To calculate the volume of a capillary tube, we use the formula:
V = πr²h
where:
- V is the volume
- r is the radius
- h is the height
Given:
- Radius (r) = 0.0300 cm with an uncertainty of ±0.0020 cm
- Height (h) = 4.00 cm with an uncertainty of ±0.10 cm
First, we need to calculate the volume using the average values of the radius and height:
V = π(0.0300 cm)²(4.00 cm)
Calculating the radius squared:
0.0300 cm × 0.0300 cm = 0.000900 cm²
Now, substituting this back into the volume formula:
V = π(0.000900 cm²)(4.00 cm)
This can be computed as:
V ≈ 3.14159 × 0.000900 cm² × 4.00 cm ≈ 0.011309733 cm³
Therefore, the approximate volume of the capillary tube is:
V ≈ 0.0113 cm³
Next, let’s calculate the uncertainty in the volume using the method of propagation of uncertainty.
The formula for the propagation of uncertainty in multiplication (and division) is:
ΔV/V = √((Δr/r)² + (Δh/h)²)
Where:
- ΔV is the absolute uncertainty in volume
- Δr is the uncertainty in radius
- Δh is the uncertainty in height
Calculating the relative uncertainties:
- Δr = 0.0020 cm, r = 0.0300 cm: Δr/r = 0.0020/0.0300 ≈ 0.06667
- Δh = 0.10 cm, h = 4.00 cm: Δh/h = 0.10/4.00 = 0.025
Now, substituting these values into the uncertainty formula:
ΔV/V = √((0.06667)² + (0.025)²)
Calculating this results in:
ΔV/V ≈ √(0.0044444 + 0.000625) ≈ √(0.0050694) ≈ 0.0712
Now, we calculate ΔV by multiplying V by this relative uncertainty:
ΔV ≈ 0.0113 cm³ × 0.0712 ≈ 0.00080516 cm³
So, the volume of the capillary tube is:
V = 0.0113 ± 0.0008 cm³