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Chapter 1 Units and Measurement In this chapter we will explore the following concepts:

Chapter 1 Units and Measurement In this chapter we will explore the following concepts: 1.1 The Scope and Scale of Physics 1.2 Units and Standards 1.3 Unit Conversion 1.4 Dimensional Analysis 1.6 Significant Figures 1.7 Solving Problems Physics. 1.1 The Scope and Scale of Physics

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Chapter 1 Units and Measurement In this chapter we will explore the following concepts:

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  1. Chapter 1 Units and Measurement In this chapter we will explore the following concepts: 1.1 The Scope and Scale of Physics 1.2 Units and Standards 1.3 Unit Conversion 1.4 Dimensional Analysis 1.6 Significant Figures 1.7 Solving Problems Physics

  2. 1.1 The Scope and Scale of Physics In Physics we carry out experiments in which we measure physical parameters. We then try to deduce the relationship between the measured quantities. We usually express this relationship in the form of a mathematical equation which we call the “Physical Law” that describes the phenomenon under study. A familiar example is Newton’s Force law, F=ma. An experiment might be to measure the acceleration due to constant force. The resulting acceleration is inversely proportional to mass. If we plot a versus F we get a straight line. This is expressed in the form: The equation is known as: “Newton’s 2nd Law” F = applied force m = is the mass a = acceleration “Model”- physical relation between parameters that are often too difficult or impossible to express in a simple relationship. a slope=1/m F

  3. 1.2 Units and Standards As scientific knowledge advanced in the 18th/19th centuries and onward scientists needed a common set of basics units so they might easily communicate ideas. This was also driven by commerce, etc. For every physical parameter we will need the appropriate units i.e. a standard by which we carry out the measurement by comparison to the standard. This does not mean that we have to define new units for all parameters. We can express all mechanical parameters, force, energy, momentum, work, in terms of three parameters: Length , Time, andMass Note: For parameters in electricity and magnetism we need to define only one more unit, that of the electric current. International System of Units (SI) or “metric system” ParameterUnit NameSymbol Length meter m Timesecond s Masskilogram kg

  4. A earth C equator B The meter (m) In 1792 the meter was defined to be one ten-millionth of the distance from the north pole to the equator. For practical reasons the meter was later defined as the distance between two fine lines on a standard meter bar made of platinum-iridium. Since 1983 the meter is based on the speed of light “c”, and defined as the length traveled by light in vacuum during the time interval of 1/299792458 of a second. The reason why this definition was adapted was that the measurement of the speed of light “c” had become extremely precise. “c” is thought to be a universal constant. 10,020 km

  5. The Second (s) The second was defined in terms of the length of a day. The solar day is getting longer as a result of the very gradual slowing of Earth’s rotation. The length of the day is not constant as is shown in the figure. Since 1967 the second is defined as the time taken by 9192631770 light oscillations of a particular wavelength emitted by a cesium-133 atom. This definition is so precise that it would take two cesium clocks 6000 years before their readings would differ more than 1 second.

  6. The kilogram(kg) The SI standard of mass is a platinum-iridium cylinder shown in the figure. The cylinder is kept at the International Bureau of Weights and Measures near Paris and assigned a mass of 1 kilogram. Accurate copies have been sent to other countries. The new method of defining the kg Is to use the universal constant h = Planck’s constant! The kg is then defined by taking the fixed numerical value of h = 6.62607015×10−34 kg⋅m2⋅s−1, and solving forkg with m=1 meter and s =1 second defined as before. Planck’s constanthis a fundamental constant in quantum theory.

  7. 1.6 Significant Figures How many significant figures are in a number: 1) Non-zero digits are always counted as significant. 2) Any zeros between two significant digits are counted as significant. 3) ONLY trailing zeros (to the rhs)in the decimal portion are counted as significant. Best to use scientific notation here.     https://www.khanacademy.org/math/arithmetic-home/arith-review-decimals/arithmetic-significant-figures-tutorial/v/more-on-significant-figures 0.00700 -> 7.00x10-3 3 sigfi 0.057 -> 5.7x10-2 2 sigfi 370. -> 3.70x102 3 sigfi 705.001 -> 6 sigfi 37,000 -> 3.7x104 2 sigfi 37,000. -> 3.7000x104 5 sigfi Case of Trailing Zeros 0.03 = 3.x10-2 1 sigfi 0.030 = 3.0x10-2 2 sigfi 0.0300 =3.00x10-2 3 sigfi

  8. 1.6 Accuracy and Precision  Accuracy reflects how well a measurement agrees with its standard value.  Precision reflects how repeatable a measure is. It depends strongly on the measurement method and measuring instrument. The uncertainty, ±ΔL, is a measure of how precise a group of measurments of L are.  If we measure L with a ruler (smallest division 1mm) I may say that I could measure to ±1mm precision and write L=1.234 m. L is written to 4 significant figures.  If on the other hand I use a caliper (precision ±0.1 mm) to measure L, I can write L=1.2345 m or 5 significant figures.  In a calculation we are only allowed to write an answer keeping the least number of significant figures.

  9. 1.7 Solving Problems in Physics Problem statement State the problem to make it clear to all. Strategy Phase Examine the problem to determine which physical principles are involved. Make a list of what is given or can be inferred from the problem as stated. Identify exactly what needs to be determined in the problem. Determine which physical principles can help you solve the problem. Solution Phase Do the algebra, calculus, geometry, or arithmetic necessary to find the unknown from the knowns. Checking Phase. Check that the answer is reasonable. Check your units.

  10. #16. Roughly how many heartbeats are there in a lifetime (80y)? We find the time for 800y in seconds. We take 72 beats per minute, the time for a heartbeat. We divide to find the number of heartbeats. TLife = 80 y (3.15 x 107 s/y) =2.52B s Tbeat= 0.83 s n = Tlife/Tbeat ~ 3B This looks reasonable! #37. Mount Everest, at 29,028 ft, is the tallest mountain on Earth. What is its height in kilometers? (Assume that 1 m = 3.281 ft.) We use a unit conversion factor. h =(29,028 ft)(1m/3.281 ft)=8847 m this looks reasonable! #67. Suppose your bathroom scale reads your mass as 65 kg with a ±3% precision. What is the error uncertainty in your mass (in kilograms)? We scale my mass reading by 3% to determine the ±Δm uncertainty. ±Δm = ±0.03 x 65 kg = ±2.0 kg (2 significant figures)

  11. 1.6 Precision in Lab Experiments In a lab experiment you will record some data readings “x” with an associated error “Δx”. Δx will depend on the measuring instrument ,person doing the measurement, lab environment, etc. Statistical theory suggests measurements of “x” will fluctuate about a mean value with a standard deviation or precision The standard deviation of the mean is how well the mean is determined. σ σ 0 1 2 3 4 5 6 x Bell or Gaussian Curve www.phy.olemiss.edu/PHYS211/~cremaldi/stdev_calculator.xlsx

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