1MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01 Supplementary Notes 1: Dimensional Analysis 1.1 International System of System of Units There are only four fundamental quantities (measurements) necessary to specify all physical phenomena: length, time, mass and charge. All other quantities are expressible in terms of these, constructed as a matter of convenience. The basic system of units used throughout science and technology today is the internationally accepted Système International (SI) (Table 1). It consists of four base quantities and their corresponding base units: length (meter), mass (kilogram), time (second), and electric current (ampere). The unit for electric charge, the coulomb, is defined in terms of the ampere, and hence is referred to as a derived unit. In addition, three other quantities, temperature, amount of substance, and luminous intensity are part of the SI base quantities with corresponding units shown in Table 1. Mechanics is based on just the first three of these quantities, the MKS or meter-kilogram-second system. An alternative metric system to this, still widely used, is the so-called CGS system (centimeter-gram-second). For distance and time measurements, British Imperial units (especially in the USA) based on the foot (ft), the yard (yd), the mile (mi), etc., as units of length, and also the minute, hour, day and year as units of time. Table 1 Système International (SI) System of Units Base Quantity Base Unit Length meter (m) Mass kilogram (kg) Time second (s) Electric Current ampere (A) Temperature Kelvin (K) Amount of Substance mole (mol) Luminous Intensity candela (cd) We shall refer to the dimension of the base quantity by the quantity itself, for example dim length [length] = L, dim mass [mass] M, dim time [time] T. (1) 1.2 Dimensions of Commonly Encountered Quantities Many physical quantities are derived from the base quantities by a set of algebraic relations defining the physical relation between these quantities. The dimension of the2derived quantity can always be written as a product of the powers of the dimensions of the base quantities. Example 1 Derived Dimensions of Mechanical Quantities (i) The dimensions of velocity are given by the relationship [velocity] = [length]/[time] = L T-1. (2) (ii) Force is also a derived quantity and using the definition of force F = maand acceleration a = dv / dt, force has dimensions [force] =[mass][velocity][time]. (3) We could express force in terms of mass, length, and time by the relationship [force] =[mass][length][time]2= M L T-2. (4) (iii) The derived dimension of kinetic energy follows from the definition that K =12mv2, thus [kineticenergy] = [mass][velocity]2, (5) which in terms of mass, length, and time is [kineticenergy] =[mass][length]2[time]2= M L2 T-2 (6) (iv) The derived dimension of work is [work] = [force][length], (7) which in terms of our fundamental dimensions is [work] =[mass][length]2[time]2= M L2 T-2 (8) So work and kinetic energy have the same dimensions.3(v) Power is defined to be the rate of change in time of work so the dimensions are [power] =[work][time]=[force][length][time]=[mass][length]2[time]3= M L2 T-3 (9) In Table 2 we list the derived dimensions of some common mechanical quantities in terms of mass, length, and time. Table 2 Dimensions of Some Common Mechanical Quantities M mass, L length, T time Quantity Dimension MKS unit Angle dimensionless1 Dimensionless = radian Steradian dimensionless Dimensionless = radian2 Area L2 m2 Volume L3 m3 Frequency T-1 s1= hertz = Hz Velocity L T-1 m s1 Acceleration L T-2 m s2 Angular Velocity T-1 rad s1 Angular Acceleration T-2 rad s2 Density M L-3 kg m3 Momentum M L T-1 kg m s1 Angular Momentum M L2 T-1 kg m2 s1 Force M L T-2 kg m s2= newton = N Work, Energy M L2 T-2 kg m2 s2= joule= J Torque M L2 T-2 kg m2 s2 Power M L2 T-3 kg m2 s3=watt=W Pressure M L-1 T-2 kg m1 s2= pascal= Pa 1 Even though angle and steradian are dimensionless quantities, it is often helpful to carry around a “unit” associated with them, like the radian, to understand their role in an expression or to determine if a result makes sense.41.3 Dimensional Analysis There are many phenomena in nature that can be explained by simple relationships between the observed phenomena. When trying to find a dimensional correct formula for a quantity from a set of given quantities, an answer that is dimensionally correct will scale properly and is generally off by a constant of order unity. Consider a simple pendulum consisting of a massive bob suspended from a fixed point by a string. Let T denote the time (period of the pendulum) that it takes the bob to complete one cycle of oscillation. How does the period of the simple pendulum depend on the quantities that define the pendulum and the quantities that determine the motion? What possible quantities are involved? The length of the pendulum l, the mass of the pendulum bob m, the gravitational acceleration g, and the initial angular amplitude of the bob 0 are all possible quantities that may enter into the formula for the period of the swing. Have we included every possible quantity? We can never be sure but let’s first work with this set and if we need more than we will have to think harder! Our problem is then to find a function f such that T = fl,m, g,0() (10) We first make a list of the dimensions of our quantities as shown in Table 3. Choose the set: mass, length, and time, to use as the base dimensions. Table 3 Dimensions of quantities that may describe the period of pendulum Name of Quantity Symbol Dimensional Formula Time of swing t T Length of pendulum l L Mass of pendulum m M Gravitational acceleration g L T-2 Angular amplitude of swing 0 No dimension We begin by writing the period as a product of these given quantities that have dimensions, (thus the initial angular amplitude of swing cannot enter into our expression), with each given quantity raised to a rational power, T = blXmYgZ (11) where b is a dimensionless constant. Our
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