PHY 115 Units and Dimensions PDF

Physics Department, University of Ilorin PHY 115 UNITS AND DIMENSION Observation and measurement of physical quantities / parameters are the core of scientific activities. Measurement is the act or process of assigning size, or value to a physical quantity. Quantities are measured when we are able...

Physics Department, University of Ilorin PHY 115 UNITS AND DIMENSION Observation and measurement of physical quantities / parameters are the core of scientific activities. Measurement is the act or process of assigning size, or value to a physical quantity. Quantities are measured when we are able to quantify it by assigning value and unit to such quantities or parameters. This is done by the use of measuring equipment- which are mostly based on one physical law or the other. Quantities are categorised into two namely: fundamental quantities and derived quantities (likewise the Units- fundamental units and derived units). Fundamental quantities are quantities upon which other quantities are based while derived quantities are quantities that are obtained from fundamental quantities. Fundamental quantities, units and dimensions Quantity Unit Dimension Length Meter(m) L Time Second(s) T Mass Kilogramme(kg) M Temperature Degree Kelvin(oK) Ɵ or K Electric current Ampere(A) I or A Electric charge Coulomb(C) Q Mole(Amount of substance) Mole(mol) N Luminous intensity Candela(cd) J or C The underlined dimensions are the one adopted for this course. Derived quantities, Units and Dimension Quantity Equation Unit Dimension Velocity 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑚 𝑠 𝐿𝑇−1 𝑡𝑖𝑚𝑒 Acceleration 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑚 𝑠2 𝐿𝑇−2 𝑡𝑖𝑚𝑒 Momentum 𝑚𝑎𝑠𝑠 × 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑘𝑔𝑚 𝑠 𝑀𝐿𝑇−1 Force 𝑚𝑎𝑠𝑠 × 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑘𝑔𝑚 𝑠2 𝑀𝐿𝑇−2 Example 𝑞1 𝑞2 Electrostatic Force = 𝐹 = 𝑘 𝑟2 1 F=[MLT -2]=[𝑘𝑄2 𝐿−2 ] This implies: [𝑘𝑄2 ] = [𝑀𝐿3 𝑇−2 ] [k] = 𝑀𝐿3 𝑇−2 𝑄−2 Work done = 𝐹 × 𝑑 = 𝑁𝑚 = 𝑀𝐿2 𝑇−2 The relationship between Work done and voltage due to separation of charges is 𝑊𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 = 𝑞𝑉 , but the dimension of work is [𝑀𝐿2 𝑇−2 ] Therefore, [𝑞𝑉] = [𝑀𝐿2 𝑇−2 ] Implying [V] = [𝑀𝐿2 𝑇−2 𝑄−1 ]. The quantity of heat energy transfer 𝑄 = 𝑚𝑐𝑇, obtain the dimension of the specific heat capacity (c). (Please note that Q is used as the dimension of charge above, quantity of heat is also represented with symbol Q - but not the same as the dimension of charge) The quantity of heat is in Joules → dimension of Q, [𝑄] = [𝑀𝐿2 𝑇−2 ] Using 𝑄 = 𝑚𝑐𝑇, implies Q  [𝑀𝑐𝐾] = [𝑀𝐿2 𝑇−2 ], or 𝑀𝐿2 𝑇−2 [c] = = 𝐿2 𝑇−2 𝐾−1 𝑀𝐾 Optics 𝑆𝑖𝑛 𝑖 Refractive index 𝑛 = 𝑆𝑖𝑛 𝑟 is dimensionless. Wavelength  [L] has the dimension of length and the unit is meter. Questions 𝐺𝑚𝑀 1. The gravitational force F is given by 𝐹 = , obtain the 𝑅2 dimension of G. 2. Acceleration due to gravity (𝑔), below the earth surface, is given 𝑏𝐺𝑀 by 𝑔 = , obtain the dimension of b. 𝑟3 2  3. The force a current carrying wire will experience when placed in a magnetic field of magnitude B is FB  BIL , where I is the current passing through the wire and L is the length of the wire that is inside the field. Obtain the dimension of B. SCALAR AND VECTOR QUANTITIES Physical quantities can be grouped into two namely: scalar and vector quantities. SCALAR – A quantity defined only by magnitude e.g. Distance, Speed, Mass. Distance – Shortest space 𝑑𝑥 𝑜𝑟 𝑑 between two points. The unit is in meter or foot. Speed(s) – Rate of change of distance with time and is equal to (distance divided by time) Mass – Quantity of matter in a body and is measured in kg VECTOR – A quantity defined by both magnitude and direction e.g. displacement, velocity, acceleration, force. Displacement – Distance in a defined direction. 𝒅⃑ θ ---------------------- Velocity – Time rate of chance of displacement or change of distance with time in a given direction. 𝑣 = 𝑑/𝑡 = 𝑑𝑠/𝑑𝑡 = 𝑑𝑥/𝑑𝑡 𝒖⃑ θ ---------------------- Acceleration - rate of change of velocity with time. 𝑎 = 𝑑𝑣/𝑑𝑡 𝑎⃑ 3 Force – Rate of change of momentum with time. This can be shown (later) to be the product of mass and acceleration. Addition and Subtraction of Scalars - Addition and subtraction of scalar quantity is done like numbers. Sum of d1 and d2 = d1 + d2 d1 d2 Difference between d4 and d3 is equal to d4-d3 d3 d4 – d3 d4 Note – Scalars have magnitude only e.g. mass, length, time, density, energy while Vectors have magnitude and direction. Displacement is characterized by length and direction. 𝐴𝐵⃗ is net the effect and is independent of the path taken to go from A to B. Vectors: Considering the net displacement say from A to C through AB followed by BC. The sum is not an algebraic sum Other example includes: b C Force, Velocity, Acceleration B r E Electric field strength a B Magnetic induction A The symbols for vectors are bold face letters or letters with arrow, i.e., 𝔸 or 𝐴⃗. The magnitude is represented by the modulus, i.e., the magnitude of vector 𝐴⃗ is |𝐴⃗| Vectors can be expressed in vector notation or in magnitude and the angle the vector makes with positive x-axis. In vector notation, a 2D vector, e.g., 𝐴⃗ can be written as 𝐴⃗ = 𝐴𝑥 𝑖 + 𝐴𝑦 𝑗 4 where Ax is the x-component of the vector, Ay is the y-component of the vector while 𝑖 and 𝑗 are the unit vectors. In terms of magnitude and angle / direction, vector 𝐴⃗ can be expressed as |𝐴⃗ |, θ. Magnitude of vector 𝐴⃗ Ay A Ax2  Ay2 ,   tan 1 Ax Ay A  Ax x- and y- components of a vector can be obtained from the magnitude and angle representation of the vector. Ax  A cos  , Ay  A sin  Therefore, vector A can also be expressed as 𝐴⃗ = 𝐴 cos 𝜃 𝑖 + 𝐴 sin 𝜃 𝑗 Addition of Vectors The addition of two vectors can be written as r a b PROPERTIES OF VECTORS 5 Cumulative Law: 𝒂⃗ + 𝒃⃗ = 𝒃⃗ + 𝒂⃗ = 𝒓⃗ b r a a b 𝑨𝒔𝒔𝒐𝒄𝒊𝒂𝒕𝒊𝒐𝒏 𝑳𝒂𝒘 − 𝑖.𝑒.𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 𝑜𝑓 𝑔𝑟𝑜𝑢𝑝𝑖𝑛𝑔, 𝑡ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟𝑠 𝑖𝑠 𝑡ℎ𝑒𝑠𝑎𝑚𝑒 𝒂⃑ + 𝒃⃑ + 𝒄⃑ = 𝒂⃑ + 𝒃⃑ + 𝒄⃑ 𝒃⃑ 𝒂⃑ + 𝒃⃑ 𝒃⃑ + 𝒄⃑ 𝒄⃑ 𝒂⃑ + 𝒃⃑ + 𝒄⃑ Subtraction - For a vector b one can define another vector  b that is of the same magnitude with b but of opposite direction as shown below. − 𝒃⃑ + 𝒃⃑ Then, a  b  a  ( b ) 𝑇ℎ𝑒𝑛 𝑎⃑ − 𝑏⃑ = 𝑎⃑ + − 𝑏⃑ 𝑉𝑒𝑐𝑡𝑜𝑟 𝑖𝑠 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑎𝑛𝑑 𝑖𝑡𝑠 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 5 𝑢𝑛𝑖𝑡𝑠 𝑣𝑒𝑐𝑡𝑜𝑟 𝑏 𝑖𝑠 450 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 Example: vector a is along the horizontal and its magnitude is 5 units, vector b makes angle 45° with the horizontal and its of magnitude 4 units while 6  vector c makes angle 30° with the vertical axis and its magnitude is 3 units. Compute a b  c. Unit Vector: This is a vector of magnitude 1 in a particular direction. For example,  in 1D, vector 𝑎⃗ = 2.5𝑎 = 2.5𝑖, where 2.5 is the magnitude and a or 𝑖 is the unit vector. 𝒂⃑ 𝒋 a 𝒊 𝑈⃗𝑎 U=1 𝒌 Sum of Two Vectors (𝒂⃗ 𝒂𝒏𝒅 𝒃⃗) 𝒓⃑ = 𝒂⃑ + 𝒃⃗ 𝑻𝒘𝒐 𝒗𝒆𝒄𝒕𝒐𝒓 𝒂⃑ + 𝒃⃗ 𝒂𝒓𝒆 𝒆𝒒𝒖𝒂𝒍 𝒊𝒇 𝒕𝒉𝒆𝒊𝒓 𝒄𝒐𝒓𝒓𝒆𝒔𝒑𝒐𝒏𝒅𝒊𝒏𝒈 𝒄𝒐𝒎𝒑𝒐𝒏𝒆𝒏𝒕𝒔 𝒂𝒓𝒆 𝒆𝒒𝒖𝒂𝒍 𝒂𝒏𝒅 𝒊𝒕𝒔 𝒓𝒆𝒔𝒖𝒍𝒕𝒂𝒏𝒕 𝒗𝒆𝒄𝒕𝒐𝒓 𝒊𝒔 𝒓⃑, i.e., 𝒓𝒙 = 𝒂𝒙 + 𝒃𝒙 , 𝒓𝒚 = 𝒂𝒚 + 𝒃𝒚 𝑟𝑦 𝒓= 𝒓𝟐𝒙 + 𝒓𝟐𝒚 , 𝑡𝑎𝑛𝜃 = (in magnitude and direction / angle) 𝑟𝑥 𝑟⃗ = 𝑟𝑥 𝑖 + 𝑟𝑦 𝑗 (in vector notation) EXAMPLE – Three coplanar vectors are expressed with respect to a certain rectangular coordinates system of a given reference frame as 𝒂⃑ = 𝟒𝒊⃑ − 𝒋 𝒃⃑ =− 𝟑𝒊 + 𝟐𝒋⃑ 𝒄⃑ =− 𝟑𝒋⃑ 𝑻𝒉𝒆 𝒄𝒐𝒎𝒑𝒐𝒏𝒆𝒏𝒕𝒔 𝒂𝒓𝒆 𝒈𝒊𝒗𝒆𝒏 𝒊𝒏 𝒂𝒓𝒃𝒊𝒕𝒓𝒂𝒓𝒚 𝒖𝒏𝒊𝒕𝒔. 𝑭𝒊𝒏𝒅 𝒕𝒉𝒆 𝒗𝒆𝒄𝒕𝒐𝒓 𝒓 𝒘𝒉𝒊𝒄𝒉 𝒊𝒔 𝒕𝒉𝒆 𝒔𝒖𝒎 𝒐𝒇 𝒕𝒉𝒆𝒔𝒆 𝒗𝒆𝒄𝒕𝒐𝒓𝒔. 𝒓𝒙 = 𝒂𝒙 + 𝒃𝒙 + 𝒄𝒙 = 𝟒 − 𝟑 + 𝟎 = 𝟏 7 𝒓𝒚 = 𝒂𝒚 + 𝒃𝒚 + 𝒄𝒚 =− 𝟏 + 𝟐 − 𝟑 =− 𝟐 𝒓⃑ = 𝒊⃑𝒓𝒙 + 𝒋⃑𝒓𝒚 𝒓⃗ = 𝒊⃑ − 𝟐𝒋⃑ and can be represented as shown below 𝑺𝒊𝒏𝒄𝒆 𝒎𝒂𝒈𝒏𝒊𝒕𝒖𝒅𝒆 𝒓 = 𝒓𝟐𝒙 + 𝒓𝟐𝒚 = 𝟏+𝟒 θ = 𝟓 = 𝟐.𝟐𝟒 -2 The angle the vector makes with positive x-axis is 360  . 1 𝐴𝑛𝑔𝑙𝑒 𝑚𝑎𝑑𝑒 𝑤𝑖𝑡ℎ 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑥 − 𝑎𝑥𝑖𝑠 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑐𝑜𝑢𝑛𝑡𝑒𝑟𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒 2 𝑇𝑎𝑛−1 ( − ) = 296.57° 1 MULTIPLICATION OF VECTORS One can not add vector and scalar in each task. One can add and multiply vectors with vectors. One can also multiple a vector by a scalar. Three Kinds – a) Multiplication of a vector by scalar b) Multiplication of 2 vectors yielding scalar (dot product) c) Multiplication of 2 vectors yielding vectors (cross product), etc. VECTOR-SCALAR MULTIPLICATION Scalar k, can be used to multiply vector 𝑎⃗ to have Product = 𝑘 𝑎⃑ A new vector with magnitude (k | a |) will have the same direction as 𝑎⃗ 1 − 𝑎 𝑛𝑒𝑤 𝑣𝑒𝑐𝑡𝑜𝑟 𝑤𝑖𝑡ℎ 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑘 𝑡𝑖𝑚𝑒𝑠 𝑎𝑠 𝑎⃗. 𝑆𝑎𝑚𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑎𝑠 𝑎⃗. 𝐶ℎ𝑒𝑐𝑘 − 𝑖𝑓 𝑘 𝑖𝑠 + 𝑣𝑒 To divide by scalar, multiply by reciprocal of k 8  1.𝑎⃗ 𝑘 VECTOR-VECTOR MULTIPLICATION I. Scalar (or dot.) II. Vector (or cross × ) SCALAR PRODUCTS 𝑎⃗.𝑏⃗ = 𝑎𝑏 cos 𝜃 𝑤h𝑒𝑟𝑒 𝑎 𝑖𝑠 𝑡h𝑒 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑎⃗, 𝑏 𝑖𝑠 𝑡h𝑒 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑏⃗ 𝑎𝑛𝑑 cos 𝜃 𝑖𝑠 𝑡h𝑒 𝑐𝑜𝑠𝑖𝑛𝑒 𝑜𝑓 𝑡h𝑒 𝑎𝑛𝑔𝑙𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡h𝑒 𝑡𝑤𝑜 𝑣𝑒𝑐𝑡𝑜𝑟𝑠. 𝑇h𝑢𝑠, 𝑠𝑐𝑎𝑙𝑎𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑐𝑎𝑛 𝑏𝑒 𝑟𝑒𝑔𝑎𝑟𝑑𝑒𝑑 𝑎𝑠 𝑡h𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑡h𝑒 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑜𝑛𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝑎𝑛𝑑 𝑡h𝑒 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝑡h𝑒 𝑜𝑡h𝑒𝑟 𝑖𝑛 𝑡h𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡h𝑒 𝑓𝑖𝑟𝑠𝑡. For example, the dot product of vector a and vector b , i.e., 𝑐 = 𝑎⃗.𝑏⃗ can be obtained if a and b are explicitly defined. For a  a x i  a y j and b  bx i  by j , a. b  ( a x i  a y j ).(bx i  by j )  a x bx  a y by Since, i.i  j. j  1, and i. j  j.i  0 This implies that a xbx  a y by  a.b cos The equation above can be used to obtain the angle between two vectors. Example 1: Calculate the angle between two vectors  3i  4 j and 2i  3 j Solution: Using 𝑎𝑥 𝑏𝑥 + 𝑎𝑦 𝑏𝑦 = 𝑎.𝑏 cos 𝜃 9   6  12  32  4 2. 2 2  32 cos  6 6  5  13 cos  ;   cos 1  cos 1 0.3328 18.0277   70.56 VECTOR PRODUCT 𝑐⃗ = 𝑎⃗ × 𝑏⃗ Magnitude of vector 𝑐⃗ can be written as c  ab sin  where θ is the angle between a and b Example Calculate the angle between two vectors  3i  4 j and 2i  3 j using magnitude c  ab sin . Solution i j k (3i  4 j )  ( 2i  3 j )   3 4 0  i (0)  j (0)  k (9  8)  17k 2 3 0 Therefore using c  ab sin  , note that magnitude of 17 k is 17. 17 17  18.0277 sin  ,   sin 1  sin 1 0.9430  70.56 18.0277  is the angle between a and b. By definition, the direction of 𝐶, the cross ⃗ product of a and b is perpendicular to the plane formed by a and b or the plane that contains a and b 10  c c 𝒃⃑ b 𝒂⃑ b 𝒂 𝒂⃑ 𝒄' = 𝒃⃑ × 𝒂⃑ 𝑎⃗ × 𝑏⃗ is pronounced as "𝑎⃗ cross 𝑏⃗" 𝑏⃗ × 𝑎⃗ ≠ 𝑎⃗ × 𝑏⃗ 𝑎⃗ × 𝑏⃗ =− 𝑏⃗ × 𝑎⃗ For cross product, i  j  k , j  k  i , k  i  j , i  k   j , j  i   k , k  j  i 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑎𝑏𝑠𝑖𝑛𝜃 = 𝑏𝑎𝑠𝑖𝑛𝜃 𝑏𝑢𝑡 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑠 𝑎𝑟𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒. Examples of such cross products include – Torque, angular momentum, force of a moving charge in a magnet, flow of electromagnetic energy. Vector Products Tensor – generated by multiplying each of three components of one vector by the three components of another vector. Tensor 2nd Rank – Has nine numbers associated with it. Vector – Three numbers Scalar – One Example of Tensors – Mechanical and Electrical Stress, Moments, Products of Inertia, Strain. PROPERTIES OF VECTORS What happens to the laws of physics when simple operations such as translation and rotation of coordinates are performed? Coordinate system 𝑥,𝑦,𝑧 𝑉𝑒𝑐𝑡𝑜𝑟𝑠 𝑎⃗, 𝑏⃗, 𝑟⃗ 11 𝑅𝑒𝑙𝑎𝑡𝑖𝑜𝑛𝑠ℎ𝑖𝑝 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒𝑚 𝑟⃗ = 𝑎⃗ + 𝑏⃗ 𝐵𝑦 𝑒𝑎𝑟𝑙𝑖𝑒𝑟 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑢𝑚 𝑟𝑥 = 𝑎𝑥 + 𝑏𝑥 𝑟𝑦 = 𝑎𝑦 + 𝑏𝑦 𝑟𝑧 = 𝑎𝑧 + 𝑏𝑧 Consider a New coordinate system 𝑥,𝑦,𝑧 with properties I. Origin does not coincide with the origin of the first coordinate system 𝑥,𝑦,𝑧 – Translation II. Its three axes are not parallel to the corresponding axes in the first system – Rotation Representation of vectors 𝑥,𝑦,𝑟 in the new system would in general prove to be different. Let’s put them in primes. The relationship between them however would be: 𝑟𝑥 ' = 𝑎𝑥 ' + 𝑏𝑧 ' 𝑟 𝑦 ' = 𝑎 𝑦 ' + 𝑏𝑦 ' 𝑟𝑧 ' = 𝑎𝑧 ' + 𝑏𝑧 ' And, the relationship 𝑟⃗ = 𝑎⃗ + 𝑏⃗ 𝑠𝑡𝑖𝑙𝑙 ℎ𝑜𝑙𝑑𝑠 Consequently, it may be said that relations among vectors are invariant (unchanged) with respect to translation or rotation of coordinates. I.e. the laws of physics are unchanged when we rotate or translate the reference system. M y x i j Z k y’ 𝑗 Z ’ 𝑘 𝑖 x’ M’ M’ 12 (a) is left handed (b) is right handed The other is a mirror image of one 𝑖𝑛 𝑎 𝑖 × 𝑗 =− 𝑘 𝑏 𝑖×𝑗=𝑘 VIOLATION (1956) Decay of some elementary particles showed that the result was independent of the handedness whether left or right. i.e., the experiment and its mirror image would yield different results. This leads to question on the symmetry of physical laws. 13