5/30/12 Chapter 1 Physical Quantities and Units « keterehsky 1/13 https://keterehsky.wordpress.com/2011/06/22/chapter-1-physical-quantities-and-units/ k eterehsky Categories A stronomi (17) B erita (7) F izik (29) M y Nikon and Me (4) U ncategorized (100) Archives M ay 2012 F ebruary 2012 J uly 2011 J une 2011 M ay 2011 A pril 2011 M arch 2011 F ebruary 2011 J anuary 2011 O ctober 2010 S eptember 2010 A ugust 2010 M ay 2010 A pril 2010 M arch 2010 F ebruary 2010 J anuary 2010 S eptember 2009 A ugust 2009 C hapter 1 Physical Quantities and Units Posted on 22/06/2011 by amimo5095 Physic is simple Simplicity is everything Introduction What is physic ? • Definition of physics – derives from Greek word means nature. • Each theory in physics involves: (a) Concept of physical quantities. (b) Assumption(andaian) to obtain mathematical model. (c) Relationship between physical concepts. - proportional (berkadar langsung) (d) Procedures to relate mathematical models to actual measurements from experiments. (e) Experimental proofs to devise explanation to nature phenomena. 1.1 Basic Quantities and International System of Units (SI units) > Physical quantity A physical quantity is a quantity that can be measured. Physical quantity consist of a
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5/30/12 Chapter 1 Physical Quantities and Units « keterehsky
> A scalar quantity is a physical quantity which has only magnitude. For example, mass, speed (laju), density, pressure, ….
> A vector quantity is a physical quantity which has magnitude and direction. For example, force, momentum, velocity (halaju), acceleration ….
Graphical representation of vectors
(http://keterehsky.files.wordpress.com/2011/06/clip_image029.gif)•A vector can be represented by a straight arrow,
The length of the arrow represents the magnitude of the vector.
The vector points in the direction of the arrow.
Basic principle of vectors
• Two vectors P and Q are equal if:
a) Magnitude of P = magnitude of Q (b) Direction of P = direction of Q
• When a vector P is multiplied by a scalar k, the product is k P and the direction remains the same as P.
The vector -P has same magnitude with P but comes in the opposite direction.
Sum of vectors
Method 1: Parallelogram of vectors
It two vectors (http://keterehsky.files.wordpress.com/2011/06/clip_image031.gif) and (http://keterehsky.files.wordpress.com/2011/06/clip_image033.gif) are represented in magnitude and direction by the adjacent sides OA and OB of a
parallelogram OABC, then OC represents their resultant(paduan).
A kite flies in still air is 4.0 ms-1. Find the magnitude and direction of the resultant velocity of the kite when the air flows across perpendicularly(serenjang) is
2.5 ms-1. If the distance of the kite is 30 m,
what is the time taken for the kite to fly? Calculate the height of the kite from the ground.
• Primary data are raw data or readings taken in an experiment. Primary data obtained using the same instrument have to be recorded to the same degree ofprecision i.e to the same number of decimal places.
• Secondary data are derived from primary data. Secondary data have to be recorded to the correct number of significant figures. The number of significantfigures for secondary data may be the same (or one more than) the least number of significant figures in the primary data. Measurement play a crucial role inphysics, but can never be perfectly precise.
It is important to specify the uncertainty or error of a measurement either by stating it directly using the ± notation, and / or by keeping only correct numberof significant figures.
Example: 51.2 ± 0.1
Processing significant figures
• Addition and subtraction
When two or more measured values are added or subtracted, the final calculated value must have the same number of decimal places as that measured valuewhich has the least number , of decimal places.
Example
1. a = 1.35 cm + 1.325 cm
= 2.675 cm
= 2.68 cm
2. b = 3.2 cm – 0.3545 cm
= 2.8465 cm
= 2.8 cm
3. c = (http://keterehsky.files.wordpress.com/2011/06/clip_image046.gif)
= 1.142 cm
= 1.14 cm
· Multiplication and division
• When two or more measured values are multiplied and/or divided, the final calculated value must have as many significant figures as that measured valuewhich has the least number of significant figures.
Example
1. Volume of a wooden block = 9.5 cm x 2.36 cm x 0.515 cm
= 11.5463 cm3
= 12 cm3
2. If the time for 50 oscillations of a simple pendulum is 43.7 s, then the period of oscillation = 43.7 ÷ 50 = 0.874 s
3. The gradient of a graph (http://keterehsky.files.wordpress.com/2011/06/clip_image0481.gif)
Note: Sometimes the final answer may be obtained only after performing several intermediate calculations. In this case, results produced in intermediatecalculations need not be rounded off. Round only the final answer.
1.4.2 Analysing error/uncertainty of a mean value
- specifically error analysing is refer to error that cause by repetition and a combining measurement to produce a derive quantity.
- Meaning that if we want to measure a volume of cube, of course we cannot just used a single measurement then we will get the answer. First we have tomeasure the length with the ruler together with the width and the height. The we need to feed in the formula length x width x height to get the volume.
- While doing the measurement and caculating the answer actually we have continually increasing the error.
- It is a good idea to mention the uncertainty for every measurement and calculation.
- In this subtopic we deal with the repetition reading or data. It’s known that if we have more than one reading so the true value is the mean of the reading.
5/30/12 Chapter 1 Physical Quantities and Units « keterehsky
Diameter ,d of a wire was measured several time to reduce the uncertainty and the reading is given in the table below. Find the true value(mean value) andthe uncertainty of the diameter.
and a =(1.83±0.01)m, b=(1.65 ±0.01) m, d=(0.00106±0.00003)m ,
q = (4.28 ± 0.05) s and T = (3.7 ± 0.1) x 103 s.
solution
First calculate the percentage uncertainties in each of the 4 terms:
(a – b) = (0.18±0.02)m 11%
d = (0.001 06 ± 0.000 03) m 3%
q = (4.28 ± 0.05) s . 1.2%
T = (3.7±0.1) x 103 s 3%
The uncertainty in (a – b) is now very large, although the readings themselves have been taken carefully. This is always the effect when subtracting two nearlyequal numbers.
The percentage uncertainty in d2 will be twice the percentage uncertainty in d;
The percentage uncertainty in (http://keterehsky.files.wordpress.com/2011/06/clip_image100.gif) will be half the percentage uncertainty in T because a
square root is a power of (http://keterehsky.files.wordpress.com/2011/06/clip_image102.gif).
This gives:
5/30/12 Chapter 1 Physical Quantities and Units « keterehsky
Uncertainty percentage in v = 11% + 2(3%) + 1.2% + (http://keterehsky.files.wordpress.com/2011/06/clip_image104.gif)(3%) = 19.7% ≈ 20%
This gives v = (7.8 ± 1.6) x 10-11 m3 s-1, a rather uncertain result which would be better expressed as:
v = (8 ± 2) x 10-11 m3 s-1
the rules for uncertainties therefore :
addition and subtraction ADD absolute uncertainties
multiplication and division ADD percentage uncertainties
powers Multiply the percentage uncertainty by the power
Note : There are some circumstances where the uncertainty in the final value is best found by working the problem through twice , once with the readings astaken and once with the limiting values which will give the maximum result. Equations containing trigonometrical ratios, or exponentials, or equations inwhich some of the terms appear both on the top and the bottom of the expression, such as
(http://keterehsky.files.wordpress.com/2011/06/clip_image106.gif) are best dealt with this way.
Example 5
The diameter of a cone is (98 ± 1)mm and the height is (224 ± 1 )mm. What is:
(a) The absolute error of the diameter.
(b) The percentage error of the diameter.
(c) The volume of the cone. Give your answer to the correct number of significant number.
Example 6
Discuss the ways of minimizing systematic and random errors
Example 7
The period of a spring is determined by measuring the time for 10 oscillations using a stopwatch. State a source of:
(a) Systematic error
(b) Random error
1.4.4. Method to find uncertainty/error from a graph
1. The usual quantities that are deduced from a straight line graph are
(a) the gradient of the graph m, and the intercept on the y-axis or the x-axis
(b) the intercepts on the axes.
First calculate the coordinates of the centroid using the formula
(http://keterehsky.files.wordpress.com/2011/06/clip_image110.gif) where n is the number of sets of readings.
2. The straight line graph that is drawn must pass through the centroid Figure . The best line is the straight line which has the plotted points closest to it. Thisline will give (http://keterehsky.files.wordpress.com/2011/06/clip_image112.gif)the best gradient together with c.
3. Two other straight lines, one with the maximum gradient (http://keterehsky.files.wordpress.com/2011/06/clip_image114.gif) and another with the
least gradient (http://keterehsky.files.wordpress.com/2011/06/clip_image116.gif), are then drawn. For a straight line graph where the intercept is not the
origin , the three lines drawn must all pass through the centroid. Here also we can find (http://keterehsky.files.wordpress.com/2011/06/clip_image118.gif)
and (http://keterehsky.files.wordpress.com/2011/06/clip_image120.gif)
4. To find the uncertainty for the gradient and intercept used this equation
5/30/12 Chapter 1 Physical Quantities and Units « keterehsky
(http://keterehsky.files.wordpress.com/2011/06/clip_image122.gif) and (http://keterehsky.files.wordpress.com/2011/06/clip_image124.gif)
Working Example
Table 1.7 shows the data collected in an experiment to determine the acceleration due to gravity using a simple pendulum. The time t for 50 oscillations of thependulum is measured for different lengths l of the pendulum. The period T is calculated using
Hence, the acceleration due to gravity, (http://keterehsky.files.wordpress.com/2011/06/clip_image130.gif)
A straight line graph would be obtained if a graph of (http://keterehsky.files.wordpress.com/2011/06/clip_image132.gif) against (http://keterehsky.files.wordpress.com/2011/06/clip_image134.gif) is plotted.
Note the various important characteristics when tabulating the data as shown in Table
(a) Name or symbol of each quantity and its unit are stated in the heading of each column. Example: Length and cm, and T(s). The uncertainty for theprimary data, such as length and t time for 50 oscillations, is also written. Example: (l ± 0.05) cm and (t ± 0.1)s.
(b) All primary data, such as length and time, should be recorded to reflect the precision of the instrument used.
For example, the length of the pendulum l is measured using a metre rule. hence it should be recorded to two decimal places of a cm, that is 10.00 cm, and not10 cm or 10.0 cm.
The time for 50 oscillations t is recorded to 0.1 s, that is 32.0 s and not 32 s.
The average value of t is also calculated to 0.1 s. The average value of 31.9 s and 32.0 s is recorded as 32.0 s and not 31.95 s.
(c) The secondary data such as T and T2, are calculated from the primary data. Secondary data should be calculated to the same number of significant figures
as I hat in the least accurate measurement. For example, T and T2, are calculated to three significant figures, the same number of significant figures as thereadings of t.
(d) For a straight line graph, there should be at least six point plotted. If the graph is a curve, then more points should be plotted, especially near themaximum and minimum points.