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Solved on Jan 11, 2024

Determine the length of a device made of two metal blocks and a copper tube, with the given dimensions and uncertainties. Calculate the length, uncertainty, and percentage uncertainty.
Length of device: 0.10±0.02 m+5.2±0.002 m+0.333±0.02 m0.10 \pm 0.02 \mathrm{~m} + 5.2 \pm 0.002 \mathrm{~m} + 0.333 \pm 0.02 \mathrm{~m} Uncertainty in length: (0.02)2+(0.002)2+(0.02)2\sqrt{(0.02)^2 + (0.002)^2 + (0.02)^2} Percentage uncertainty in length: Uncertainty in lengthLength of device×100%\frac{\text{Uncertainty in length}}{\text{Length of device}} \times 100\%

STEP 1

Assumptions
1. The length of the first metal block is 0.10±0.02 m0.10 \pm 0.02 \mathrm{~m}.
2. The length of the second metal block is 0.333±0.02 m0.333 \pm 0.02 \mathrm{~m}.
3. The length of the copper tube is 5.2±0.002 m5.2 \pm 0.002 \mathrm{~m}.
4. The lengths are added linearly to form the device.
5. The uncertainties in the lengths add linearly as well.

STEP 2

Calculate the total length of the device by adding the lengths of the two metal blocks and the copper tube.
Total length=Length of first block+Length of second block+Length of copper tubeTotal\, length = Length\, of\, first\, block + Length\, of\, second\, block + Length\, of\, copper\, tube

STEP 3

Plug in the given lengths to calculate the total length.
Total length=0.10 m+0.333 m+5.2 mTotal\, length = 0.10\, m + 0.333\, m + 5.2\, m

STEP 4

Calculate the total length of the device.
Total length=0.10 m+0.333 m+5.2 m=5.633 mTotal\, length = 0.10\, m + 0.333\, m + 5.2\, m = 5.633\, m

STEP 5

Calculate the total uncertainty by adding the uncertainties of the lengths of the two metal blocks and the copper tube.
Total uncertainty=Uncertainty of first block+Uncertainty of second block+Uncertainty of copper tubeTotal\, uncertainty = Uncertainty\, of\, first\, block + Uncertainty\, of\, second\, block + Uncertainty\, of\, copper\, tube

STEP 6

Plug in the given uncertainties to calculate the total uncertainty.
Total uncertainty=0.02 m+0.02 m+0.002 mTotal\, uncertainty = 0.02\, m + 0.02\, m + 0.002\, m

STEP 7

Calculate the total uncertainty of the device.
Total uncertainty=0.02 m+0.02 m+0.002 m=0.042 mTotal\, uncertainty = 0.02\, m + 0.02\, m + 0.002\, m = 0.042\, m

STEP 8

To find the percentage uncertainty in the length, we divide the total uncertainty by the total length and then multiply by 100 to convert it to a percentage.
Percentage uncertainty=(Total uncertaintyTotal length)×100%Percentage\, uncertainty = \left(\frac{Total\, uncertainty}{Total\, length}\right) \times 100\%

STEP 9

Plug in the values for the total uncertainty and the total length to calculate the percentage uncertainty.
Percentage uncertainty=(0.042 m5.633 m)×100%Percentage\, uncertainty = \left(\frac{0.042\, m}{5.633\, m}\right) \times 100\%

STEP 10

Calculate the percentage uncertainty.
Percentage uncertainty=(0.0425.633)×100%≈0.745%Percentage\, uncertainty = \left(\frac{0.042}{5.633}\right) \times 100\% \approx 0.745\%
The length of the device is 5.633 m5.633 \mathrm{~m}. The uncertainty in the length is 0.042 m0.042 \mathrm{~m}. The percentage uncertainty in the length is approximately 0.745%0.745\%.

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