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Measurement

Measurement. 1.1 Standards of Length, Mass, and Time 1.2 Matter and Model Building 1.3 Dimensional Analysis 1.4 Conversion of Units 1.5 Estimates and Order-of- Magnitude Calculations 1.6 Significant Figures. 1.1 Standards of Length, Mass, and Time.

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Measurement

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  1. Measurement 1.1 Standards of Length, Mass, and Time 1.2 Matter and Model Building 1.3 Dimensional Analysis 1.4 Conversion of Units 1.5 Estimates and Order-of- Magnitude Calculations 1.6 Significant Figures

  2. 1.1 Standards of Length, Mass, and Time • If we are to report the results of a measurement to someone who wishes to reproducethis measurement, a standard must be defined. • In 1960, an international committee established a set of standards for the fundamentalquantities of science. It is called the SI (Système International), and itsfundamental units of length, mass, and time are the meter, kilogram, and second,respectively. Other standards for SI fundamental units established by the committeeare those for temperature(the kelvin), electric current (the ampere), luminousintensity (the candela), and the amount of substance (the mole).

  3. Length ;We can identify length as the distance between two points in space.

  4. Time ? • Before 1967, the standard of time was defined in terms of the mean solar day. (A solarday is the time interval between successive appearances of the Sun at the highest pointit reaches in the sky each day.) The fundamental unit of a second (s) was defined as 1/60 1/60 1/24of a mean solar day. This definition is based on the rotation of one planet,the Earth. Therefore, this motion does not provide a time standard that is universal.

  5. UNITS (SystémeInternationale)

  6. Dimensional Analysis • In physics, the word dimension denotes the physical nature of a quantity. The distancebetween two points, for example, can be measured in feet, meters, or furlongs,which are all different ways of expressing the dimension of length. Dimensions & units can be treated algebraically.

  7. Dimensional Analysis Checking equations with dimensional analysis: (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

  8. Example The expression yields: 40.11 m 4011 cm A or B Impossible to evaluate (dimensionally invalid) -1.5 m -1.5 kg m2 -1.5 kg Impossible to evaluate (dimensionally invalid)

  9. PROBLEM Find a relationship between a constant acceleration a, speed v, and distance r from the origin for a particle traveling in a circle. STRATEGY Start with the term having the most dimensionality, a. Find its dimensions, and then rewrite those dimensionsin terms of the dimensions of v and r. The dimensions of time will have to be eliminated with v, because that’s theonly quantity (other than a, itself) in which the dimension of time appears.

  10. Example 1.1 Check the equation for dimensional consistency: Here, m is a mass, g is an acceleration,c is a velocity, h is a length

  11. Example 1.2 Consider the equation: Where m and M are masses, r is a radius andv is a velocity. What are the dimensions of G ? L3/(MT2)

  12. 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? Yes Is this equation dimensionally valid? No

  13. Conversion of Units The speed limit is given in both kilometersper hour and miles per hour on this road sign. How accurate is the conversion? Sometimes it’s necessary to convert units from one system to another. Conversionfactors between the SI and U.S. customary systems for units of length are as follows: 1 mi=1 609 m =1.609 km 1 ft =0.304 8 m =30.48 cm 1 m = 39.37 in. = 3.281 ft 1 in. = 0.025 4 m = 2.54 cm 1 in. = 2.54 cm,

  14. PROBLEM If a car is traveling at a speed of 28.0 m/s, is the driver exceeding the speed limit of 55.0 mi/h? STRATEGY Meters must be converted to miles and seconds to hours, using the conversion factors listed on the front endsheetsof the book. Here, three factors will be used. PROBLEM The traffic light turns green, and the driver of a high-performance car slams the accelerator to the floor. Theaccelerometer registers 22.0 m/s2. Convert this reading to km/min2.

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