Mastering Measurement in General Physics

1. Phy 201: General Physics I Chapter 1 Measurement Lecture Notes

2. Observations • The sciences are ultimately based on observations of the natural (& unnatural) world • There are 2 types of observations: • Qualitative • Subjective, touchy-feely Example: the outside temperature is hot today • Quantitative • Objective, based on a number and a reference scale • Quantitative observations are referred to as measurements Example: the outside temperature is 80oF today Notes: • Quantitative observations are only as reliable as the measurement device and the individual(s) performing the measurement • The accuracy associated with a measurement (or set of measurements) is the often specified as the % Error: • The precision associated with a set of measurements is the often specified as the % Range:

3. Systems of Measurement • There are several units systems for measurement of physical quantities • The most common unit systems are the metric and the USCS systems • For consistency, the l’Systeme Internationale (or SI) was adopted • The SI system is a special set of metric units • International System (SI) base units: Mass Kilogram kg Length meter m Time second s Temperature Kelvin K Current Ampere A Luminous Intensity candela cd Amount of substance mole mol • All of the other SI units are derived from these base units Examples of derived units: 1 Newton = 1 N = 1 kg.m/s2 1 Joule = 1 J = 1 kg.m2/s2 1 Coulomb = 1 C = 1 A.s

4. Common Metric Prefixes • Using Metric prefixes: • 1 mm = 1x10-3 m  35 mm = 35 x 10-3 m or 3.5 x 10-2 m • 1 kg = 1x103 g  12 kg = 12 x 103 g or 1.2 x 104 g

5. Unit Conversion In physics, converting units from one unit system to another (especially within the Metric system) can appear daunting at first glance. However, with a little guidance, and a lot of practice, you can develop the necessary skill set to master this process Example: How is 25.2 miles/hour expressed in m/s? • Eliminate: {assign mi units to the denominator and hr units to the numerator of the conversion factor} • Replace: {assign m units to the numerator and s units to the denominator of the conversion factor} 3. Relate: {assign the corresponding value to its unit, 1 mi = 1609 m & 1 hr = 3600 s}

6. Length, Time & Mass Length is the 1-D measure of distance • Quantities such as area and volume, and their associated units, are ultimately derived from measures of length Definition of SI Unit: The meter is the length of the path traveled by light (in vacuum) during a time interval of 1/299,792,458 s (or roughly 3.33564 ns) Examples of units derived from length (in this case radius, r): • Area of a circle: {units are m2} • Volume of a sphere: {units are m3} Time is the physical quantity that measures either: __ when an event took place __ the duration of the event Definition of the SI Unit: The second is the time taken by 9,192,631,770 oscillations of the light emitted by a cesium-133 atom Massis the measure of inertia for a body (or loosely speaking the amount of matter present) Definition of the SI Unit: The kilogram is the amount of mass in a platinum-iridium cylinder of 3.9 cm height and diameter.

7. C A q B Trigonometry(remember: SOHCAHTOA) • The relationships between sides and angles of right triangles • Consider the following right triangle: The 3 primary relations between the sides and angles are: • The Sine: • The Cosine: • The Tangent:

8. Vectors & Scalars Most physical quantities can categorized as one of 2 types (tensors notwithstanding): • Scalars: • described by a single number & a unit (s). Example: the length of the driveway is 3.5 m • Vectors: • described by a value (magnitude) & direction. Example: the wind is blowing 20 m/s due north • Vectors are represented by an arrow: • the length of the arrow is proportional to the magnitude of the vector. • The direction of the arrow represents the direction of the vector

9. A Ay Ay = + = Ax Ax + = A B R + = A B R Properties of Vectors • Only vectors of the same kind can be added together • 2 or more vectors can be added together to obtain a “resultant” vector • The “resultant” vector represents the combined effects of multiple vectors acting on the same object/system • Direction as well as magnitude must be taken into account when adding vectors • When vectors are co-linear they can be added like scalars • Any single vector can be treated as a “resultant” vector and represented as 2 or more “component vectors • To add vectors of this type requires sophisticated mathematics or use of graphical techniques

10. q R B A Vector Addition A. Graphic Method • To add 2 vectors, place them tail-to-head, without changing their direction; the sum (resultant) is the vector obtained by connecting the tail of the first vector with the head of the second vector • R=A+B means“the vector R is the sumof vectors A and B” • R¹ A+B : the magnitude of the vector R isNOT equal to the sum of the magnitudes of vectors A and B R2 = A2 + B2+ 2AB cos q • Note: • For co-linear vectors pointing in the same direction, R = A + B • For co-linear vectors pointing in opposite directions, R = A - B

11. Vector Addition (cont.) B. Component Method • Express each vector as the sum of 2 perpendicular vectors. The direction of each component vector should be the same for both vectors. It is common to use the horizontal and vertical directions (These vectors are the horizontal and vertical components of the vector) Example: vector AAx (horizontal) and Ay(vertical) or vector BBx (horizontal) and By(vertical) or Note: The unit vectors x and y indicate the directions of the vector components • The common component vectors for A & B can now be added together like scalars to obtain the component vectors for the resultant vector: Rx = Ax + Bxand Ry = Ay + Bythus:

12. y R Ry q x Rx Vector Addition (cont.) • To calculate the magnitude of the resultant from the component vectors by using the Pythagorean Theorem: • The angle of the resultant vector (from the x axis) obtained from the ratio of component vectors: • To calculate the components from the magnitude R and the angle it makes with the horizontal direction:

13. J. Willard Gibbs (1839-1903) • American mathematician & physicist • Considered one of the greatest scientists of the 19th century • Major contributions in the fields of: • Thermodynamics & Statistical mechanics • Formulated a concept of thermodynamic equilibrium of a system in terms of energy and entropy • Chemistry • Chemical equilibrium, and equilibria between phases • Mathematics • Developed the foundation of vector mathematics "A mathematician may say anything he pleases, but a physicist must be at least partially sane."