1
• Determine the mass, length, inner and outer diameters of a hollow cylinder of unknown metal.
• Calculate the density of the hollow cylinder.
• Determine the uncertainty in the value of the calculated density caused by the uncertainties in the measured mass, length and diameters.
2
• Ruler
• Vernier caliper
• Micrometer screw gauge
• Balance
3
The most general definition of density is mass per unit volume. Density can vary throughout the body if the mass is not distributed uniformly. If the mass of an object is distributed uniformly throughout the object, the density ρ is defined as the total mass M divided by the total volume V of the object. In equation form this is
ρ =
M
V
(1)
The goal of this experiment is to measure the density of a hollow cylinder. In fact, the process involves measuring directly the mass, length, inner and outer diameters of the hollow cylinder, and calculating the density from these directly measured quantities. For a hollow cylinder, the volume is given by
V =
π
4 d
2
2
− d
2
1
L (2)
Level 1 Laboratory Page 1 of 6 Department of Physics
National University of Singapore

Measurement of Density Page 2 of 6 where L is the length of the hollow cylinder, and d
1 and d
2 are its inner and outer diameters respectively. The density of the hollow cylinder is then given by
ρ =
4 M
π ( d 2
2
− d 2
1
) L
(3)
The quantities M , L , d
1 and d
2 will be determined by measuring each of them ten times independently and calculating the mean and standard error for each quantity. Using the mean of each measured quantity in equation (3) leads to the best value for the measured density
ρ . The uncertainty in the density σ
ρ
(standard error) is related to the standard errors in the mass, length, inner and outer diameters by:
σ
ρ
= ρ v u u t
σ
M
M
2
+
σ
L
L
2
+ 4 d
1
σ d
1 d
2
2
− d
1
2
!
2
+ 4 d
2
σ d
2 d
2
2
− d
1
2
!
2
(4)
4
The mass of the hollow cylinder will be determined by a digital balance. With nothing on the balance pan, the balance reading should be zero. Otherwise, zero the digital balance by pressing the ‘ZERO’ button before any measurement is taken.
P1.
Using the digital balance, determine the mass of the hollow cylinder. Make TEN independent measurements for the mass of the hollow cylinder and record the results in
Data Table 1 as M .
The length of the hollow cylinder will be determined with a meter stick. In general, DO NOT attempt to line up either edge of the hollow cylinder with one end of the meter stick or with any certain mark on the meter stick.
P1.
Place the meter stick along the length of the hollow cylinder.
P2.
Read the scale on the meter stick that is aligned with one end of the hollow cylinder and record that measurement in Data Table 1 as X
1
.
Note: The stick has 1 millimeter as the smallest marked scale division. Therefore, each measurement should be estimated to the nearest 0 .
1 millimeter (nearest 0 .
01 cm).
P3.
Read the scale that is aligned with at the other end of the hollow cylinder and record that measurement in Data Table 1 as X
2
.
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Measurement of Density Page 3 of 6
P4.
Repeat steps P2 – P4 nine more times for a total of TEN measurements of the length of the hollow cylinder. Make the measurements at different places on the hollow cylinder in order to sample the variation in length of the hollow cylinder.
The inner diameter of the hollow cylinder will be measured with a vernier caliper (see Figure 1).
A caliper is actually any device used to determine the thickness, the diameter of an object or the distance between two surfaces. Often, calipers consist of a fixed rule containing one jaw and a second jaw with a vernier scale that slides along the fixed-rule scale. Each of the two jaws has two parts pointing in opposite directions. The span between the upper jaw is used to measure the inside diameter between two surfaces. The distance between the lower jaws is a measure of the outside diameter of objects over which it is placed.
Figure 2: A vernier scale reading.
Figure 1: Vernier caliper.
The vernier caliper shown in the Figure 2 has marked on the main scale major divisions of 5 mm for which there are both a mark and a number. On the main scale, there are also marked five divisions, each 1 mm apart between the 5 mm divisions. The 1 mm marks are not labeled with a number. This vernier is marked with a scale that, when aligned with different marks on the fixed rule scale, allows interpolation between the 1 mm marks on the fixed scale to 0 .
1 mm precision. A vernier caliper can measure distances precisely to the nearest 0 .
1 mm.
A measurement is made by closing the jaws on some object and noting the position of the zero mark on the vernier and which one of the vernier marks is aligned with some mark on the fixed rule scale. This is illustrate in Figure 2. The position of the zero mark of the vernier scale gives 1 .
0 mm and the interpolation between 1 .
0 mm and 2 .
0 mm for this case is given by the the fact that the seventh mark beyond the vernier zero is best aligned with a mark on the fixed-rule scale. The reading in this example is 1 .
7 mm.
Before making any measurements, it should be determined whether the vernier calipers read zero when the jaws are closed. If the calipers do not read zero when the jaws are closed, they are said to have a zero error. A correction must be made for each measurement made with the calipers. If the vernier zero is to the right of the fixed-scale zero when the jaws are closed, the zero error is positive. A positive zero error in the instrument gives a larger measurement
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Measurement of Density Page 4 of 6 than the actual measurement. In order to obtain the exact measurement, positive zero error is subtracted from the total reading. On the other hand, if the vernier zero is to the left of the fixed-scale zero, then the zero error is negative. In this case, the instrument gives a smaller measurement than the actual measurement and the exact measurement is obtained by adding the negative zero error to the total reading.
P1.
Close the jaws of the vernier caliper. Make a reading of the zero correction for the vernier caliper and record it in Data Table 1. Record the zero correction as positive if the vernier zero is to the right of the fixed scale zero and record it as negative if the vernier zero is to the left of the fixed scale zero.
Note: The exact measurement is obtained by subtracting the zero correction from the total reading.
P2.
Using the vernier caliper, measure the inner diameter of the hollow cylinder. Make TEN separate trials of the measurement of the inner diameter of the hollow cylinder. Measure the inner diameter at TEN different positions along the length of the hollow cylinder in order to sample the variation in inner diameter of the hollow cylinder. Record the results as d
1 in Data Table 1.
The outer diameter of the hollow cylinder will be determined by a micrometer screw gauge. A micrometer screw gauge is used to measure even smaller dimensions than the vernier calipers such as diameter of fine wire, diameters of spheres, the thickness of paper and similar small lengths. The micrometer screw gauge also uses a vernier scale which is marked on a rotary thimble. The main features of a micrometer screw gauge are shown in Figure 3.
Figure 4: A micrometer reading.
Figure 3: Micrometer screw gauge.
The precision of micrometer screw gauge depends on its screw. A pitch is the distance between two neighbouring threads of the screw. When the thimble is moved through one
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Measurement of Density Page 5 of 6 revolution, the gap between anvil and spindle will change by a distance equal to the pitch of the screw. If the thimble has a scale of n equal divisions, the thimble will move 1 /n of a pitch by turning the thimble through 1 /n of a revolution. The precision of micrometer screw gauge is 1 /n of the pitch.
In order to measure an object, the object is placed between the anvil and spindle (jaws).
The thimble is then rotated using the ratchet until the object os lightly gripped. DO NOT
OVER TIGHTEN. Note that the ratchet knob must be used to secure the object firmly between the jaws, otherwise the instrument could be damaged or given an inconsistent reading.
The lock must be used to ensure that the thimble does not rotate while the reading is taken.
Both the main scale and thimble readings are taken into account while taking measurement.
The first part of the measurement is taken from the last graduation showing on the sleeve directly to the left of the revolving thimble and the second part of the measurement is taken from the thimble which is in line with the main scale. The total reading is therefore the summation of the first and second part. As an illustration, in Figure 4, the main scale reading is 5 .
50 mm since the additional half scale below the main scale is visible. The thimble reading which is in line with the main scale is 28. Therefore, the total reading is (5 .
50 + 28 × 0 .
01) =
5 .
78 mm.
P1.
Close the jaws of the micrometer screw gauge. Make a reading of the zero correction for the micrometer and record it in Data Table 1. Record the zero correction as positive if the revolving thimble is to the right of the main scale zero and record it as negative if the revolving thimble is to the left of the main scale zero.
Note: The exact measurement is obtained by subtracting the zero correction from the total reading.
P2.
Using the micrometer screw gauge, measure the outer diameter of the hollow cylinder.
Make TEN separate trials of the measurement of the outer diameter of the hollow cylinder. Measure the outer diameter at TEN different positions along the length of the hollow cylinder in order to sample any variation in outer diameter of the hollow cylinder.
Record the results as d
2 in Data Table 1.
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National University of Singapore

Measurement of Density
5
Page 6 of 6
D1.
Calculate the mean M , the standard deviation σ
M and the standard error σ
M
TEN measurements of the mass of the hollow cylinder in Data Table 1.
for the
D2.
Determine the measured length L of the hollow cylinder for each trial for the scales of the ends of the hollow cylinder in Data Table 1. Calculate the mean L , the standard deviation σ
L and the standard error σ
L hollow cylinder.
for the TEN measurements of the length of the
D3.
Determine the measured diameters ( d
1 and d
2
) of the hollow cylinder for each trial in
Data Table 1 by making the appropriate zero correction. Calculate the means, the standard deviations and the standard errors for the TEN measurements of the inner and outer diameters of the hollow cylinder.
D4.
Determine your best experimental value for the density ρ of the hollow cylinder with the corresponding uncertainty.
Level 1 Laboratory Department of Physics
National University of Singapore