Structural Analysis
6th Edition
ISBN: 9781337630931
Author: KASSIMALI, Aslam.
Publisher: Cengage,
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Transcribed Image Text:**Understanding Hydrometer Measurements and the Float Level**
A hydrometer is an instrument used to measure the specific gravity (SG) of a liquid, which is the ratio of the density of the liquid to the density of water. The hydrometer consists of a calibrated stem and a weighted bulb that allows it to float.
**Problem Statement:**
Given:
- The float level \( h \) of the hydrometer is a measure of the specific gravity of the liquid.
- For stem diameter \( D \) and total weight \( W \),
- If \( h = 0 \) represents SG = 1.0,
**Question:**
What is the formula for \( h \) as a function of \( W, D, \) SG, and \( \gamma_0 \) for water?
**Diagram Explanation:**
The diagram provided showcases a hydrometer floating in a liquid of specific gravity greater than 1 (SG > 1).
- The vertical component \( h \) is the floating height measured from the fluid's surface to a marked point on the stem.
- The diameter of the stem where the measurement takes place is \( D \).
- The label SG = 1.0 corresponds to the point where \( h = 0 \).
- The total weight of the hydrometer is \( W \).
- The fluid in which the hydrometer is immersed has a specific gravity greater than one compared to water.
**Visual Guide:**
- The hydrometer is partially submerged, with its weighted bulb below and the stem extending upwards.
- The diagram indicates fluid with specific gravity greater than one around the hydrometer's bulb.
- The stem diameter \( D \) is clearly marked near the top of the fluid level.
This setup helps learners understand how changes in specific gravity of the fluid affect the float level \( h \) of the hydrometer. The relationship between the hydrometer’s weight, the diameter of its stem, and the specific gravity of the fluid are crucial in determining the formula for \( h \).
![### Multiple Choice Questions
#### Question:
Calculate the value of \( h \) using the correct formula.
Please select the correct formula for \( h \) from the options below:
1.
\[ h = \frac{W(SG - 1)}{SG \gamma_0 \left(\frac{\pi}{4}D^2 \right)} \]
2.
\[ h = \frac{(SG + 1)}{W \cdot SG \gamma_0 \left(\frac{\pi}{4}D^2 \right)} \]
3.
\[ h = \frac{W \left(\frac{\pi}{4} D^2 \right)}{(SG - 1) \gamma_0} \]
4.
\[ h = \frac{W \left(\frac{\pi}{4} D^2 \right)}{(SG + 1) \gamma_0} \]
### Explanation of Diagram:
The image contains four options for calculating the value of \( h \). Each option includes a formula that involves the following variables:
- \( W \) : Weight
- \( SG \) : Specific Gravity
- \( \gamma_0 \) : Unit Weight of Water
- \( D \) : Diameter
- \(\pi\) : Pi (approximately 3.14159)
Each formula uses these variables with different operations and combinations to solve for \( h \).
Please carefully review the options and choose the one that correctly calculates the value of \( h \).](https://content.bartleby.com/qna-images/question/bab71d29-f3d9-43ce-83f4-a263357daa0d/d9484e50-9e8d-43c2-be34-db6e05bcc9f5/2k78ocb_processed.jpeg)
Transcribed Image Text:### Multiple Choice Questions
#### Question:
Calculate the value of \( h \) using the correct formula.
Please select the correct formula for \( h \) from the options below:
1.
\[ h = \frac{W(SG - 1)}{SG \gamma_0 \left(\frac{\pi}{4}D^2 \right)} \]
2.
\[ h = \frac{(SG + 1)}{W \cdot SG \gamma_0 \left(\frac{\pi}{4}D^2 \right)} \]
3.
\[ h = \frac{W \left(\frac{\pi}{4} D^2 \right)}{(SG - 1) \gamma_0} \]
4.
\[ h = \frac{W \left(\frac{\pi}{4} D^2 \right)}{(SG + 1) \gamma_0} \]
### Explanation of Diagram:
The image contains four options for calculating the value of \( h \). Each option includes a formula that involves the following variables:
- \( W \) : Weight
- \( SG \) : Specific Gravity
- \( \gamma_0 \) : Unit Weight of Water
- \( D \) : Diameter
- \(\pi\) : Pi (approximately 3.14159)
Each formula uses these variables with different operations and combinations to solve for \( h \).
Please carefully review the options and choose the one that correctly calculates the value of \( h \).
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