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UNITS, ERROR CALCULATION, AND GRAPHING 1. UNITS In order that measurement may be meaningful to everybody, units of measurements are standardized internationally. Such a system of units is called System Internationale (S.I). The following are some of the commonly used S.I units. Quantity measured Length Mass Time Electric Current Temperature Unit used Meter Kilogram Second Ampere Kelvin abbreviation m kg s A K Length is measured using a meter ruler. For measuring small lengths or distances a meter is divided into smaller units such as the centimeter (cm), millimeter (mm), micrometer (m), nanometer (nm), picometer (pm) etc. Long distances are measured in kilometers (km) 102 cm = 1 m 103 mm = 1 m 106 m = 1 m 109 nm = 1 m 1012 pm = 1 m 1 km 1 cm = 10-2 m 1 mm = 10-3 m 1 m = 10-6 m 1 nm = 10-9 m 1 pm = 10-12 m = 1000 m In a similar way a kilogram is divided into grams (g), milligram (mg), microgram (g), nanogram (ng) and so on. There are different forms of laboratory scales that you may use to measure mass. The most popular is the electronic balance that you see along the side benches. . Question: Perform the following conversions: 1.1. 1.2 mm =……………..cm 1.2. 1.2 mm =……………..m 1.3. 17.6 cm =……………..m 1.4. 356 g =……………..kg 1.5. 586 mg =……………..kg 1.6. 15 m =……………..cm 1.7. 1.51 m =……………..mm 1.8. 252 m =……………..km 1.9. 500 km =……………..m 1.10. 0.5 km =……………..cm The units of length, mass, time, temperature and electric current are called FUNDAMENTAL UNITS. The units of a lot of other quantities that we measure in the lab are obtained by suitably combining these fundamental units. For example, the unit of area is m2, the unit of speed is meter per second (m.s-1) and so on. Such units are called DERIVED UNITS. Units are written in the scientific notation form. For example m/s is written as ms-1. Question: Perform the following conversions: 2.1. 15 cm2 =…………….m2 2.2. 226 mm2 =…………….m2 2.3. 1 m2 =…………….cm2 2.4. 1m3 =…………….cm3 2.5. 151 cm3 =…………….m3 2.6. 1 m3 =…………….mm3 Question: Convert 80 kmh-1 to ms-1 Question: Convert 3 gcm-3 to kgm-3 Question: 2. What is the easiest way to convert gcm-3 to kgm-3? ERROR IN MEASUREMENT The accuracy of a measurement is limited by the accuracy of the measuring instrument. Every instrument has a least count which is the least measurement that can be accurately measured by it. We will take the smallest division on the instrument as its least count. The smallest division on a meter rule is 1 mm. When you measure the length of a table with a meter rule, the maximum accuracy is 1 mm. If you obtain the length of the table as 2.72 m, the actual length may be anywhere between 2.71m and 2.73 m. The possible error in this case is 1 cm. The difference between the actual value and the measured value is called the error. The error could be positive or negative. A better way to understand error is to obtain the percent error. Percent error = actual value − measured value actual value ×100 Some times the actual value of a measurement may not be known. In such a case, we find the percent difference between measurements from two sources. Percent difference = difference of measured values average value ×100 Question: John measured the length of the table as 2.53 m. If the actual length of the table is 2.55 m, what is the percent error in the measurement? Question: Newton measured the speed of sound in air as 310 ms-1. If the actual value of the speed of sound in air at room temperature is 340 ms-1, calculate the percent error in Newton’s measurement. Question: Amy measured the density of copper as 7.2 × 103 kgm-3. Jamie measured it as 8.1 × 103 kgm-3. (a) Calculate the percent difference between the two measurements. (b) If the actual density of copper is 8.93 × 103 kgm-3, calculate the percent error in each of the measurement. Question: 3. Joyce measured the thickness of a sheet of paper and obtained the value 1.2 × 10-3 m. Mark measured the thickness of the same sheet of paper and obtained the value 0.98 × 10-4 m. Calculate the percent difference between these two measurements. GRAPHING We use a graph to study the relationship between two or more measured quantities. Very often we study the relation between a dependent variable and an independent variable. For example the relation between the position of a moving object and time can be written as: y=2t+5 Here y is the position of the object which depends on the time t. Therefore, t is the independent variable and y is the dependent variable. We graph the independent variable along the x axis and the dependent variable along the y axis. Question: y = mx + b is a linear relation between x and y. What does this mean? What is the shape of the graph of y against x? What does m and b represent in this relation? Question: What is the shape of the graph of y against t of y = 2t + 5? What do the numbers 2 and 5 represent? Make a data table for t and y and perform a linear fit using excel. Also, the equation for linear fit must be displayed on the graph. t y 1 7 2 3 4 5 1 T gives the relation between frequency (f) and 2l m tension (T) of a stretched string of length l and linear density m. If T is the independent variable and f the dependent variable with l and m constant, (a) write this equation in the form y = mx + b and identify y, x, m and b. (b) Sketch a graph and show these on the graph. The equation f = Question: (a)………………………………………………………………………… m =………………………….. b =……………………………………… (b) Question: The distance (x) traveled by an object moving with a constant acceleration a for a time t is given by the relation: 1 x = at 2 2 (a) Write the equation in the form y = mx + b and identify m and b. What is the independent variable and what is the dependent variable? (b) How will you draw a graph using this relation to obtain a straight line? (c) What is the slope of this straight line graph? The following table lists the distance traveled against time of a car that starts from rest. Use this data to obtain a straight line graph on a graph paper. Use your graph to obtain the acceleration a of the car. xm ts 0 0 2 1 8 2 18 3 32 4 50 5 72 6 98 7 128 8 162 9 200 10 Tabulate t2 values in the data table and plot a graph of x on the vertical axis and t2 on the horizontal axis (You need to include this graph with the report and graph must be plotted using excel or any other software. Hand drawn graphs will not be accepted.) Measure and record its slope with correct unit. Use the measured slope to find the value of a. The value of a obtained from the graph =…………………………………… …………………………………………………………………………………………….
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UNITS, ERROR CALCULATION, AND GRAPHING
1.

UNITS

In order that measurement may be meaningful to everybody, units of measurements are
standardized internationally. Such a system of units is called System Internationale (S.I).
The following are some of the commonly used S.I units.
Quantity measured
Length
Mass
Time
Electric Current
Temperature

Unit used
Meter
Kilogram
Second
Ampere
Kelvin

abbreviation
m
kg
s
A
K

Length is measured using a meter ruler. For measuring small lengths or distances a meter
is divided into smaller units such as the centimeter (cm), millimeter (mm), micrometer
(m), nanometer (nm), picometer (pm) etc. Long distances are measured in kilometers
(km)
102 cm = 1 m
103 mm = 1 m
106 m = 1 m
109 nm = 1 m
1012 pm = 1 m
1 km

1 cm = 10-2 m
1 mm = 10-3 m
1 m = 10-6 m
1 nm = 10-9 m
1 pm = 10-12 m

= 1000 m

In a similar way a kilogram is divided into grams (g), milligram (mg), microgram (g),
nanogram (ng) and so on. There are different forms of laboratory scales that you may use
to measure mass. The most popular is the electronic balance that you see along the side
benches.
.
Question:

Perform the following conversions:

1.1.

1.2 mm

= 12 cm

1.2.

1.2 mm

= 0.012 m

1.3.

17.6 cm

= 0.176 m

1.4.

356 g

= 0.356 kg

1.5.

586 mg

= 0.000568 kg

1.6.

15 m

= 1500 cm

1.7.

1.51 m

= 1510 mm

1.8.

252 m

= 0.252 km

1.9.

500 km

= 500 000 m

1.10. 0.5 km

= 50 000 cm

The units of length, mass, time, temperature and electric current are called
FUNDAMENTAL UNITS. The units of a lot of other quantities that we measure in the
lab are obtained by suitably combining these fundamental units. For example, the unit of
area is m2, the unit of speed is meter per second (m.s-1) and so on. Such units are called
DERIVED UNITS. Units are written in the scientific notation form. For example m/s is
written as ms-1.
Question:

Perform the following conversions:

2.1.

15 cm2

= 0.0015 m2

2.2.

226 mm2

= 0.000226 m2

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