# Deskmate

## Ask the Genius => Physics => A-level => Topic started by: Deskmate on February 10, 2016, 05:32:32 PM

Title: Measurement
Post by: Deskmate 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 Quantities SI Units Name Symbol Length metre m Mass kilogram kg Time second s Amount of substance mole mol Temperature Kelvin K Current ampere A Luminous intensity candela cd
Derived units as products or quotients of the base units:

 Derived Quantities Equation Derived Units Area (A) A = L2 m2 Volume (V) V = L3 m3 Density (ρ) ρ = m / V kg m-3 Velocity (v) v = L / t ms-1 Acceleration (a) a = Δv / t ms-1 / s = ms-2 Momentum (p) p = m x v (kg)(ms-1) = kg m s-1

 Derived Quantities Equation Derived Unit Derived Units Special Name Symbol Force (F) F = Δp / t Newton N [(kg m s-1) / s = kg m s-2 Pressure (p) p = F / A Pascal Pa (kg m s-2) / m2 = kg m-1 s-2 Energy (E) E = F x d joule J (kg m s-2)(m) = kg m2 s-2 Power (P) P = E / t watt W (kg m2 s-2) / s = kg m2 s-3 Frequency (f) f = 1 / t hertz Hz 1 / s = s-1 Charge (Q) Q = I x t coulomb C A s Potential Difference (V) V = E / Q volt V (kg m2 s-2) / A s = kg m2 s-3 A-1 Resistance (R) R = V / I ohm Ω (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 Factor Prefix Symbol 10-12 pico p 10-9 nano n 10-6 micro μ 10-3 milli m 10-2 centi c 10-1 decid d 103 kilo k 106 mega M 109 giga G 1012 tera T

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 Quantity Reasonable Estimate Mass of 3 cans (330 ml) of Coke 1 kg Mass of a medium-sized car 1000 kg Length of a football field 100 m Reaction time of a young man 0.2 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
s-1
EXAMPLE 2: Which estimate is realistic?

 Option Explanation A The kinetic energy of a bus travelling on an expressway is 30000J A 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. B The 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. C The temperature of a hot oven is 300 K. 300K = 27 0C. Not very hot. D The volume of air in a car tyre is 0.03 m3. (http://www.xtremepapers.com/images/a-level/physics/measurement/volume_of_air_in_a_car_tyre_is_0_03_m_3.png)[/t][/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(R2 – r2)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 HigherLess Accurate↓ ↓ ↓[/t][/t]
[/t]
 (http://www.xtremepapers.com/images/a-level/physics/measurement/true_value_01.png) (http://www.xtremepapers.com/images/a-level/physics/measurement/true_value_02.png) (http://www.xtremepapers.com/images/a-level/physics/measurement/true_value_03.png) (http://www.xtremepapers.com/images/a-level/physics/measurement/true_value_04.png)
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:
 Scalar Vector Definition A 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. Examples Distance, 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.
(http://www.xtremepapers.com/images/a-level/physics/measurement/representation_of_a_vector.png)
 By accurate scale drawing By 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.(http://www.xtremepapers.com/images/a-level/physics/measurement/representation_of_a_vector_by_scale_drawing.png)[/t][/t]
R = 3.2 N (≡ 3.2 cm)
at θ = 112° to the 4 N vector.[/t]
 (http://www.xtremepapers.com/images/a-level/physics/measurement/representation_of_a_vector_using_sine_and_cosine_rules.png)Using cosine rule, a2 = b2 + c2 – 2bc cos A R2 = 42 + 62 -2(4)(6)(cos 30°) R = 3.23 NUsing 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 (http://www.xtremepapers.com/images/a-level/physics/measurement/representation_of_a_vector_summing_components.png)[/t][/t]
[/t]
 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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