Sample Notes: Physical Quantities and Measurement Techniques

Chapter 1.1: Physical Quantities and Measurement Techniques

Measuring Length with Precision

• Length is a scalar quantity measured in metres (m), using standard instruments such as:
  • Tapes – used for longer distances (e.g., measuring the length of a classroom).
  • Rulers – precision of up to 1 mm; useful for everyday measurements.
  • Vernier Calipers – accurate to 0.01 cm (0.1 mm); used for internal/external diameters and depths.
  • Micrometers – used for very small measurements (e.g., wire thickness), accurate to 0.01 mm.
• Reading an analogue micrometer:
  • Main scale on the sleeve gives full mm and half mm.
  • Rotating scale (thimble) gives additional fractions of mm.
  • Total length = sleeve reading + thimble reading.

Measuring Volume

• Liquids:
  • Measured using a measuring cylinder.
  • Ensure eye level is at the bottom of the meniscus for accurate reading.
• Solids:
  • Regular shapes: use mathematical formulas.
    • Volume of cube = length³
    • Volume of cylinder = πr²h
  • Irregular shapes: use displacement method.
    • Fill measuring cylinder partially with water.
    • Note initial volume.
    • Submerge object fully.
    • Final volume – initial volume = volume of solid.

Measuring Time

• Instruments:
  • Stopwatch or digital timer (accurate to 0.01 s).
  • Wall clock (less accurate, suitable for long durations).
• For short intervals:
  • Repeat the event multiple times (e.g., 10 oscillations of a pendulum).
  • Measure total time and divide to get average time per event.
  • Reduces error and improves precision.

Using Average Values

• Helps minimize random errors.
• Example:
  • Measuring the period of a pendulum:
    • Record time for 10 oscillations.
    • Period = total time ÷ 10
• Averaging is crucial when time/distance is small.

Scalar vs Vector Quantities

Scalar Quantities

• Have magnitude only, no direction.
• Examples:
  • Distance
  • Speed
  • Time
  • Mass
  • Energy
  • Temperature

Vector Quantities

• Have both magnitude and direction.
• Must include direction when describing.
• Examples:
  • Displacement
  • Velocity
  • Acceleration
  • Force
  • Weight
  • Momentum
  • Electric field strength
  • Gravitational field strength

Resultant of Two Vectors at Right Angles

• When two vectors (e.g., 3 N and 4 N) act perpendicularly, use the Pythagorean theorem:

  Resultant R = √(A² + B²)

  Example:

  • A = 3 N, B = 4 N
  • R = √(3² + 4²) = √(9 + 16) = √25 = 5 N
• To find the direction, use trigonometry:

  tan(θ) = opposite / adjacent

  θ = tan⁻¹(B/A)

Graphical Determination of Resultant

• Use the vector triangle or parallelogram method.
• Place the two vectors head to tail.
• Draw the resultant from the tail of the first to the head of the second.
• For vectors at right angles, draw a right-angled triangle and use scale drawing to measure resultant and angle.

Errors and Accuracy in Measurements

• Systematic error: due to faulty equipment (e.g., zero error).
• Random error: unpredictable variations (e.g., human reaction time).
• Parallax error: wrong eye angle while taking reading.

Tips to Minimize Errors

• Use instruments with appropriate least count.
• Repeat readings and use average.
• View scales perpendicularly to avoid parallax.
• Calibrate instruments before use.