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The data in the following table represent measurements of the masses and dimensions of solid cylinders of aluminum, copper, brass, tin, and iron. (a) Use these data to calculate the densities of these substances. (b) State how your results compare with those given in Table 14.1.
(a)
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The densities of each substance.
Answer to Problem 1.61AP
The density of aluminum solid cylinders is 2.75 g/cm3, density of copper solid cylinders is 9.36 g/cm3, brass solid cylinders is 8.91 g/cm3, tin solid cylinders is 7.68 g/cm3 and iron solid cylinders is 7.88 g/cm3.
Explanation of Solution
Given info: The mass, diameter and length of each substance are given below,
Substance | Mass (g) | Diameter (cm) | Length (cm) |
Aluminum | 51.5 | 2.52 | 3.75 |
Copper | 56.3 | 1.23 | 5.06 |
Brass | 94.4 | 1.54 | 5.69 |
Tin | 69.1 | 1.75 | 3.74 |
Iron | 216.1 | 1.89 | 9.77 |
Formula to calculate the density of substance is,
ρ=mV (1)
Here,
m is the mass of substance.
V is the volume of substance.
Write the expression for the volume of solid cylinder,
V=π(d2)2l
Here,
d is the diameter of the substance.
l is length of the substance.
Substitute π(d2)2l for V in the equation (1).
ρ=mπ(d2)2l=4mπd2l (2)
For aluminum:
Substitute 51.5 g for m, 2.52 cm for d and 3.75 cm for l in the equation (2).
ρ=4(51.5 g)π(2.52 cm)2(3.75 cm)=2.75 g/cm3
Thus, the density of aluminum solid cylinders is 2.75 g/cm3.
For copper:
Substitute 56.3 g for m, 1.23 cm for d and 5.06 cm for l in the equation (2).
ρ=4(56.3 g)π(1.23 cm)2(5.06 cm)=9.36 g/cm3
Thus, the density of copper solid cylinders is 9.36 g/cm3.
For brass:
Substitute 94.4 g for m, 1.54 cm for d and 5.69 cm for l in the equation (2).
ρ=4(94.4 g)π(1.54 cm)2(5.69 cm)=8.91 g/cm3
Thus, the density of brass solid cylinders is 8.91 g/cm3.
For tin:
Substitute 69.1 g for m, 1.75 cm for d and 3.74 cm for l in the equation (2).
ρ=4(69.1 g)π(1.75 cm)2(3.74 cm)=7.68 g/cm3
Thus, the density of tin solid cylinders is 7.68 g/cm3.
For iron:
Substitute 216.1 g for m, 1.89 cm for d and 9.77 cm for l in the equation (2).
ρ=4(216.1 g)π(1.89 cm)2(9.77 cm)=7.88 g/cm3
Thus, the density of iron solid cylinders is 7.88 g/cm3.
Conclusion:
Therefore, the density of aluminum solid cylinders is 2.75 g/cm3, density of copper solid cylinders is 9.36 g/cm3, brass solid cylinders is 8.91 g/cm3, tin solid cylinders is 7.68 g/cm3 and iron solid cylinders is 7.88 g/cm3.
(b)
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The comparison between results of part (a) and table 14.1.
Answer to Problem 1.61AP
The density of aluminum from table is 2% less than the density of aluminum from result of part (a), the density of copper from table is 5% less than the density of copper from result of part (a), the density of brass from table is 6% less than the density of brass from result of part (a), the density of tin from table is 5% less than the density of tin from result of part (a) and the density of iron from table is 0.3% less than the density of iron from result of part (a).
Explanation of Solution
Given info:
Formula to calculate the percentage error is,
percentage error=(ρ−ρ′ρ′)×100 (3)
Here,
ρ is the density of the aluminum from the result.
ρ′ is the density of aluminum from table 14.1.
For aluminum:
From part (a), the density of the aluminum is 2.75 g/cm3 and from table 14.1 the density of aluminum is 2.70 g/cm3.
Substitute 2.75 g/cm3 for ρ and 2.70 g/cm3 for ρ′ in the equation (3).
percentage error=(2.75 g/cm3−2.70 g/cm32.70 g/cm3)×100=1.85%≈2%
Thus, the density of aluminum from table is 2% less than the density of aluminum from result of part (a).
For copper:
From part (a), the density of the copper is 9.37 g/cm3 and from table 14.1 the density of copper is 8.92 g/cm3.
Substitute 9.37 g/cm3 for ρ and 8.92 g/cm3 for ρ′ in the equation (3).
percentage error=(9.37 g/cm3−8.92 g/cm38.92 g/cm3)×100=5%
Thus, the density of copper from table is 5% less than the density of copper from result of part (a).
For brass:
From part (a), the density of the brass is 8.91 g/cm3 and from table 14.1 the density of brass is 8.4 g/cm3.
Substitute 8.91 g/cm3 for ρ and 8.4 g/cm3 for ρ′ in the equation (3).
percentage error=(8.91 g/cm3−8.4 g/cm38.4 g/cm3)×100=6%
Thus, the density of brass from table is 6% less than the density of brass from result of part (a).
For tin:
From part (a), the density of the tin is 7.68 g/cm3 and from table 14.1 the density of tin is 7.30 g/cm3.
Substitute 7.68 g/cm3 for ρ and 7.30 g/cm3 for ρ′ in the equation (3).
percentage error=(7.68 g/cm3−7.30 g/cm37.30 g/cm3)×100=5.2%≈5%
Thus, the density of tin from table is 5% less than the density of tin from result of part (a).
For iron:
From part (a), the density of the iron is 7.88 g/cm3 and from table 14.1 the density of iron is 7.86 g/cm3.
Substitute 7.88 g/cm3 for ρ and 7.86 g/cm3 for ρ′ in the equation (3).
percentage error=(7.88 g/cm3−7.86 g/cm37.86 g/cm3)×100≈0.3%
Thus, the density of iron from table is 3% less than the density of iron from result of part (a).
Conclusion:
Therefore, the density of aluminum from table is 2% less than the density of aluminum from result of part (a), the density of copper from table is 5% less than the density of copper from result of part (a), the density of brass from table is 6% less than the density of brass from result of part (a), the density of tin from table is 5% less than the density of tin from result of part (a) and the density of iron from table is 0.3% less than the density of iron from result of part (a).
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