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Quantities, Mass, and Weight

QUANTITIES
  • A physical quantity is made up of magnitude and unit.

Base Units
  • The following are base units:
  • All units (not from the table above) can be broken to base units.
  • Homogeneity can be used prove equations.
  • An equation is homogeneous if the base units on the left hand side are the same as the base units on the right hand side.
  • This may not work all the time due to the fact that it does not take pure numbers into account.
Multiples and Sub-multiples

Scalar and Vector
  • A scalar has magnitude only, it cannot be -ve. E.g. Speed, energy, power, work, mass, distance, etc.
  • A vector has magnitude and direction, it can be -ve. E.g. Displacement, acceleration, force, velocity, momentum, weight, electric field strength, etc.

Vectors

Measurement Techniques

Using a Cathode Ray Oscilloscope (C.R.O.)

  1. A supply of peak value, 5.0 V, and of frequency 50 Hz, is connected to a c.r.o with time-base at 10 ms per division, and Y-gain at 5.0 V per division. Which trace is obtained?

    • The maximum value is 5.0 V; therefore eliminate A and B
    • F = 1/T and T = Time-base × Divisions
    • ∴ F = 1/Time-base × Divisions
    • Divisions = 1/F × Time-base = 1/50 × 10 × 10-3 = 2
    • The trace must have a period of 2 divisions and a height of 1 division, therefore D.

Systematic and Random Errors

  • Systematic Error:
    • Constant error in one direction; too big or too small
    • Cannot be eliminated by repeating or averaging
    • If systematic error is small, the measurement is accurate.
    • Accuracy refers to the degree of agreement between a result of measurement and the true value of the quantity.
  • Random Error:
    • Random fluctuations or scatter about a true value
    • Can be reduced by repeating and averaging
    • When random errors are small, the measurement is precise
    • Precision refers to the degree of agreement of repeated measurements of the same quantity (regardless of whether it is correct or not)

Calculations Involving Errors

  • For a quantity x = (2.0 ± 0.1) mm
    • Absolute uncertainty = Δx = ± 0.1 mm
    • Fractional uncertainty = Δx/x = 0.05
    • Percentage uncertainty = Δx/x × 100% = 5%
  • Combining errors:
    • When values are added or subtracted, add the absolute error.
    • If p = 2x + y/3 or p = 2x – y/3, then Δp = 2Δx + Δy/3
    • When the values are multiplied or divided, add percentage errors
    • When values are powered (e.g. squared), multiply the percentage error with the power.
      • If r = 2xy3 or r = 2x/y3, then Δr/r = Δx/x + 3Δy/y

Treatment of Significant Figures

  • Actual error is recorded to only 1 significant figure.
  • The number of decimal places for a calculated quantity is equal to the number of decimal places in the actual error.
  • During a practical, when calculating using a measured quantity, give answers to the same significant figure as the measurement or one less.

Micrometer Screw Gauge

  • Measures objects up to 0.01 mm
  • Place the object between the anvil and spindle
  • Rotate thimble until the object is firmly held by the jaws.
  • Add the values from the main scale and the rotating scale.

Vernier Scale

  • Measures objects up to 0.1 mm
  • Place the object on the rule
  • Push the slide scale to the edge of the object
  • The sliding scale is 0.9 mm long and is divided into 10 equal divisions
  • Check which line division on the sliding scale matches with a line division on the rule
  • Subtract the value from the sliding scale (0.09 × divisions) by the value from the rule
MASS AND WEIGHT
  • Mass is the amount of matter an object contains, and is a property that ‘resists’ change in motion.
  • Weight is the force of gravity acting on an object, measured in Newtons, and given by the formula: Weight = Mass × Gravity
  • Weights (and hence masses) may be compared using a balance.
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