Study Notes on Physical Quantities and Measurement

UNIT 1: PHYSICAL QUANTITIES AND MEASUREMENT

ACTIVITY: Determi

ing Height

• Students will determine their height in metres and millimetres by constructing a paper scale.

• The paper scale should be 2 m long with markings in metres, centimetres, and millimetres.

• Students can partner up to measure each other's heights using the constructed scale.

1.7.2: VERNIER CALIPER

• Definition: A Vernier caliper is a precise measuring tool used for measuring small lengths fractions of a millimetre.

• It consists of a main scale and a sliding secondary scale, called the Vernier scale, which allows for measurements that can go beyond the smallest division on the main scale.

• Functionality: It can measure the thickness, diameter, or width of objects, as well as the inner and outer diameters of hollow cylinders.
  

Components of the Vernier Caliper

• Main Scale:

  • Has usually 1 mm divisions.

  • Contains jaws at its left end.

• Vernier Scale:

  • Has markings based on multiples of the main scale's divisions.

Least Count (Vernier Constant)

• The minimum length that can be accurately measured with a Vernier caliper is termed the least count.

• Calculation of Least Count:
  $\text{Least Count} = \frac{\text{Smallest division on main scale}}{\text{Total number of divisions on Vernier scale}}$

Example Calculation of Least Count

• If the smallest division on the main scale is 1 mm and there are 10 divisions on the Vernier scale:
  $\text{Least Count} = \frac{1 \text{ mm}}{10} = 0.1 \text{ mm} = 0.01 \text{ cm}$

Understanding Zero Error

• When the jaws of the Vernier calipers close, the zero of the Vernier scale should align with the zero of the main scale. If this is not the case, it indicates zero error:

  • No Zero Error: The two zeros coincide.

  • Positive Zero Error: The Vernier scale zero extends beyond the main scale zero (to the right).

  • Negative Zero Error: The Vernier scale zero is before the main scale zero (to the left).

Measuring with Vernier Calipers

• Steps for Measurement:

  1. Note the least count and check for zero error.

  2. Fix the object in the jaws and note the divisions passed by the Vernier scale (this is the main scale reading).

  3. Consult the division of the Vernier scale that aligns with any main scale divisions (this is the Vernier scale reading).

  4. Calculate the total reading by adding the main scale reading and the product of the least count and the Vernier scale reading.

DIGITAL VERNIER CALIPER

• Digital Vernier calipers offer greater precision than mechanical ones, with a least count of 0.01 mm.

1.7.3: SCREW GAUGE

• Definition: A screw gauge, or micrometer, is used to measure even smaller lengths than a Vernier caliper.

• Structure: The device consists of a main scale, a circular scale that can be rotated, and a ratchet mechanism.

• Pitch: The distance that the circular scale moves along the main scale per rotation is termed the pitch.

• Least Count Calculation:
  $\text{Least Count} = \frac{\text{Pitch of Screw Gauge}}{\text{Total Number of Divisions on Circular Scale}}$

• Example Calculation:

  • If pitch is 0.5 mm and there are 50 divisions:
    $\text{Least Count} = \frac{0.5 \text{ mm}}{50} = 0.01 \text{ mm}$

• Zero Error Identification:

  • Align the datum line with the zero mark on the thimble scale. If they do not coincide, zero error exists.

Taking Measurements with Screw Gauge

• Steps:

  1. Determine pitch and least count, correcting for any zero error.

  2. Place the object between spindle and anvil; close gently using the ratchet.

  3. Read the main scale value shown and the circular scale aligned with the datum line.

  4. Total reading is the sum of the main scale and the product of the circular scale reading and least count.

1.7.4: PHYSICAL BALANCE

• Description: A physical balance is a sensitive instrument used to measure mass, capable of measuring in milligram increments.

• Comprises a vertical pillar, horizontal beam resting on a knife edge, two pans, and a long pointer.

• Operation: Masses are determined by comparing a standard weight to the weight of the body being measured.

1.7.5: MEASURING CYLINDER

• Functionality: Used in laboratories to measure the volume of liquids, chemicals, or solutions.

• Structure: Made of glass or clear plastic and marked with vertical scales in millilitres (ml) or cubic centimeters (cm³).

• Least Count: Generally 1 cm³, so volume changes smaller than this cannot be accurately measured.

Measuring Volume of Irregular Solids

• When measuring the volume of an irregular solid, the initial water level is noted, and then the water level is measured after the object is submerged. The volume of the solid is found by subtracting the initial level from the new level.

1.7.6: STOP WATCH

• Definition: A stopwatch measures time intervals.

• Types:

  • Mechanical (Analogue) Stopwatch: Features circular dials (second and minute hands) and operates with a start/stop knob.

  • Digital Stopwatch: Controlled by buttons, displays elapsed time, and can record split/lap times.

• Least Counts: Typically, 1 second for mechanical and 0.1 seconds for digital stopwatches.

1.8: ERRORS IN MEASUREMENT

• Definition: Errors refer to uncertainties inherent in measurements.

Types of Errors:

1. Systematic Errors:

   • Cause consistent deviations in one direction.

   • Sources:
     a. Instrumental errors from faulty design/calibration.
     b. Imperfections in experimental technique or environmental conditions (e.g., temperature).
     c. Personal errors due to biases or careless observations.

2. Random Errors:

   • Arise from unpredictable variables that can affect measurements, leading to variability in results.

1.9: PRECISION AND ACCURACY

• Precision: Refers to the closeness of repeated measurements to each other.

• Accuracy: Indicates how close a measured value is to the true or accepted value.

• Examples:

  • Accurate and Precise: Measurements cluster tightly around the true value.

  • Accurate but not Precise: Measurements may vary but average is close to true value.

  • Precise but not Accurate: Measurements cluster together but are off from the true value.

• Aiming can be analogized with darts: consistency vs. hitting the bull's-eye.

1.10: SIGNIFICANT FIGURES

• Definition: Significant figures convey the precision of measurements.

• General Rules:

  1. All digits from measurements are significant.

  2. Nonzero digits (1-9) are always significant.

  3. In large numbers, trailing zeros are not significant unless specified.

  4. In small decimal numbers, leading zeros are not significant.

Rounding Off Numbers

• Rounding is essential for expressing measurements with appropriate precision.

• Whole Number Rounding Rules:

  1. Identify the digit for rounding. Examine the next smallest place.

  2. If that digit is < 5, leave it; otherwise, add 1 to the target digit.

• Decimal Rounding Rules:

  1. Identify the digit for rounding.

  2. If the digit beyond it is < 5, treat it as zero; if ≥ 5, add 1 to the target digit.

Example Rounding Off

• Usage of significant figures ensures meaningful representation in scientific contexts.

  • For instance, measuring an object as 3.5678 cm can be effectively rounded to 3.57 cm for practicality without losing significant accuracy.

END OF NOTES