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UNITS (Systéme Internationale) Dimension SI (mks) Unit Definition Length meters (m) Distance traveled by light in 1/(299,792,458) s Mass kilogram (kg) Mass of a specific platinumiridium allow cylinder kept by Intl. Bureau of Weights and Measures at Sèvres, France Time seconds (s) 9,192,631,700 oscillations of cesium atom Standard Kilogram at Sèvres Conversion Factors • They are simply ratios that are equal to 1 e.g. • These ratios make converting units easy! e.g. Conversion Factors • The most holy of conversion factors: 1 inch = 2.54 centimeters Dimensional Analysis Dimensions & units can be treated algebraically. Variable from Eq. x m t v=(xf-xi)/t a=(vf-vi)/t dimension L M T L/T L/T2 Dimensional Analysis Checking equations with dimensional analysis: 1 x f xi vi t at 2 2 (L/T2)T2=L L (L/T)T=L • Each term must have same dimension • Two variables can not be added if dimensions are different • Multiplying variables is always fine • Numbers (e.g. 1/2 or p) are dimensionless Example 1.1 Check the equation for dimensional consistency: 2 mc 2 mgh mc 2 1 (v / c ) Here, m is a mass, g is an acceleration, c is a velocity, h is a length Example 1.2 Consider the equation: v2 Mm m G 2 r r Where m and M are masses, r is a radius and v is a velocity. What are the dimensions of G ? L3/(MT2) Example 1.3 Given “x” has dimensions of distance, “u” has dimensions of velocity, “m” has dimensions of mass and “g” has dimensions of acceleration. Is this equation dimensionally valid? (4 / 3)ut x 1 (2gt 2 / x) Yes Is this equation dimensionally valid? vt x 1 mgt 2 No Units vs. Dimensions Dimensions: L, T, M, L/T … Units: m, mm, cm, kg, g, mg, s, hr, years … When equation is all algebra: check dimensions When numbers are inserted: check units Units obey same rules as dimensions: Never add terms with different units • Angles are dimensionless but have units (degrees or radians) • In physics sin(Y) or cos(Y) never occur unless Y is dimensionless • • • • • Example 1.3 Grandma traveled 27 minutes at 44 m/s. How many miles did Grandma travel? 44.3 miles Prefixes In addition to mks units, standard prefixes can be used, e.g., cm, mm, mm, nm Example 1.4a 40 m 11cm ? The above expression yields: a) b) c) d) 40.11 m 4011 cm A or B Impossible to evaluate (dimensionally invalid) Example 1.4b 1.5 m 3.0 kg ? The above expression yields: a) b) c) d) 4.5 m kg 4.5 g km A or B Impossible to evaluate (dimensionally invalid) Example 1.4b 1.5 m-3.0 kg m/s ? The above expression yields: a) b) c) d) -1.5 m -1.5 kg m2 -1.5 kg Impossible to evaluate (dimensionally invalid)