vector symbol with vertical bars on both sides: ADDITION OF VECTORS Suppose a particle undergoes a displacement A followed by a second displacement B (Fig. a). The final result is the same as if the particle had started at the same initial point and undergone a single displacement C as shown. We call displacement C the vector sum, or resultant, of displacement and B. we express this relationship symbolically as If we make the displacements A and B in reverse order, with B first and A second, the result is the same (Fig. b). Thus This shows that the order of terms in a vector sum doesn’t matter. In other words, vector addition obeys the commutative law. Figure c shows another way to represent the vector sum: If vectors A and B are both drawn with their tails at the same point, vector C is the diagonal of a parallelogram constructed with A and B as two adjacent sides. (a) We can add two vectors by placing them head to tail. (b) Adding them in reverse order gives the same result. (c) We can also add them by constructing a parallelogram. Fig a, b, c: Addition of vectors There are three methods of adding vectors: i. Arithmetic method ii. Geometric method iii. Analytic method. Arithmetic method: If two or more vectors act on an object in the same or opposite direction, then the resultant is obtained by adding the vector arithmetically. For example; consider a force P = 10N in the positive x-direction and another force Q = 30N acting in the same direction. The resultant of P and Q is R = P + Q = 40N (positive x-direction). If the forces P and Q acts in opposite direction, say; P = 10N in the positive x-direction and Q = 30N in the negative x-direction, then, the resultant R = Q – P = 20N (negative x-direction). GEOMETRICAL METHOD: this is done by scale drawing. The technique for this method is that the tail of one of the vectors in the system is placed on the head of the other vector. This technique utilizes the equality of vectors. The geometric method is done in three forms: i. Triangle method ii. Parallelogram method iii. Polygon method Parallelogram method: this method involves only two vectors. ‘If two vectors are represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is the diagonal drawn from a common point. Ex; if the angle between the vectors, 40N and 59N is 600, obtain the resultant vector. Polygon method: this method of finding the resultant R of several vectors (A, B, and C) consists in beginning at any convenient point and drawing (to scale and in the proper directions) each vector arrow in turn. They may be taken in order of succession: A + B + C = C + A + B = R. the tail end of each arrow is positioned at the tip end of the preceding one, as shown below: The Figure below shows three vectors A, B, and C We first add A and B to give a vector sum D; we then add vectors C and D by the same process to obtain the vector sum R. Alternatively, we can first add B and C and to obtain vector E, and then add A and E to obtain R. ANALYTIC METHOD: the geometric method is not very useful for vectors in three dimensions; rather the analytic method is used. The method involves resolution of vectors into components with respect to a coordinate system. RESOLUTION OF VECTORS Vectors can be resolved into horizontal and vertical components. The horizontal component is represented by Ax, and the vertical component by Ay. Each of the component vector lies along a coordinate-axis direction. Ax = A cos and Ay = A sin. The angle describing the direction of the component is given by Tan = When the component of vector Ax points in the positive x-direction, we define the number Ax to be equal to the magnitude of Ax. When the component vector points in the negative direction, we define the number Ax to be equal to the negative of that magnitude. Note: however, that the magnitude of a vector is never negative but the direction the vector points RESOLUTION OF MORE THAN ONE VECTOR Suppose you have a system of three forces as shown below, then, Ax = A cos , Ay = A sin , Bx = B cos , By = -B sin , Cx = -C cos Cy = -C sin The horizontal component is then sum to obtain Fx = A cos + B cos - C cos Also, the vertical component is then sum to obtain Fy = A sin - B sin - C sin If R is the resultant of the vector, then we can write that R = √ 2 + 2 and the angle the resultant makes with the x-axis is tan = / In general, we can write the resultant of n-vectors acting at an angle 𝜃𝑛 where n = 1, 2, 3 as; Fx = F1cos𝜃1 + F2cos𝜃2 + F3cos𝜃3 +…… + F1cos𝜃𝑛, and Fy = F1sin𝜃1 + F2sin𝜃2 + F3sin𝜃3 + …… + Fnsin𝜃𝑛 NOTE: the sign convection is very important in the Analytic method Example 1: find the magnitude of the resultant of five coplanar forces and direction of the forces shown below. Assignment 3 Question 1: find the magnitude of the resultant of four coplanar forces and direction of the forces shown below Question 2: find the resultant of four vectors as shown in the diagram below: UNIT VECTOR A unit vector is a vector that has a magnitude of one (1) with no units, and it gives the direction of the main vector in space. Throughout this lecture, we shall adopt by way of convention i, j, k to represent the unit vector in space, with the space coordinate as x, y, z. In an x-y coordinate system we can define a unit vector î that points in the direction of the positive x-axis and a unit vector ĵ that point in the direction of the positive y-axis (Fig. below). Then we can express the relationship between component vectors and components The unit vectors i and j point in the directions of the x- and y-axes and have a magnitude of 1. Similarly, we can write a vector in terms of its components as When two vectors and are represented in terms of their components, we can express the vector sum using unit vectors as follows: If the vectors do not all lie in the xy-plane, then we need a third component. We introduce a third unit vector ĸ that points in the direction of the positive z-axis depicted below. ADDITION AND SUBTRACTION OF UNIT VECTORS Given the vectors, = + + and = + + , to perform the operation of addition and subtraction, we simply identify corresponding components, then sum or subtract appropriately. i.e. + THE DOT (  ) OR SCALAR PRODUCT The dot product of two vectors say A and B is defined as A  B = /A//B/ cos NOTE: i. The dot product of two vectors is a scalar quantity. ii. Two vectors are perpendicular if their dot product is equal to zero iii. i i  j  j  k k 1 iv. i  j  j k  k i  0 The scalar product or dot product of two vectors and , denoted as  is a product that results to a scalar quantity. Given the vectors = + + and = + + , then  = The dot product is the sum of the product of corresponding components. Generally,  =| | | |c , where is the angle between the vectors, Example 1: Example 2: Assignment 4 Question 1 Given the two displacements; Question 2 Find the angle between the two vectors; Question 3 Given = 2 +3 − and = − + +2 , find the angle between the vectors. Question 4 THE CROSS (×) OR VECTOR PRODUCT The cross product of a vector A and B is defined as A × B = /A//B/ sin. If = 0 (when A is parallel B) then A × B = 0 and if = 90 then A × B = 1. It is a vector quantity. For cross product i × i = j × j = k × k = 0 and i × j = j × k = k × i = 1. The cross product of a vector is not commutative, i.e. A × B = -B × A. Given that vector = + + and = + + , then; Example 2: Find the cross product of the vectors and Example 5: Obtain the vector product of the vectors and. Assignment 5 1. Two vectors are given by A = 3.0i + 5.0j and B = -2.0i + 4.0j. Find (a) (A × B) (b) A.B (c) (A + B). B 2. If the vectors = 2 + + , = + + 7 and =− +2 are coplanar, find the value of. Hint: since the vectors are co-planar, their vector triple product must be zero. i.e. ( × )=0 3. Show that  A  B    A  B   A  B  = A B 2 2 2