Exp 1

Experiment 1 Basic Measurement
EXPERIMENT 1a: BASICS MEASUREMENT AND UNCERTAINTY
Objectives: i. To develop skills experience in handling of measuring devices ii. To use appropriate instruments to measure the physical quantities iii. To determine the errors associated with the measuring instruments and measuring procedure iv. To determine the density of substance and its uncertainty Theory: Measuring some physical quantities is part and parcel of any physics experiment. It is important to realise that, not all measured values are exactly the same as the actual values. This could be due to the errors that we made during the measurement, or perhaps the apparatus that we used may not be accurate or sensitive enough. Therefore, as a rule, the uncertainty of a measurement must be taken and has to be recorded together with the measured value. The uncertainty of a measurement depends on the type of measurement and on how it is done. For a quantity x with the uncertainty x , the reading written as follows:
x x
Furthermore, we can calculate, Relative uncertainty and, Percentage of uncertainty, 1.1 Single Reading (a) If the reading is taken from a single point or at the end of the scale we use: =
x x x 100% x
x % =
x =
1 (the smallest division from the scale) 2
(b) If the apparatus uses a vernier scale:
x = 1 (the smallest division from the vernier scale)

(c) If the readings are taken from two points on the scale:
1 x = 2 [ [ ] (the smallest division from the scale)] 2

= the smallest division from the scale

This applies to measurements made using electric meters, stopwatch, vernier calipers and micrometer screw gauge.
1.2
Repeated Readings
For a set of n repeated measurements of x, the best value is the average value:
xi
x=
where
i =1
1.1
n = the number of readings taken xi= the ith reading
The uncertainty is given by:
x =
i =1
xx n
1.2
The result should be written as:
x = x x
1.3 Density
1.3
The density, of a substance is given by:
=
where,
m V
1.4
m = mass of the substance V = volume of the substance
This may be determined experimentally by measuring the mass and volume of a given sample. The volume of regularly shaped object can be calculated from length measurements; for example: (i) Cuboid
V = l w h
V l w h = + + V l w h
where,
1.5 h= height of the rectangle
l = length , w= width
and

(ii) Cylinder
V = (r 2 )l
V r l =2 + V r l
where, r =radius and l = length of the cylinder
1.6
(iii) Sphere
4 V = r 3 3
V r =3 V r
where r = radius of the sphere 1.7
The uncertainty of density is given by:
m V = + m V
Apparatus: i) ii) iii) iv) v) vi) vii) viii) ix) x) xi) A wooden block A meter ruler A vernier caliper A boiling tube A graduated cylinder A pebble A string (40 cm) A micrometer screw gauge A 20 cm copper wire SWG 16 (or lower) A ball bearing A balance (preferably electronic balance)
1.8
Instructions: 1. Using a meter ruler, record the dimensions of the wooden block including their uncertainties. 2. Using a vernier calliper, determine the thickness of the boiling tube wall including its uncertainty. 3. Using instruments and materials listed above, determine the density and its uncertainty of the wooden block, the pebbles, the copper wire and the ball bearing.
Experiment 1b Experimental Uncertainty (Error) and Data Analysis
1. Significant Figures (a) Express the numbers listed in Table 1 to three significant figures, writing the numbers in normal notation in the first column and the numbers in power-of-10 (scientific) notation in the second column.
Table 1
0.524 ____________________ 15.08 ____________________ 1444 ____________________
5280 0.060 82.453 0.00010
____________________ ____________________ ____________________ ____________________
0.0254 ____________________ 83,909 ____________________
2,700,000,000 ____________________
(b)
In an experiment to determine the value of , a cylinder is measured to have an average value of 4.25 cm for its diameter and an average value of 13.39 cm for its circumference. What is the experimental value of to the correct number of significant figures? Given that: = circumference, C / diameter, D.
Calculations (show work)
Experimental Value of :

2. Expressing Experimental Error If the accepted value of is 3.1416, what is the fractional error and the percent error of the experimental value found in 1 (b)?
(a)
Fractional Error : Percent Error : (b) In an experiment to measure the acceleration g due to gravity, two values, 9.96 ms-2 and 9.72 ms-2 are determined. Find (i) the percentage difference of the measurements (ii) the percentage error of each measurement and (iii) the percenaget error of their mean (Accepted value: g = 9.81 ms-2 ) Calculations (show work)
Percent Difference : Percent Error of E1: Percent Error of E2: Percent Error of Mean : (c) Table 2 shows data taken in a free fall experiment. Measurement were made of the distance of fall (y) at each of four precisely measured times. Complete the table. Use only the proper number of significant figures in your table entries, even if you carry extra digits during your intermediate calculations.
Table 2 Time t (s) 0.00 0.50 0.75 1.00 1.25 Distance (m) y1 y2 y3 y4 y5 (Optional)
t2
0.0 0.0 0.0 0.0 0.0 1.0 1.4 1.1 1.4 1.5 2.6 3.2 2.8 2.5 3.1 4.8 4.4 5.1 4.7 4.8 8.2 7.9 7.5 8.1 7.4
(d)
Plot a graph of y versus t (optional: with 2 d ) error bars for the free fall data in part (c). Remember that t = 0 is a known point. The equation of motion for an object in free fall starting from rest is y =
(e)
1 2 gt , where g 2
is the acceleration due to gravity. This is the equation of a parabola, which has the general form y = ax 2 . Convert the curve to a straight line by plotting y versus t2. That is, plot the square of the time on the abscissa. (Optional: with 2 d error bars). Determine the slope of the line and compute the experimental value of g from the slope value.
Experimental Value of g from graph :
(f)
Compute the percent error of the experimental value of g determined from the graph in part (e). (Accepted value: g = 9.81 ms-2 ).
Percent Error :

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Experiment 1 Basic Measurement

EXPERIMENT 1a: BASICS MEASUREMENT AND UNCERTAINTY

1 (the smallest division from the scale) 2

(b) If the apparatus uses a vernier scale:

x = 1 (the smallest division from the vernier scale)

1 x = 2 [ [ ] (the smallest division from the scale)] 2

Experiment 1 Basic Measurement

n = the number of readings taken xi= the ith reading

The uncertainty is given by:

The result should be written as:

The density, of a substance is given by:

m = mass of the substance V = volume of the substance

1.5 h= height of the rectangle

Experiment 1 Basic Measurement

The uncertainty of density is given by:

Experiment 1 Basic Measurement

Experiment 1 Basic Measurement

Experiment 1b Experimental Uncertainty (Error) and Data Analysis

0.524 ____________________ 15.08 ____________________ 1444 ____________________

5280 0.060 82.453 0.00010

____________________ ____________________ ____________________ ____________________

0.0254 ____________________ 83,909 ____________________

Calculations (show work)

Experiment 1 Basic Measurement

Calculations (show work)

Experiment 1 Basic Measurement

Experimental Value of g from graph :

Calculations (show work)

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0.524 15.08 1444

0.0254 83,909