To Determine Volume of an Irregular Lamina Using Screw Gauge
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1. Aim
To determine the volume of an irregular lamina using a screw gauge by measuring its thickness at different points and calculating its area.
2. Apparatus Used
- Screw gauge
- Irregular lamina (metal sheet)
- Graph paper
- Pencil
- Scale
- Calculator
3. Diagram
Figure 1: Screw Gauge and measurement of lamina thickness
4. Theory
The volume of an irregular lamina can be determined by measuring its area and thickness. The area can be found by tracing the outline of the lamina on a graph paper and counting the number of squares within the outline. The thickness is measured at various points using a screw gauge to get an average value.
A screw gauge is a precision instrument used to measure very small dimensions with high accuracy. It works on the principle of a micrometer screw, where the advancement of the spindle is directly proportional to the rotation of the thimble.
The least count of a screw gauge is given by:
$\text{Least Count} = \frac{\text{Pitch}}{\text{Number of divisions on circular scale}}$
Where:
- Pitch is the distance moved by the spindle for one complete rotation of the thimble
- Typically, the pitch is 0.5 mm or 1 mm
- The number of divisions on the circular scale is usually 50 or 100
For accurate measurements, the zero error of the screw gauge must be determined and appropriate corrections applied to the readings.
5. Formula
The following formulas are used for calculations:
a) Screw Gauge Reading:
$\text{Total Reading} = \text{Main Scale Reading} + (\text{Circular Scale Reading} \times \text{Least Count})$
b) Corrected Reading (considering zero error):
$\text{Corrected Reading} = \text{Observed Reading} - \text{Zero Error}$
c) Average Thickness:
$\text{Average Thickness} = \frac{\sum_{i=1}^{n} t_i}{n}$
d) Area of the Lamina:
$\text{Area} = \text{Number of complete squares} + \frac{\text{Number of partial squares}}{2}$
e) Volume of the Lamina:
$\text{Volume} = \text{Area} \times \text{Average Thickness}$
6. Procedure
- Determination of Zero Error:
- Close the screw gauge gently until the ratchet starts slipping.
- Note the reading on both the main scale and circular scale.
- If the zero of the circular scale coincides with the reference line on the main scale, there is no zero error.
- If the zero of the circular scale is above the reference line, the zero error is positive.
- If the zero of the circular scale is below the reference line, the zero error is negative.
- Mark the Points for Thickness Measurement:
- Mark at least 10 points distributed evenly throughout the lamina.
- Number these points for reference.
- Measure Thickness at Each Point:
- Place the lamina between the anvil and the spindle of the screw gauge.
- Rotate the thimble until the ratchet slips.
- Note the main scale reading and the circular scale reading.
- Calculate the total reading by adding the main scale reading and the product of circular scale reading and least count.
- Apply zero error correction if necessary.
- Repeat for all marked points.
- Determine the Area of the Lamina:
- Trace the outline of the lamina on a graph paper.
- Count the number of complete squares enclosed within the outline.
- Count the number of partial squares (squares that are more than half covered).
- Calculate the area using the formula mentioned above.
- Multiply by the area of one square on the graph paper (typically 1 mm² or 1 cm²).
- Calculate the Volume:
- Calculate the average thickness from all the measurements.
- Multiply the average thickness by the area to get the volume.
7. Observation Table
a) Determination of Zero Error:
| Zero Error Determination | |
|---|---|
| Main Scale Reading | ______ mm |
| Circular Scale Reading | ______ divisions |
| Zero Error | ±______ mm |
b) Measurement of Thickness:
| Point No. | Screw Gauge Reading | Total Reading (mm) | Corrected Reading (mm) | |
|---|---|---|---|---|
| Main Scale (mm) | Circular Scale (divisions) | |||
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 | ||||
| 6 | ||||
| 7 | ||||
| 8 | ||||
| 9 | ||||
| 10 | ||||
| Average Thickness (tavg) | ______ mm | |||
c) Determination of Area:
| Parameter | Value |
|---|---|
| Number of Complete Squares (N1) | |
| Number of Partial Squares (N2) | |
| Area of One Square on Graph Paper | ______ mm² |
| Total Area (A) | ______ mm² |
8. Calculations
a) Least Count of Screw Gauge:
$\text{Least Count} = \frac{\text{Pitch}}{\text{Number of divisions on circular scale}}$
For a screw gauge with 0.5 mm pitch and 50 divisions:
$\text{Least Count} = \frac{0.5 \text{ mm}}{50} = 0.01 \text{ mm}$
b) Zero Error Correction:
$\text{Zero Error} = \pm x \text{ mm}$
c) Total Reading for each point:
$\text{Total Reading} = \text{Main Scale Reading} + (\text{Circular Scale Reading} \times 0.01 \text{ mm})$
d) Corrected Reading for each point:
$\text{Corrected Reading} = \text{Total Reading} - \text{Zero Error}$
e) Average Thickness:
$t_{avg} = \frac{t_1 + t_2 + t_3 + ... + t_{10}}{10}$
f) Area Calculation:
$\text{Area} = N_1 + \frac{N_2}{2} \times \text{Area of one square}$
g) Volume Calculation:
$\text{Volume} = \text{Area} \times t_{avg}$
$\text{Volume} = A \times t_{avg}$
9. Result
The volume of the given irregular lamina is ______ mm³ or ______ cm³.
10. Precautions
- Handle the screw gauge carefully to avoid damage to the spindle and anvil.
- Do not apply excessive pressure while measuring the thickness of the lamina.
- Ensure that the ratchet is used for tightening the screw gauge to apply uniform pressure.
- Take readings with proper alignment of eyes to avoid parallax error.
- Determine the zero error correctly and apply the appropriate correction.
- Measure thickness at sufficient number of points distributed evenly throughout the lamina.
- Keep the lamina clean and free from dust.
- Trace the outline of the lamina on graph paper carefully to get accurate area measurement.
- Count the squares on graph paper accurately.
- Calculate all values with proper units and significant figures.
11. Sources of Error
- Instrumental Errors:
- Imperfection in the screw gauge leading to backlash error.
- Irregularities in the pitch of the screw.
- Uneven anvil and spindle faces.
- Observational Errors:
- Parallax error in reading the scales.
- Incorrect application of pressure while measuring thickness.
- Error in counting squares on the graph paper.
- Personal Errors:
- Incorrect tracing of the lamina's outline on graph paper.
- Improper selection of points for thickness measurement.
- Calculation errors.
- Environmental Errors:
- Temperature variations affecting the dimensions of the screw gauge and lamina.
- Dust or dirt affecting the measurements.
12. Viva Voice Questions
Q1: What is a screw gauge and what is its principle of operation?
A screw gauge is a precision measuring instrument used to measure dimensions with high accuracy. It works on the principle of a micrometer screw, where the advancement of the spindle is directly proportional to the rotation of the thimble. One complete rotation of the thimble causes the spindle to move by a distance equal to the pitch of the screw.
Q2: How is the least count of a screw gauge determined?
The least count of a screw gauge is calculated by dividing the pitch of the screw by the number of divisions on the circular scale:
$\text{Least Count} = \frac{\text{Pitch}}{\text{Number of divisions on circular scale}}$
Q3: What is zero error in a screw gauge and how is it corrected?
Zero error in a screw gauge is the deviation of the zero mark on the circular scale from the reference line on the main scale when the anvil and spindle are in contact. If the zero mark is above the reference line, the error is positive; if below, it's negative. To correct this error, it is subtracted from all measurements if positive and added if negative.
Q4: Why is the ratchet mechanism used in a screw gauge?
The ratchet mechanism in a screw gauge ensures that uniform pressure is applied when measuring an object. It prevents over-tightening which could damage the object or the instrument and ensures consistent measurements.
Q5: How is the area of an irregular lamina determined?
The area of an irregular lamina can be determined by tracing its outline on a graph paper and counting the number of complete squares (N₁) and partial squares (N₂) within the outline. The area is calculated as:
$\text{Area} = (N_1 + \frac{N_2}{2}) \times \text{Area of one square}$
Q6: What is the importance of taking thickness measurements at multiple points?
Taking thickness measurements at multiple points is important because the lamina may not have uniform thickness throughout. By measuring at multiple points and averaging the readings, we get a more accurate representation of the lamina's overall thickness.
Q7: How does temperature affect the measurements taken with a screw gauge?
Temperature changes can cause thermal expansion or contraction of both the screw gauge and the object being measured. This can introduce errors in the measurement. Generally, precision measurements should be conducted at a standard temperature (usually 20°C) to minimize such errors.
Q8: What is backlash error in a screw gauge and how can it be avoided?
Backlash error occurs when there is a gap between the threads of the screw and the nut, causing inaccurate readings when the direction of rotation is changed. To avoid this error, measurements should always be taken by rotating the thimble in the same direction (usually clockwise) and using the ratchet to ensure proper contact.
Q9: What are the parallax errors and how can they be minimized?
Parallax errors occur when the observer's line of sight is not perpendicular to the scale, resulting in incorrect readings. To minimize parallax errors, the eye must be positioned directly above the scale marking, and measurements should be taken with proper alignment.
Q10: What are the applications of volume measurement of irregular lamina?
Volume measurement of irregular lamina has applications in:
- Material science for determining density and other properties
- Quality control in manufacturing
- Metallurgy for studying material properties
- Archaeology for artifact analysis
- Engineering for component design and analysis