Mechanics Lecture PDF

Mechanics Dr. Yasmin Mohamed Yousef Bakier Physics Department - Faculty of Science - Assiut University - Egypt 𝟓𝒕𝒉 floor, Room no. 510 Physics and Measurements  Content ▪ Basic and derived quantities ▪ Systems of units ▪ Density an...

Mechanics Dr. Yasmin Mohamed Yousef Bakier Physics Department - Faculty of Science - Assiut University - Egypt 𝟓𝒕𝒉 floor, Room no. 510 Physics and Measurements  Content ▪ Basic and derived quantities ▪ Systems of units ▪ Density and atomic mass ▪ Dimensional Analysis ▪ Conversion of units ▪ Vector & Scalar Quantities ▪ Operations on vectors 4  Basic and derived quantities ▪The laws of physics are expressed in terms of basic quantities that require a clear definition and cannot be explained in terms of other physical quantities. Basic Quantities Length Mass Time L M T ▪Any other mechanical quantities can be expressed in terms of the basic quantities and are called derived quantities Derived Mechanical Quantities Force Velocity Volume ……etc. 4  Different systems of units C.G.S Units The C.G.S. system of units (Centimeter, Gram, Second system) is a French system. This system deals with only three fundamental units – the Centimeter, Gram and the Second for length, mass and time respectively. F.P.S Units The F.P.S. system of units (Foot, Pound, Second system) is a British system. This system deals with only three fundamental units – the Foot, Pound and the Second for length, mass and time respectively. M.K.S Units The M.K.S. system of units (Meter, Kilogram, Second system) was set up by France. This system also deals with three fundamental units – the Meter, kilogram and the Second for length, mass and time respectively. This system is also called the metric system of 4 units and is closely related to C.G.S system of units.  Different systems of units International System (SI) of units In 1960, an International committee established a set of standards for the three basic quantities as follows: Mechanics Basic Quantities Other Quantities Quantity Unit Quantity Unit Mass Kilogram (kg) Electric current Ampere (A) Length Meter (m) Temperature Kelvin (K) Time Second (s) Luminous Intensity Candela Amount of substances Mole (mol) Other Systems 7  SI Units The measurement system which is internationally accepted now is suggested by the Eleventh general conference of weights and Measures in 1960 - France and is known as SI units of measurement. There are seven basic or fundamental units and three supplementary units. The basic units are the meter (m) for length Mechanical The supplementary units are Quantities the kilogram (kg) for mass the radian (rad) for angle the second (s) for time the steradian (sr) for solid angle the becquerel (Bq) for radioactivity the Kelvin (K) for temperature Non-Mechanical Quantities the ampere (A) for electric current the candela (cd) for luminous intensity the mole (mol) for the amount of substance 4  Standards: Length, Mass and Time The International committee established a set of standards for the three basic quantities as follows: 𝑳 (𝒎) Standards of length: is based on speed of light in vacuum. 𝒄 = 𝒕 (𝒔) = 𝟐𝟗𝟗 𝟕𝟗𝟐 𝟒𝟓𝟖 (𝒎/𝒔) Since 1983: Meter: distance travelled by light in vacuum during a time of 𝟏 𝒎 𝟏 𝒕 𝒔 = 𝒎 ≅ 𝟖 𝒔 𝟐𝟗𝟗 𝟕𝟗𝟐 𝟒𝟓𝟖 𝒔 𝟑 × 𝟏𝟎 Standards of mass: is based on the mass of an alloy. Since 1887:Kilogram: mass of a Pt-Ir alloy cylinder. (Pt-Ir alloy is very stable) Standards of time: is based on the atomic clock of Cs133 atom Second (s): obtained by the characteristic frequency (period of vibration ) of radiation from the Cs133 atom. 𝟏 Periodic time is 𝑻 = 𝒔 T 𝟗𝟏𝟗𝟐𝟔𝟑𝟏𝟕𝟕𝟎 6 𝟏 𝒔 = 𝟗 𝟏𝟗𝟐 𝟔𝟑𝟏 𝟕𝟕𝟎 × 𝑻 Time  Density and Atomic mass Density(𝜌) is defined as a mass (m) per unit volume (V). 𝒎 𝜌= 𝒗 ρ𝐴𝑙 = 2.7 gm/𝑐𝑚 3 ρ𝑃𝑏 = 11.3 gm/𝑐𝑚3 Two different pieces of Al and Pb of the same volume =10 𝑐𝑚3 has a masses 𝑚𝐴𝑙 =ρV=2.7x10 = 27 gm 𝑚𝑃𝑏 =ρV=113x10= 113 gm Why they are different? Because, atomic masses are different  Al AAl = 27 Note : the atomic mass   Pb APb = 207  Pb 11.3 A 207  Pb APb = = 4.19  Pb = = 7.67 mean while   Al 2.7 AAl 27  Al AAl The ratios of densities and atomic masses aren’t the same because of atomic spacing and atomic arrangements in both materials (Al & Pb) are different. 9  Density and Atomic mass Mole One mole of a substance is that amount of it that consists of Avogadro’s number of atoms or molecules. Avogadro’s Number (Na) The constant is named in honor of Avogadro. Avogadro discovered that the volume of a gas is directly proportional to the amount of the gas at constant pressure and temperature. Na = 6.02 x 1023 atom /mol One mole of Al has a mass of AAL=27 gm One mole of Pb has a mass of APb=207 gm A (gm/mol) Mass of any atom = Na (atom/mol) 10 Where A is the atomic mass in gm (mass of one mole)  Density and Atomic mass Examples: 1. A solid cube of Al (density=2.7g/ 𝑐𝑚3 ) has a volume of 0.2 𝑐𝑚3. If the atomic mass of Al is 27g/mol, how many Al atoms are contained in the cube? 2. Two spheres are cut from a certain uniform rock. One has radius 4.50cm. The mass of the second sphere is five times greater. Find the radius of the second sphere. 3. The standard kilogram is a Platinum-Iridium cylinder 39mm in height and 39mm in diameter. What is the density of this material? 10  Density and Atomic mass Answers: 1. 10  Density and Atomic mass Answers: 2. 10  Density and Atomic mass Answers: 3. 10  Dimensional Analysis Dimension denotes the physical nature of a quantity and is described using the basic quantities like: Dimension of some physics quantities: Area[A] =L.L = L2 Mechanics Basic Quantities Volume[V] =L.L.L = L3 Length (L) Velocity[ν] =L / T = LT −1 Mass (M) Acceleration[a] =L / T 2 = LT −2 Time (T) Momentum[P] = [m]. [ν] = ML T −1 Kinetic Energy[K.E] = [1/2].[m]. [v 2 ] = M(L T −1 )2 = M L2 T −2 Potential Energy[P.E] = [m]. [g]. [h] = M(L T −2 ) L = M L2 T −2 Note that [K.E] = [P.E] 11  Dimensional Analysis Importance Checking the validity of a specific formula Deriving some formulas Examples: 1. A car started from rest and moves with a constant acceleration a. What is the distance x travelled in a time t. (a) Check the accuracy of the formula x=½ a t 2 (b) Derive the formula that linked between the variables. 2. Suppose we are told that the acceleration of a particle moving with a uniform speed v in a circular orbit of radius r is proportional to some power of r say 𝒓𝒏 and some power of v say 𝒗𝒎. How can we determine the power of r and v? 12  Dimensional Analysis Answers: (a) (b) 12