Question
A density bottle has a mas of 0.04kg when empty, a mass of 0.20kg when added some quantity of steel ball bearing and 0.24 when the remainder of the bottle is filled with water. If the denskty bottle weighs 0.1kg when filled with water, calculate the relative density of the steel ball bearing
Answers
GPT-4o mini
To find the relative density of the steel ball bearings, we can follow these steps:
1. **Calculate the mass of the water in the bottle:**
- The mass of the empty density bottle is \(0.04 \, \text{kg}\).
- When filled with water, the total mass of the density bottle is \(0.1 \, \text{kg}\).
- Therefore, the mass of water in the bottle is:
\[
\text{Mass of water} = \text{Mass of filled bottle} - \text{Mass of empty bottle} = 0.1 \, \text{kg} - 0.04 \, \text{kg} = 0.06 \, \text{kg}
\]
2. **Calculate the mass of the steel ball bearings:**
- The mass of the bottle with some steel ball bearing added is \(0.20 \, \text{kg}\).
- Therefore, the mass of the steel ball bearings alone can be calculated as:
\[
\text{Mass of steel ball bearings} = \text{Mass of bottle with steel ball bearings} - \text{Mass of empty bottle} = 0.20 \, \text{kg} - 0.04 \, \text{kg} = 0.16 \, \text{kg}
\]
3. **Calculate the volume of water:**
- The density of water is approximately \(1000 \, \text{kg/m}^3\).
- Using the mass of the water, we can find its volume:
\[
\text{Volume of water} = \frac{\text{Mass of water}}{\text{Density of water}} = \frac{0.06 \, \text{kg}}{1000 \, \text{kg/m}^3} = 6 \times 10^{-5} \, \text{m}^3
\]
4. **Calculate the density of the steel ball bearings:**
- The density of the steel ball bearings can be calculated using its mass and the volume of water (since the steel ball bearings will displace the same volume of water):
\[
\text{Density of steel ball bearings} = \frac{\text{Mass of steel ball bearings}}{\text{Volume of water}} = \frac{0.16 \, \text{kg}}{6 \times 10^{-5} \, \text{m}^3} \approx 2666.67 \, \text{kg/m}^3
\]
5. **Calculate the relative density of steel ball bearings:**
- The relative density is calculated as the ratio of its density to the density of water:
\[
\text{Relative Density} = \frac{\text{Density of steel ball bearings}}{\text{Density of water}} = \frac{2666.67 \, \text{kg/m}^3}{1000 \, \text{kg/m}^3} \approx 2.67
\]
Thus, the relative density of the steel ball bearings is approximately \(2.67\).
1. **Calculate the mass of the water in the bottle:**
- The mass of the empty density bottle is \(0.04 \, \text{kg}\).
- When filled with water, the total mass of the density bottle is \(0.1 \, \text{kg}\).
- Therefore, the mass of water in the bottle is:
\[
\text{Mass of water} = \text{Mass of filled bottle} - \text{Mass of empty bottle} = 0.1 \, \text{kg} - 0.04 \, \text{kg} = 0.06 \, \text{kg}
\]
2. **Calculate the mass of the steel ball bearings:**
- The mass of the bottle with some steel ball bearing added is \(0.20 \, \text{kg}\).
- Therefore, the mass of the steel ball bearings alone can be calculated as:
\[
\text{Mass of steel ball bearings} = \text{Mass of bottle with steel ball bearings} - \text{Mass of empty bottle} = 0.20 \, \text{kg} - 0.04 \, \text{kg} = 0.16 \, \text{kg}
\]
3. **Calculate the volume of water:**
- The density of water is approximately \(1000 \, \text{kg/m}^3\).
- Using the mass of the water, we can find its volume:
\[
\text{Volume of water} = \frac{\text{Mass of water}}{\text{Density of water}} = \frac{0.06 \, \text{kg}}{1000 \, \text{kg/m}^3} = 6 \times 10^{-5} \, \text{m}^3
\]
4. **Calculate the density of the steel ball bearings:**
- The density of the steel ball bearings can be calculated using its mass and the volume of water (since the steel ball bearings will displace the same volume of water):
\[
\text{Density of steel ball bearings} = \frac{\text{Mass of steel ball bearings}}{\text{Volume of water}} = \frac{0.16 \, \text{kg}}{6 \times 10^{-5} \, \text{m}^3} \approx 2666.67 \, \text{kg/m}^3
\]
5. **Calculate the relative density of steel ball bearings:**
- The relative density is calculated as the ratio of its density to the density of water:
\[
\text{Relative Density} = \frac{\text{Density of steel ball bearings}}{\text{Density of water}} = \frac{2666.67 \, \text{kg/m}^3}{1000 \, \text{kg/m}^3} \approx 2.67
\]
Thus, the relative density of the steel ball bearings is approximately \(2.67\).