The standard kilogram (Fig. 1.1a) is a platinum–iridium cylinder 39.0 mm in height and 39.0 mm in diameter. What is the density of the material?
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Reference
Serway & Jewett's "Physics For Scientists And Engineers With Modern Physics 9ed" is a comprehensive textbook covering various physics topics. It provides in-depth explanations and problem-solving techniques for mechanics, thermodynamics, electromagnetism, optics, and modern physics. The book is widely used in undergraduate physics courses and is known for its clear explanations and numerous examples.
Question
The standard kilogram (Fig. 1.1a) is a platinum–iridium cylinder 39.0 mm in height and 39.0 mm in diameter. What is the density of the material?
Explain Question
This question asks us to calculate the density of the material used to create the standard kilogram. We are given the following information:
- The standard kilogram is a cylinder made of platinum-iridium alloy
- The cylinder's height is 39.0 mm
- The cylinder's diameter is 39.0 mm
- The mass of the cylinder is 1 kg (as it is the standard kilogram)
- Calculate the volume of the cylinder
- Use the formula for density: ρ = m / V, where ρ is density, m is mass, and V is volume
Explain Solve
Let's solve this problem step by step: 1. Calculate the volume of the cylinder: - The formula for the volume of a cylinder is V = πr²h, where r is the radius and h is the height - The diameter is 39.0 mm, so the radius is 19.5 mm - V = π * (19.5 mm)² * 39.0 mm 2. Convert the volume to cubic meters: - 1 m³ = (1000 mm)³ = 10⁹ mm³ - We'll need to divide our result by 10⁹ to convert from mm³ to m³ 3. Calculate the density: - Use the formula ρ = m / V - m = 1 kg (the standard kilogram) - Divide 1 kg by the volume in m³ 4. Simplify and calculate the final result
Summary
To solve this problem, we calculated the volume of the standard kilogram cylinder using its dimensions, converted the volume to cubic meters, and then used the density formula (ρ = m / V) to find the density of the platinum-iridium alloy. The key steps involved using the cylinder volume formula, unit conversion, and the density equation.
Solve
With V = (base area)(height), V = πr²h and ρ = m/V, we have:
ρ = m / (πr²h) = 1 kg / (π * (19.5 mm)² * 39.0 mm) * (10⁹ mm³ / 1 m³)
ρ = 2.15 × 10⁴ kg/m³
To solve this problem: 1. We used the cylinder volume formula: V = πr²h 2. We substituted the given values: r = 19.5 mm, h = 39.0 mm 3. We used the density formula: ρ = m / V 4. We converted the volume from mm³ to m³ 5. We calculated the final result: 2.15 × 10⁴ kg/m³
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