A standard 1 kilogram weight is a cylinder 42.0 mm in height and 51.0 mm in diameter. What is the density of the material?

To find the density of the material, we can use the formula for density:

\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]

### Step 1: Calculate the Volume of the Cylinder
The volume \( V \) of a cylinder can be calculated using the formula:

\[
V = \pi r^2 h
\]

where:
- \( r \) is the radius of the base of the cylinder (half of the diameter),
- \( h \) is the height of the cylinder.

Given:
- Diameter \( d = 51.0 \) mm, so the radius \( r = \frac{51.0}{2} = 25.5 \) mm,
- Height \( h = 42.0 \) mm.

First, convert the dimensions to meters to maintain SI units:

\[
r = 25.5 \text{ mm} = 0.0255 \text{ m}
\]
\[
h = 42.0 \text{ mm} = 0.0420 \text{ m}
\]

Now calculate the volume:

\[
V = \pi (0.0255)^2 (0.0420) \text{ m}^3
\]

\[
V \approx \pi \times 0.00065025 \times 0.0420 \text{ m}^3
\]

\[
V \approx 8.588 \times 10^{-5} \text{ m}^3
\]

### Step 2: Calculate the Density
Now, use the mass of the cylinder (which is 1 kg) to find the density:

\[
\text{Density} = \frac{1 \text{ kg}}{8.588 \times 10^{-5} \text{ m}^3}
\]

\[
\text{Density} \approx 11645 \text{ kg/m}^3
\]

### Final Answer:
The density of the material is approximately \( 11645 \text{ kg/m}^3 \).

Would you like more details or have any questions?

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Here are some related questions you might find interesting:

1. How would the density change if the height of the cylinder was doubled?
2. What would be the density if the mass was only 500 grams?
3. How can the concept of density help in identifying different materials?
4. If the cylinder was hollow with a uniform thickness, how would that affect the density?
5. How do temperature changes affect the density of a material?

**Tip:** Always ensure that all units are converted to the SI system before performing any calculations for consistency and accuracy.

Learn how to calculate the density of a cylinder with dimensions 51.0 mm diameter and 42.0 mm height, weighing 1 kilogram. Includes step-by-step calculation using volume formula and density formula. Suitable for high school students learning about geometry and density.

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