Tute UV1 : MEASUREMENT 1 We measure physical quantities. To achieve this we firstly define the quantity, then secondly we define units in terms of which that quantity can be measured. Definition of a Quantity: This is a detailed set of steps that would need to be followed to measure the quantity. A unit is a particular amount of that quantity. Quantities are specified as the number of units of that quantity, namely quantity = n unit where n is a numeric (that is, just a number). Some examples are Potential Difference = 5 Volts Length = 7 metres Mass = 99 kilograms. Here Volts, metres and kilograms are the units and 5, 7 and 99 are the numerics. UNITS There are two types, fundamental and derived. Fundamental Units are defined in terms of a specific physical experiment that could be used to measure the quantity of which they are a unit. Derived Units are those defined in terms of the fundamental units. 1 October 11, 2017
Fundamental Units of the SI (metric) System Quantity Unit Symbol Dimensional Symbol Length metre m L Mass kilogram kg M Time second s T Current Ampere A A Temperature Kelvin K K Luminosity candela cd Substance mole mol Supplementary Units Quantity Unit Symbol Dimensional Symbol Phase angle radian rad 1 Solid angle steradian sr 1 Note that both phase angle and solid angle are dimensionless quantities.
FUNDAMENTAL UNIT DEFINITIONS Fundamental units are always defined in terms of a standard. Standards should be accessible, reproducible and indestructible. Some examples are given below: Mass (1901) 1 kg mass of a specific platinum iridium alloy cylinder that is kept at the International Bureau of Weights and Measures at Sévres, France. Length (redefined in 1983) 1 m distance travelled by light in a vacuum during a time interval of (299792458) 1 seconds. Time (1967) 1 s time required for a cesium-133 atom to undergo 9192631770 vibrations (atomic clock). It has also been suggested that time may be kept using the rapidly rotating remnants of stars known as pulsars. Their accuracy is similar to that of atomic clocks (an error of less than about one billionth of a second). Some Derived Units: These are defined in terms of the fundamental units, namely Velocity = distance/time and so the derived unit of velocity is m/s = m s 1 Acceleration = velocity/time. The derived unit of acceleration is then (m/s)/s = m s 2 Force = mass acceleration. The derived unit of force is (kg) m s 2 = kg m s 2 Many derived units are given a special name. For example, the unit of force (kg m s 2 ) is also called Newton (N).
Prefixes used in the SI (metric) System Prefix Factor Symbol deci 10 1 d centi 10 2 c milli 10 3 m micro 10 6 µ nano 10 9 n pico 10 12 p femto 10 15 f atto 10 18 a deka 10 1 da hecto 10 2 h kilo 10 3 k mega 10 6 M giga 10 9 G tera 10 12 T peta 10 15 P exa 10 18 E
DIMENSIONS Fundamental units are represented by a special symbol. Let the notation [x] mean the dimension of x. Therefore [mass] = M where M is the symbol used to represent the dimension of mass. The others are [length] = L [time] = T [current] = A [temperature] = K Dimensions are very useful indeed in testing the (in)correctness of formulae and for deriving formulae. Dimensional Formulae: This involves the expression of a unit in terms of the fundamental units from which it is defined. For example [velocity] = [distance/time] = LT 1 [acceleration] = [velocity/time] = LT 2 [force] = [mass acceleration] = MLT 2 [pressure] = [force/area] = MLT 2 L 2 = ML 1 T 2 [momentum] = [mass velocity] = MLT 1 [work] = [force distance] = MLT 2 L = ML 2 T 2 [angle] = [arc length/radius] = LL 1 = 1 [angular velocity] = [angle/time] = 1T 1 = T 1 [torque] = [force distance] = ML 2 T 2 = [work] [Electric charge] = [current time] = AT [Voltage] = [electric field distance] =[force distance / charge] = ML 2 T 3 A 1 [Electrical resistance] = [voltage / current] = ML 2 T 3 A 2 Note that both torque and work have the same dimensions but are quite different quantities.
1. Find the dimensional formulae for (a) Energy (ML 2 T 2 ) (b) Power (ML 2 T 3 ) 2. Planck s equation relating the energy E of a photon of light to its frequency f is given by E = hf where h is is a constant called Planck s constant. Find the dimensions and units for h. (Dimensions: ML 2 T 1, units: Joule second)
Use of Dimensions in Checking Formulae In any formula, terms on both sides of the = must have the same dimensions. All equations must be dimensionally homogeneous. For example, consider the equation s = ut + 1 2 at2. Checking the dimensions of each term [s] = L [ut] = [velocity time] = LT 1 T = L [ 1 2 at2 ] = [acceleration (time) 2 ] = LT 2 T 2 = L [left hand side] = [s] = L and the [right hand side] = [ut + 1 2 at2 ] = L. Consequently [left hand side] = [right hand side] as expected and this equation is dimensionally homogeneous as required. Note that if an equation is NOT dimensionally homogeneous, then it is wrong. However, if an equation IS dimensionally homogeneous, then there is no guarantee that the equation is in fact correct. Although the dimensions match, this does not necessarily mean that the dimensionless scalings are correct. For example, if the previous equation had been written s = 1 3 ut + 3 7 at2. then this would have been both dimensionally homogeneous and incorrect.
Use of Dimensions in Deriving Formulae Consider any mass m rotating in a horizontal circle of radius r and with speed v. Find a formula for the force (tension) in the string holding the mass. Assume the tension force F depends on mass m, velocity v and radius r in some unknown manner. A general expression can be written in the following form: F = kr x m y v z where k is a dimensionless constant of proportionality and x, y and z are arbitrary, unknown constants still to be determined. This equation must be dimensionally homogeneous. Hence [LHS] = [F ] = MLT 2 [RHS] = [kr x m y v z ] = L x M y (LT 1 ) z = M y L (x+z) T z Since [LHS] = [RHS] then y = 1 and z = 2. Also x + z = 1 and so x = 1. Accordingly the equation for the tension in the string is given by F = k mv2 r This is the correct mathematical form for the equation describing this situation. The constant of proportionality k is dimensionless and can be measured experimentally. Its value is one..
The speed of a wave along a string may depend on the tension F in the string and the mass per unit length µ of the string. Find a formula for the speed v of the wave in terms of F and µ. Assume a general expression as before v = kf x µ y. Using dimensional analysis yields [LHS] = [v] = LT 1 [RHS] = [kf x µ y ] = (MLT 2 ) x (ML 1 ) y = M (x+y) L (x y) T 2x Since [LHS] = [RHS] then 2x = 1 and so x = 1 2. Similarly x + y = 0 and so y = 1 2. As a further check, note that x y = 1 which is true. v = kf 1/2 µ 1/2 = k F µ where the constant of proportionality k can be determined experimentally. Its value is one.
1. Show that Einstein s equation E = mc 2 is dimensionally correct. Note that E is energy, m is mass and c is the speed of light. 2. Given that the period P of a pendulum may depend on the (a) length of the pendulum l (b) mass of the pendulum m (c) acceleration due to gravity g, derive a formula for P in terms of l, m and g using dimensional analysis. [ Answer: P = k where k can be found experimentally and is equal to 2π. l g Note that the period of a pendulum does NOT depend on the mass of the bob. ] 3. Observations suggest that the speed of ocean waves v has a functional dependence on the wavelength λ of the waves (the distance between one wave crest and the next), the density of the water ρ (the mass per unit volume) and the acceleration due to gravity g. Use dimensions to find an expression for v in terms of λ, ρ and g. [ Answer: v = k λg ] 4. Given that the fundamental frequency f 0 of vibration of a violin string depends on the length of the string l, the mass per unit length of the string µ (also known as the linear mass density) and the tension force in the string F, use dimension analysis to derive an expression for f 0 in terms of l, µ and F. [ Answer: f 0 = k l F µ ] 5. The pressure P experienced by a deep sea diver is expected to depend on the density of the water ρ, the acceleration due to gravity g and the depth h of the diver below the surface of the water. Use dimensional analysis to find an expression for P in terms of ρ, g and h. [ Answer: P = kρgh ]
6. The power output P produced by a wind generator depends on the total area A swept by the propeller blades, the velocity v of the wind and the density ρ of the wind at the blades. Use dimensions to find a relationship giving the power output P in terms of A, v and ρ. [ Answer: P = kaρv 3 ] 7. A telecommunications satellite orbits the Earth with a speed v. Given that the dimension of the universal gravitational constant G is [G] = L 3 M 1 T 2 and assuming the speed of the satellite depends on G, on the mass of the Earth M E and on the distance r of the satellite from the centre of the Earth, use dimensional analysis to derive the form of the equation for v in terms of G, M E and r. GM [ Answer: v = k E ] r 8. Electrons are the charge carriers in metals and they follow an erratic path, bouncing from atom to atom, but generally drifting in the opposite direction of the electric field. The speed at which they drift v is found to be dependent on the current I, the cross-sectional area of the wire A and the total charge per unit volume Q. Using dimensional analysis, find the relationship between v, I, A and Q. [ Answer: v = k I AQ ] 9. The second law of thermodynamics is a general principle which states that any closed system or a system which is free of external influences becomes more disordered with time. This disorder can be expressed in terms of the quantity called entropy. The entropy or disorder δs (units : J/K) of a closed system may depend on the temperature T (units : K) of the system, the mass m of the system (units : kg) and the amount of heat δq (units : J) transfered to the system. Use dimensions to find an expression for δs in terms of parameters given above. [ Answer: δs = kδq/t ]