The volume of a steel rod is measured as follows: The diameter of the rod is measured with a
micrometer to be D = 5.080 ± 0.002 mm. The length is measured to be
L = 88.90 ± 0.02 mm. What is the uncertainty in the volume?

To calculate the uncertainty in the volume of the steel rod, we need to use the formula for the volume of a cylinder, which is given by:

\[
V = \pi \left(\frac{D}{2}\right)^2 L
\]

Here:
- \(D = 5.080 \, \text{mm}\) with an uncertainty of \(\pm 0.002 \, \text{mm}\)
- \(L = 88.90 \, \text{mm}\) with an uncertainty of \(\pm 0.02 \, \text{mm}\)

### Step 1: Calculate the nominal volume

First, let's calculate the nominal (most likely) volume using the central values of \(D\) and \(L\):

\[
V = \pi \left(\frac{5.080 \, \text{mm}}{2}\right)^2 \times 88.90 \, \text{mm}
\]

### Step 2: Calculate the relative uncertainties

The relative uncertainty in the volume can be calculated using the formula:

\[
\frac{\Delta V}{V} = \sqrt{\left(2 \times \frac{\Delta D}{D}\right)^2 + \left(\frac{\Delta L}{L}\right)^2}
\]

Where:
- \(\Delta D = 0.002 \, \text{mm}\) is the uncertainty in the diameter.
- \(\Delta L = 0.02 \, \text{mm}\) is the uncertainty in the length.

### Step 3: Calculate the absolute uncertainty in the volume

Finally, the absolute uncertainty in the volume is given by:

\[
\Delta V = V \times \frac{\Delta V}{V}
\]

Now, let's compute these values.The nominal volume of the steel rod is approximately \(1801.85 \, \text{mm}^3\). The uncertainty in the volume is approximately \( \pm 1.48 \, \text{mm}^3\).

### Summary:
- **Nominal Volume**: \(1801.85 \, \text{mm}^3\)
- **Uncertainty in Volume**: \( \pm 1.48 \, \text{mm}^3\)

Would you like to explore more details or ask any questions?

Here are some related questions you might find interesting:
1. How does the uncertainty in the diameter affect the overall uncertainty in the volume compared to the length?
2. How would the volume uncertainty change if the rod were twice as long?
3. What would happen to the uncertainty if we used a micrometer with a smaller uncertainty for the diameter measurement?
4. How do the relative uncertainties compare for the diameter and length measurements?
5. Can this approach be applied to other shapes, such as spheres or cubes?

**Tip:** When calculating uncertainties in measurements, the component with the larger relative uncertainty often contributes most to the overall uncertainty.

Learn how to calculate the uncertainty in the volume of a steel rod based on given measurements. The problem involves using measurement uncertainties for diameter and length to determine the overall uncertainty in volume. Suitable for advanced high school students studying measurement uncertainties and error propagation in mathematics.

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