Physics 111 NMU Lecture 1 PDF

9/29/2024 Physics I PHY111 Hamdi M. Abdelhamid Physics Department Faculty of Science New Man...

9/29/2024 Physics I PHY111 Hamdi M. Abdelhamid Physics Department Faculty of Science New Mansoura University 29/09/2024 1 1. Introduction Physics is fundamental science that based on experimental observations and quantitative measurements. Objectives of physics To find the limited number of fundamental laws that govern natural phenomena To use these laws to develop theories that can predict the results of future experiments Express the laws in the language of mathematics  Mathematics provides the bridge between theory and experiment. 29/09/2024 22 1 2  9/29/2024 1. Introduction  Physics divided into six major areas: Classical Mechanics Relativity Thermodynamics Electromagnetism Optics Quantum Mechanics 29/09/2024 33 3 1.2 Physical property Physical property can be defined as a characteristic that can be observed or measured without changing the identity or composition of the substance. Example include:  Color  Size  Physical state (liquid, gas, or solid)  Boiling point  Melting point  Density 29/09/2024 44 2 4  9/29/2024 1.2 Physical property Physical properties used to describe matter can be classified as: 1. Extensive – depends on the amount of matter in the sample e.g. Mass, volume, length. 2. Intensive – depends on the type of matter, not the amount present e.g. Hardness, density, boiling point. 29/09/2024 55 5 1.2 Physical property 29/09/2024 66 3 6  9/29/2024 1.3 Fundamental Quantities and Their Units SI – Systéme International Main system used in this course Quantity SI Unit Length meter Mass kilogram Time second Temperature Kelvin Electric Current Ampere Amount of Substance mole 29/09/2024 77 7 1.3 Fundamental Quantities and Their Units Quantities Used in Mechanics In mechanics, three fundamental quantities are used: Length Mass Time All other quantities in mechanics can be expressed in terms of the three fundamental quantities. 29/09/2024 88 4 8  9/29/2024 1.3 Fundamental Quantities and Their Units Length Length is the distance between two points in space. Defined in terms of a meter – the distance traveled by light in a vacuum during a given time. Units SI – meter, m Time Defined in terms of the oscillation of radiation from a cesium atom Units seconds, s 29/09/2024 99 9 1.3 Fundamental Quantities and Their Units Mass Defined in terms of a kilogram, based on a specific cylinder kept at the International Bureau of Standards Units SI – kilogram, kg 29/09/2024 10 10 5 10  9/29/2024 1.4 Prefixes Prefixes correspond to powers of 10. Each prefix has a specific name. Each prefix has a specific abbreviation. The prefixes can be used with any basic units. They are multipliers of the basic unit. Examples: 1 mm = 10-3 m 1 mg = 10-3 g 29/09/2024 11 11 11 1.4 Prefixes 29/09/2024 12 12 6 12  9/29/2024 1.5-Dimensional Analysis Dimensional analysis: Technique to check the correctness of an equation or to assist in deriving an equation  Dimensions (length, mass, time, combinations) can be treated as algebraic quantities. Add, subtract, multiply, divide  Both sides of equation must have the same dimensions.  Any relationship can be correct only if the dimensions on both sides of the equation are the same. 29/09/2024 13 13 13 1.5-Dimensional Analysis Ex (1)  Given the equation: x = ½ at 2  Check dimensions on each side: L L  T2  L T 2  The T2’s cancel, leaving L for the dimensions of each side.  The equation is dimensionally correct.  There are no dimensions for the constant. 29/09/2024 14 14 7 14  9/29/2024 1.5-Dimensional Analysis Ex (2) The expressions for Kinetic energy 𝐸 = 𝑚𝑣 (where 𝑚 is the mass of body and 𝑣 is its speed) and potential energy 𝐸 = 𝑚𝑔ℎ (where 𝑔 is acceleration due to gravity and h is the height of the body) look very different but both describe energy. Prove that the two expression can be added to each other. Solution Dimension of kinetic energy Dimension of potential energy 1 𝑚𝑔ℎ = 𝑀 𝐿𝑇 𝐿 = 𝑀𝐿 𝑇 𝑚𝑣 = 𝑀(𝐿𝑇 ) = 𝑀𝐿 𝑇 2 The two expressions have the same dimensions. They can therefore 29/09/2024 be added and subtracted from each other. 15 15 15 1.5-Dimensional Analysis Ex (3) Hooke’s law states that the force,F in a spring extended by a length x is given by 𝐅 = −𝐊𝐱. From Newton’s second law 𝐅 = 𝐦𝐚, where m is mass and a is the acceleration. Calculate the dimension of the spring constant 𝐊. Solution F K= x Now, F = ma , so the dimension of the force are given by F =M∗L∗T Therefore, the spring constant has dimension M∗L∗T 29/09/2024 K = =M∗T 16 16 8 L 16  9/29/2024 1.5-Dimensional Analysis Ex (4) Using the dimensional analysis to set up an expression of the form 𝐱 ∝ 𝐚𝐭 𝐭 𝐦 Where, n and m are exponents that must be determined and the symbol ∝ indicates a proportionality. Solution This relationship is correct only if the dimensions of both sides are the same. Because the dimension of the left side is length, the dimension of the right side must also be length.That is L. H. S = x = [L] = [L T ] 29/09/2024 17 17 17 1.5-Dimensional Analysis Because the dimensions of acceleration are L/T2 and the dimension of time is T, we have R. H. S = [a t ] = (L/T ) T R. H. S = L T T =L T L. H. S = R. H. S L T = LT The exponents of L and T must be the same on both sides of the equation. The exponents of L, give n=1. The exponents of T, give m−2n = 0, and substituting for n, gives m=2. 29/09/2024 18 18 9 18  9/29/2024 1.5-Dimensional Analysis Ex (5) The force attraction between two masses m1 and m2 lying at a distance r is given by 𝐦𝟏 𝐦𝟐 𝐅=𝐆 𝐫𝟐 Where, G is the constant of gravitation. What is the dimensions of G? Solution [r ] r G = [F] =ma [m ][m ] m m Substituting the dimensions of various quantities, we have L [G] = M L T 29/09/2024 M M 19 19 19 1.5-Dimensional Analysis L [G] = M L T M M [G] = M L T 29/09/2024 20 20 10 20  9/29/2024 1.6 Conversion of Units Sometimes it is necessary to convert units from one measurement system to another or convert within a system (for example, from kilometers to meters). Conversion factors between SI and U.S. customary units of length are as follows: 1 mile = 1.609 m 1 m = 39.37 in = 3.218 ft 1ft = 0.304,8 m = 30.48 cm 1 in =0.025, 4 = 2.54 cm 29/09/2024 21 21 21 1.6 Conversion of Units Like dimensions, units can be treated as algebraic quantities that can cancel each other. For example, suppose we wish to convert 15.0 in. to centimeters. Because 1 in. is defined as exactly 2.54 cm, we find that 2.54 𝑐𝑚 15.0 𝑖𝑛. = 15.0 𝑖𝑛. = 38.1 𝑐𝑚 1 𝑖𝑛. Where, the ratio in parentheses is equal to 1. We express 1 as 2.54 cm/1 in. (rather than 1 in./2.54 cm) so that the unit “inch” in the denominator cancels with the unit in the original quantity. The remaining unit is the centimeter, our desired result. 29/09/2024 22 22 11 22  9/29/2024 1.6 Conversion of Units Ex (6) On an interstate highway in a rural region of Wyoming, a car is traveling at a speed of 38.0 m/s. Is the driver exceeding the speed limit of 75.0 mi/h? Solution Convert meters in the speed to miles: 𝑚 1 𝑚𝑖 38 = 2.36 ∗ 10 𝑚𝑖/𝑠𝑒𝑐 𝑠 1.609 𝑚. Convert seconds to hours: 60 𝑠 60 𝑚𝑖𝑛. 2.36 ∗ 10 𝑚𝑖/ sec = 85 𝑚𝑖/ℎ𝑟 29/09/2024 1 𝑚𝑖𝑛. 1 ℎ𝑟. 23 23 23 1.6 Conversion of Units The driver is indeed exceeding the speed limit and should slow down. What if the driver were from outside the United States and is familiar with speeds measured in kilometers per hour? What is the speed of the car in km/h? We can convert our final answer to the appropriate units: 1.609 𝑘𝑚. 85 𝑚𝑖/ℎ𝑟 = 137 𝑘𝑚/ℎ𝑟 1 𝑚𝑖. 29/09/2024 24 24 12 24  9/29/2024 1.7 Vectors In our study of physics, we often need to work with physical quantities that have both numerical and directional properties. Vector quantities is physical quantities that have both numerical and directional properties A scalar quantity is completely specified by a single value with an appropriate unit and has no direction. Mathematical operations of vectors in this chapter Addition Subtraction 29/09/2024 25 25 25 1.7.1 Coordinate Systems Used to describe the position of a point in space. Common coordinate systems are: Cartesian Polar 29/09/2024 26 26 13 26  9/29/2024 Cartesian Coordinate System Also called rectangular coordinate system. x- and y- axes intersect at the origin Points are labeled (x,y). 29/09/2024 27 27 27 Polar Coordinate System Origin and reference line are noted Point is distance r from the origin in the direction of angle , from reference line The reference line is often the x- axis. Points are labeled (r,) 29/09/2024 28 28 14 28  9/29/2024 Polar to Cartesian Coordinates Based on forming a right triangle from r and θ  x = r cos θ  y = r sin θ If the Cartesian coordinates are known: y tan  x r  x2  y 2 29/09/2024 29 29 29 Polar to Cartesian Coordinates Ex (7) The Cartesian coordinates of a point in the xy plane are (x,y) = (-3.50, -2.50) m, as shown in the figure. Find the polar coordinates of this point. Solution r  x y 2 2  ( 3.50 m)2  ( 2.50 m)2  4.30 m y 2.50 m tan    0.714 x 3.50 m   216 (signs give quadrant) 29/09/2024 30 30 15 30  9/29/2024 Vector Notation   Text uses bold with arrow to denote a vector: A  Also used for printing is simple bold print: A When dealing with just the magnitude ofa vector in print, an italic letter will be used: A or | A | The magnitude of the vector has physical units. The magnitude of a vector is always a positive number. 29/09/2024 31 31 31 Some Properties of Vectors I. Equality of Two Vectors Two vectors are equal if they have the same magnitude and the same direction.   A  B if A = B and they point along parallel lines. All of the vectors shown are equal. Allows a vector to be moved to a position parallel to itself 29/09/2024 32 32 16 32  9/29/2024 Some Properties of Vectors II. Adding Vectors Vector addition is very different from adding scalar quantities. When adding vectors, their directions must be taken into account. Units must be the same All of the vectors must be of the same type of quantity. For example, you cannot add a displacement to a velocity. The rules for adding vectors are conveniently described by a graphical method. 29/09/2024 33 33 33 Some Properties of Vectors Adding Vectors Graphically   Draw the first vector, A , with the appropriate length and in the direction specified, with respect to a coordinate system. Draw the next vector with the appropriate length and in the direction specified, with respect to  a coordinate system whose origin is the end of vector  A and parallel to the coordinate system used for A. Continue drawing the vectors “tip-to-tail” or “head-to- tail”. The resultant is drawn from the origin of the first vector to the end of the last vector. Measure the length of the resultant and its angle. 29/09/2024 34 34 17 34  9/29/2024 Some Properties of Vectors Adding Vectors Graphically Use the scale factor to convert length to actual magnitude. 29/09/2024 35 35 35 Some Properties of Vectors Adding Vectors Graphically o When you have many vectors, just keep repeating the process until all are included. o The resultant is still drawn from the tail of the first vector to the tip of the last vector. 29/09/2024 36 36 18 36  9/29/2024 Some Properties of Vectors Adding Vectors Graphically When two vectors are added, the sum is independent of the order of the addition. This is the Commutative Law of Addition.     AB B A 29/09/2024 37 37 37 Some Properties of Vectors Adding Vectors Graphically When adding three or more vectors, their sum is independent of the way in which the individual vectors are grouped. This is called the Associative Property of Addition.          A  BC  A B C 29/09/2024 38 38 19 38  9/29/2024 Some Properties of Vectors III. Negative of a Vector The negative of a vector is defined as the vector that, when added to the original vector, gives a resultant of zero.  Represented as A     A  A  0 The negative of the vector will have the same magnitude, but point in the opposite direction. 29/09/2024 39 39 39 Some Properties of Vectors IV. Subtracting Vectors The operation of vector subtraction makes use of the definition of the negative of a vector.     If A B, then use   A  B Continue with standard vector addition procedure. 29/09/2024 40 40 20 40  9/29/2024 Some Properties of Vectors IV. Subtracting Vectors (method 2) Another way to look at subtraction is to find the vector that, added to the second vector gives you the first vector      A  B  C As shown, the resultant vector points from the tip of the second to the tip of the first. 29/09/2024 41 41 41 Some Properties of Vectors V. Multiplying or Dividing a Vector by a Scalar The result of the multiplication or division of a vector by a scalar is a vector. The magnitude of the vector is multiplied or divided by the scalar. If the scalar is positive, the direction of the result is the same as of the original vector. If the scalar is negative, the direction of the result is opposite that of the original vector. 29/09/2024 42 42 21 42  9/29/2024 Some Properties of Vectors  Graphical addition is not recommended when: High accuracy is required If you have a three-dimensional problem Component method is an alternative method It uses projections of vectors along coordinate axes 29/09/2024 43 43 43 1.7.2 Components of a Vector A component is a projection of a vector along an axis. Any vector can be completely described by its components. It is useful to use rectangular components. These are the projections of the vector along the x- and y-axes. 29/09/2024 44 44 22 44  9/29/2024 1.7.2 Components of a Vector    A x and A y are the component vectors of A. They are vectors and follow all the rules for vectors. Ax and Ay are scalars,  and will be referred to as the components of A.  Assume you are given a vector A It can be expressed  in terms  of two other vectors, A x and A y These three vectors form a right triangle. 29/09/2024 45 45 45 1.7.2 Components of a Vector The y-component is moved to the end of the x-component. This is due to the fact that any vector can be moved parallel to itself without being affected. The x-component of a vector is the projection along the x-axis. Ax  A cos The y-component of a vector is the projection along the y-axis. Ay  A sin 46 29/09/2024 46 23 46  9/29/2024 1.7.2 Components of a Vector This assumes the angle θ is measured with respect to the x-axis. The components are the legs of the right triangle whose hypotenuse is the length of A Ay A  Ax2  Ay2 and   tan1 Ax 29/09/2024 47 47 47 1.7.2 Components of a Vector The components can be positive or negative and will have the same units as the original vector. The signs of the components will depend on the angle. 29/09/2024 48 48 24 48  9/29/2024 1.7.3 Unit Vectors A unit vector is a dimensionless vector with a magnitude of exactly 1 Unit vectors are used to specify a direction and have no other physical significance. The symbols î , ĵ, and k̂ represent unit vectors. They form a set of mutually perpendicular vectors in a right- handed coordinate system 29/09/2024 49 49 49 1.7.3 Unit Vectors The magnitude of each unit vector is 1 ˆi  ˆj  kˆ  1 Ax is the same as Ax î and Ay is the same as Ay ĵ etc. The complete vector can be expressed as:  A  Ax ˆi  Ay ˆj 29/09/2024 50 50 25 50  9/29/2024 1.7.3 Unit Vectors Position Vector, Example A point lies in the xy plane and has Cartesian coordinates of (x, y). The point can be specified by the position vector. rˆ  x ˆi  yˆj This gives the components of the vector and its coordinates. 29/09/2024 51 51 51 1.7.3 Unit Vectors Adding Vectors Using Unit Vectors    Using R  A  B Then     R  Ax iˆ  Ay ˆj  Bx ˆi  By ˆj   R   Ax  Bx  ˆi   Ay  By  ˆj  R  Rx ˆi  Ry ˆj So Rx = Ax + Bx and Ry = Ay + By Ry R  Rx2 Ry2   tan1 Rx 29/09/2024 52 52 26 52  9/29/2024 1.7.3 Unit Vectors Three-Dimensional Extension    Using R  A  B Then     R  Ax ˆi  Ay ˆj  Azkˆ  Bx ˆi  By ˆj  Bzkˆ   R   Ax  Bx  ˆi   Ay  By  ˆj   Az  Bz  kˆ  R  R ˆi  R ˆj  R kˆ x y z So Rx= Ax+Bx, Ry= Ay+By, and Rz = Az+Bz Rx R  Rx2  Ry2  Rz2  x  cos1 , etc. R 29/09/2024 53 53 53 1.7.3 Unit Vectors Adding Three or More Vectors The same method can be extended to adding three or more vectors. Assume     R  A BC And  R   Ax  B x  C x  ˆi   Ay  B y  C y  ˆj   Az  Bz  C z  kˆ 29/09/2024 54 54 27 54  9/29/2024 1.7.3 Unit Vectors Ex (9) A hiker begins a trip by first walking 25.0 km southeast from her car. She stops and sets up her tent for the night. On the second day, she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower. Solution Draw a sketch as in the figure Denote the displacement vectors on the first and second days by and respectively. Use the car as the origin of 29/09/2024 55 55 coordinates. 55 1.7.3 Unit Vectors  Drawing the resultant R , we can now categorize this problem as an addition of two vectors. The first displacement has a magnitude of 25.0 km and is directed 45.0° below the positive x axis. Its components are:. Ax  A cos( 45.0)  (25.0 km)(0.707) = 17.7 km Ay  A sin( 45.0) 29/09/2024 (25.0 km)( 0.707)  17.7 km 56 56 28 56  9/29/2024 1.7.3 Unit Vectors The second displacement has a magnitude of 40.0 km and is 60.0° north of east. Its components are:. Bx  B cos 60.0  (40.0 km)(0.500) = 20.0 km By  B sin 60.0  (40.0 km)(0.866)  34.6 km The negative value of Ay indicates that the hiker walks in the negative y direction on the first day. The signs of Ax and Ay also are evident from the figure. 29/09/2024 57 57 57 1.7.3 Unit Vectors Determine the components of the hiker’s resultant displacement for the trip. Find an expression for the resultant in terms of unit vectors. The resultant displacement for the trip has components given by Rx = Ax + Bx = 17.7 km + 20.0 km = 37.7 km Ry = Ay + By = -17.7 km + 34.6 km = 16.9 km In unit vector form  R = (37.7 ˆi + 16.9ˆj) km 29/09/2024 58 58 29 58  9/29/2024 1.7.3 Unit Vectors The resultant vector has a magnitude of 41.3 km and is directed 24.1° north of east.  The units of R are km, which is reasonable for a displacement. From the graphical representation , estimate that the final position of the hiker is at about (38 km, 17 km) which is consistent with the components of the resultant. Both components of the resultant are positive, putting the final position in the first quadrant of the coordinate 29/09/2024 59 59 system. 59 30