Topic : Measurement

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Offline Deskmate

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« on: February 10, 2016, 05:32:32 PM »
Base quantities and their units; mass (kg),  length (m), time (s), current (A), temperature (K), amount of substance (mol):
Base QuantitiesSI Units
Amount of substancemolemol
Luminous intensitycandelacd
Derived units as products or quotients of the base units:
DerivedQuantities EquationDerived Units
Area (A)A = L2m2
Volume (V)V = L3m3
Density (ρ)ρ = m / Vkg m-3
Velocity (v)v = L / tms-1
Acceleration (a)a = Δv / tms-1 / s = ms-2
Momentum (p)p = m x v(kg)(ms-1) = kg m s-1

Derived QuantitiesEquationDerived UnitDerived Units
Special NameSymbol
Force (F)F = Δp / tNewtonN[(kg m s-1) / s = kg m s-2
Pressure (p)p = F / APascalPa(kg m s-2) / m2 = kg m-1 s-2
Energy (E)E = F x djouleJ(kg m s-2)(m) = kg m2 s-2
Power (P)P = E / twattW(kg m2 s-2) / s = kg m2 s-3
Frequency (f)f = 1 / thertzHz1 / s = s-1
Charge (Q)Q = I x tcoulombCA s
Potential Difference (V)V = E / QvoltV(kg m2 s-2) / A s = kg m2 s-3 A-1
Resistance (R)R = V / IohmΩ(kg m2 s-3 A-1) / A = kg m2 s-3 A-2
Prefixes and their symbols to indicate decimal sub-multiples or multiples of both base and derived units:
Multiplying FactorPrefixSymbol
 Estimates of physical quantities: When making an estimate, it is only reasonable to give the figure to 1 or at most 2 significant figures since an estimate is not very precise.
Physical QuantityReasonable Estimate
Mass of 3 cans (330 ml) of Coke1 kg
Mass of a medium-sized car1000 kg
Length of a football field100 m
Reaction time of a young man0.2 s
  • Occasionally, students are asked to estimate the area under a graph. The usual method of counting squares within the enclosed area is used. (eg. Topic 3 (Dynamics), N94P2Q1c)
  • Often, when making an estimate, a formula and a simple calculation may be involved.
EXAMPLE 1: Estimate the average running speed of a typical 17-year-old‟s 2.4-km run.
velocity = distance / time = 2400 / (12.5 x 60) = 3.2 ≈3 m
  EXAMPLE 2: Which estimate is realistic?
AThe kinetic energy of a bus travelling on an expressway is 30000JA bus of mass m travelling on an expressway will travel between 50 to 80 kmh-1, which is 13.8 to 22.2 ms-1. Thus, its KE will be approximately ½ m(182) = 162m. Thus, for its KE to be 30000J: 162m = 30000. Thus, m = 185kg, which is an absurd weight for a bus; ie. This is not a realistic estimate.
BThe power of a domestic light is 300W.A single light bulb in the house usually runs at about 20W to 60W. Thus, a domestic light is unlikely to run at more than 200W; this estimate is rather high.
CThe temperature of a hot oven is 300 K.300K = 27 0C. Not very hot.
DThe volume of air in a car tyre is 0.03 m3.[/t][/t]
Estimating the width of a tyre, t, is 15 cm or 0.15 m, and estimating R to be 40 cm and r to be 30 cm,
volume of air in a car tyre is
 = π(R2r2)t
 = π(0.42 – 0.32)(0.15)
 = 0.033 m3
 ≈ 0.03 m3 (to one sig. fig.)
 Distinction between systematic errors (including zero errors) and random errors and between precision and accuracy: Random error: is the type of error which causes readings to scatter about the true value.
 Systematic error: is the type of error which causes readings to deviate in one direction from the true value.
 Precision: refers to the degree of agreement (scatter, spread) of repeated measurements of the same quantity. {NB: regardless of whether or not they are correct.}
 Accuracy: refers to the degree of agreement between the result of a measurement and the true value of the quantity.
→ → R Error Higher → → →
 → → → Less Precise → → →

S Error

  Assess the uncertainty in a derived quantity by simple addition of actual, fractional or percentage uncertainties (a rigorous statistical treatment is not required).
For a quantity x = (2.0 ± 0.1) mm,
 Actual/ Absolute uncertainty, Δ x = ± 0.1 mm
 Fractional uncertainty, Δxx = 0.05
 Percentage uncertainty, Δxx  100% = 5 %
 If p = (2x + y) / 3 or p = (2x - y) / 3 , Δp = (2Δx + Δy) / 3
 If r = 2xy3 or r = 2x / y3 , Δr / r = Δx / x + 3Δy / y
 Actual error must be recorded to only 1 significant figure, &
 The number of decimal places a calculated quantity should have is determined by its actual error.
For eg, suppose g has been initially calculated to be 9.80645 ms-2 & Δg has been initially calculated to be 0.04848 ms-2. The final value of Δg must be recorded as 0.05 ms-2 {1 sf }, and the appropriate recording of g is (9.81 ± 0.05) ms-2.
 Distinction between scalar and vector quantities:
DefinitionA scalar quantity has a magnitude only. It is completely described by a certain number and a unit.A vector quantity has both magnitude and direction. It can be described by an arrow whose length represents the magnitude of the vector and the arrow-head represents the direction of the vector.
ExamplesDistance, speed, mass, time, temperature, work done, kinetic energy, pressure, power, electric charge etc.
 Common Error:
 Students tend to associate kinetic energy and pressure with vectors because of the vector components involved. However, such considerations have no bearings on whether the quantity is a vector or scalar.
Displacement, velocity, moments (or torque), momentum, force, electric field etc.
Representation of vector as two perpendicular components: In the diagram below, XY represents a flat kite of weight 4.0 N. At a certain instant, XY is inclined at 30° to the horizontal and the wind exerts a steady force of 6.0 N at right angles to XY so that the kite flies freely.
By accurate scale drawingBy calculations using sine and cosine rules, or Pythagoras‟ theorem
Draw a scale diagram to find the magnitude and direction of the resultant force acting on the kite.
R = 3.2 N (≡ 3.2 cm)
 at θ = 112° to the 4 N vector.[/t]
Using cosine rule, a2 = b2 + c2 – 2bc cos A
 R2 = 42 + 62 -2(4)(6)(cos 30°)
 R = 3.23 N
Using sine rule: a / sin A = b / sin B
 6 / sin α = 3.23 / sin 30°
 α = 68° or 112°
  = 112° to the 4 N vector
Summing Vector Components
Fx = - 6 sin 30° = - 3 N
 Fy = 6 cos 30° - 4 = 1.2 N
 R = √(-32 + 1.22) = 3.23 N
 tan θ = 1.2 / 3 = 22°
R is at an angle 112° to the 4 N vector. (90° + 22°)
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