Understanding lab

1

Provided data for Exp 1 and instructions for data analysis and lab report

1. Provided data for Experiment 1 are given in Tables 1 to 4.

Table 1 Steel ball

Table 2 Aluminum block

Table 3 Brass cylinder

1 2 3 4 5 average

D (mm) 15.885 15.880 15.880 15.880 15.880 3

/ 6V D=

Vi i d V V= −

m (g) 16.650 16.650 16.650 16.650 16.650

 (g/mm3)

i i d

  = −

% error between the measured Fe and the accepted 3 3

:17.8 0 /Fe kg m = 

1 2 3 4 5 average

L1 (cm) 1.550 1.550 1.550 1.550 1.550

L2 (cm) 1.560 1.550 1.540 1.550 1.540

L3 (cm) 4.880 4.890 4.880 4.870 4.880

1 2 3 V L L L=  

Vi i d V V= −

m (g) 32.685 32.685 32.685 32.685 32.685

 (g/cm3)

i i d

  = −

% error between the measured Al and the accepted 3 3

:2.7 10 /Al kg m = 

1 2 3 4 5 average

D (mm) 12.670 12.665 12.670 12.660 12.660

L (cm) 3.890 3.880 3.890 3.880 3.880 2

/ 4V D L=

Vi i d V V= −

m (g) 43.800 43.790 43.790 43.790 43.790

 (g/mm3)

i i d

  = −

% error between the measured Brace and the accepted 3 3

8.9 :10 /Brass kg m = 

2

Table 4 Aluminum annular cylinder

2. Data analysis for Experiment 1

(a) To reduce the measurements errors five measurements have been made for each object as

shown in each of above 4 tables. Calculate the average (mean) dimensions of each object

and record your calculated data in each Table.

(b) Calculate the volume of each object (V  dV ), where V is the mean of volume and dV

is the mean deviation of volume. Record your calculated data in each Table.

(c) Calculate the density (   d ) of the material of each object, where  is the mean of

density and d is the mean deviation of density. Record your calculated data in each Table.

(d) Compare the measured  with accepted  of each object and calculate the percent %.

Record your calculated data in each Table.

(e) Put attention to the significant figures of your calculated data.

3. Lab report on Experiment 1

(a) Tables 1 to 4 with analyzed data must be included in your lab report.

(b) Answers to the questions #1 & #2 at the end of Exp 1 lab manual must be included in your lab

report.

(c) The required other contents and format for your lab report can be found in the syllabus.

1 2 3 4 5 average

L (cm) 0.940 0.940 0.940 0.940 0.940

D1 (cm) 1.880 1.890 1.870 1.880 1.870

D2 (cm) 0.960 0.970 0.970 0.970 0.960

( )2 2

1 2 4

L V D D

 = −

VVd iVi −=

m (g) 5.720 5.720 5.720 5.720 5.720

 (g/cm3)

 −= ii

d

% error between the measured Al and the accepted 3 3

:2.7 10 /Al kg m = 

Significant figures (sig. fig.)

Least Count: Smallest marked division of an instrument. Illustration:

Counting # of significant figures in a direct measurement: # of sig. fig in a reading = # of figures read directly from instrument

+ 1 for an estimate figure Illustration: Using a standard ruler, you measure some object’s

length to be 2.5cm. But, actual length is between the 2.5cm and 2.6cm marks on ruler. So, real length = 2.5cm + (extra). You then eye estimate the (extra) to be 0.04cm. Then you have: final length reading = 2.54cm. Then this reading have 3 sig. fig., with the last ‘4’ as the estimate figure.

Calculations with sig. fig.

Multiplication and Division: The # of sig. fig. in calculated result is same as that in the measured number with the least # of sig. fig. Decimal place of last sig. fig in each factor is irrelevant. Illustration: If the sides of a rectangle are measured to be

2.03cm and 1.234cm, measured with different instruments. Then, if we calculate its area, then it can only have 3 sig. fig. Area= (1.234 × 2.03)cm2 = 2.5 0̄502cm2≃ 2.51cm2

Addition and Subtraction: The last significant decimal place in calculated result is same as that of the term with least # of sig. fig. after the decimal place. Total # of sig. fig. in each term is irrelevant. Illustration: Say we add Potential energy 1.234J with kinetic

Energy 100.0 J, with correct sig fig each. Then total energy E is: E= (1.234+100.0)J = 101. 2̄34 J ≃ 101.2 J

Calculations with sig. fig. (contd.)

Sig fig. for constants: Constants do not contribute to determining # of sig. fig. A constant is assumed to have infinite # of sig. fig. and hence can be truncated at any necessary decimal place. Illustration: If the radius of a ring is measured to be

r = 2.035cm , then its area is Area = π r2. Then we can choose value of as 3.142, as that is the number of figures in the radius. Then,π

we have: Area= (3.142× 2.0352)cm2= 13.0 1̄172895 ≃ 13.01 cm2

About 0’s: Any 0’s on the left of 1st non-zero digit are not sig. fig. Any 0’s on the right of and in between non-zero digits are sig. fig. Illustration: 0.00010, 0.10 and 1.0 → all have 2 sig. fig.

Scientific Notation → 0.00010 J→1.0×10−4 J (2 sig. fig.)

Mass, volume and Density

Density: Density of a body is obtained through equation:

ρ = m

V

where m is mass and V is volume of the object.

Volume determination: To determine volume of an object, you need its dimensions (such as length, diameter, etc.) and plug them in correct equation.

Dimension measurements: To measure dimensions of the objects, we use vernier calipers and micrometer.

Mass measurements: To measure mass, we utilize the laboratory balance.

Vernier Calipers

Least count: The vernier calipers we use have least count of 0.01cm.

Bottom prongs measure length or outer diameters. Top prongs measure inner diameters.

Vernier Calipers: Taking Measurements

Key Idea of taking reading: Total reading = Main scale reading

+ (Least count) X (smaller Marking # on vernier scale that line up or almost line up with some main scale marking)

Offset Error

In some instruments, the 0 of main scale do not match with the 0 of vernier scale even when the prongs are touching each other, causing error. This is offset error.

Micrometer

Least count: The vernier calipers we use have least count of 0.01mm.

Micrometer: Taking Measurements

Key Idea of taking reading: Total reading = Main scale reading

+ (Least count) X (Marking # on circular Scale that is just below central line on main scale)

Percent error

When average experimentally measured value for some quantity is̄x , and its true accepted standardized value is known to be A , either through literature or through theoretical considerations, then the percent error in the measurement done through the experiment is given by:

% error = |x̄ − A| A

× 100%

End of Theory