Physics: Units, Dimensions and Error Analysis

Units and Measurements

• If the units of length and force increase four times, the energy unit increases 16 times.
• Dimensionally, E = ML²T⁻²; if length and force are scaled by 4, the energy becomes 16(ML²T⁻²).
• When velocity (V), force (F), and time (T) are fundamental quantities, the dimensional formula for mass is KV⁻¹FT.

Dimensional Analysis

• Assume mass M = f(V, F, T) = KVˣFʸTᶻ where k is a dimensionless constant.
• Dimensionally, [M] = [LT⁻¹]ˣ[MLT⁻²]ʸ[T]ᶻ simplifies to [M] = [Mʸ][Lˣ⁺ʸ][T⁻ˣ⁻²ʸ⁺ᶻ].
• Solving for x, y, and z, yields y = 1, x = -1, and z = 1, so M = KV⁻¹FT.

Error Calculation

• A rectangular plate with length ( l = 2 \pm 0.02 ) cm and width ( b = 1 \pm 0.01 ) cm has a maximum percentage error in area of 2%.
• For A = lb, ( \frac{\Delta A}{A} \times 100 = \frac{\Delta l}{l} \times 100 + \frac{\Delta b}{b} \times 100 ).

Resonance Tube Experiment

• In a resonance tube with a tuning fork frequency of 512 Hz, the first resonance at 30.3 cm and second at 63.7 cm give a maximum possible error in speed of 204.8 cm/sec.
• Calculated using ( v = 2n(l_2 - l_1) ) and ( \Delta v = 2n(\Delta l_2 + \Delta l_1) ).

Density Measurement

• If the maximum errors in measuring mass and length of a cube are 4% and 3% respectively, the maximum error in density is 13%.
• Percentage error in density ( d ) is calculated as ( % \text{ error in mass} + 3 \times [% \text{ error in length}] ).

Planck's Constant and Electronic Charge

• Dimensions of ( \frac{h}{e} ) (Planck's constant ( h ) divided by electronic charge ( e )) are the same as those of magnetic flux.
• Based on Faraday's law ( v = \frac{d\phi}{dt} ) and photoelectric effect ( ev = h\nu ).

Dimensions of Physical Quantities

• The dimensions of ( \frac{E^2}{\mu_0} ) (where ( E ) is the electric field and ( \mu_0 ) is the permeability of free space) are [MLT⁻⁴].
• Calculated from ( \frac{\epsilon_0 E^2}{\epsilon_0 \mu_0} = \frac{\text{energy/volume}}{(1/\text{speed of light})^2} ).
• The dimensions of ( \sigma b^4 ) (( \sigma ) is Stefan’s constant, ( b ) is Wien’s constant) are [ML⁻¹T⁻³].
• Kilowatt hour is a unit of energy.

Dimensional Formula and Kinetic Energy

• The dimensional formula of kinetic energy is the same as that of work.
• The frequency of vibration ( f ) of a mass ( m ) suspended from a spring with spring constant ( k ) is given by ( f = cm^x k^y ), where ( x = -\frac{1}{2} ) and ( y = \frac{1}{2} ).
• If ( s = \frac{f}{t^3} ), then ( f ) has dimensions of [ML¹T⁻³].

Error in Measurement

• A rectangular plate with length ( (4 \pm 0.04) ) cm and width ( (2 \pm 0.02) ) cm has a maximum percentage error in area of 6%.
• A wire with mass ( (0.3 \pm 0.003) ) gm, radius ( (0.5 \pm 0.005) ) cm, and length ( (6 \pm 0.6) ) cm has a maximum percentage error in density of 4%.
• A pressure of ( 10^6 ) dyne/cm² is equivalent to ( 10^5 ) N/m².

Latent Heat and Energy

• The dimensional formula for latent heat is [M⁰L²T⁻²].
• ( Q = mL ) indicates heat is a form of energy.

Circular Motion

• To keep an object moving in a circle at constant speed requires a force ( F \propto m^a v^b r^c ), where ( a = 1 ), ( b = 2 ), and ( c = -1 ).
• Via dimensional analysis, ( [F] = [m]^a [v]^b [r]^c ) leads to ( a=1 ), ( b+c=1 ), and ( -b=-2 ).
• In the formula ( X = 3YZ^2 ), where ( X ) and ( Z ) have dimensions of capacitance and magnetic induction, respectively, the dimensions of ( Y ) in the MKSA system are ( M^{-3}L^{-2}A^4T^8 ).
• Via ( [Y] = \frac{[X]}{[Z]^2} = \frac{[M^{-1}L^{-2}T^4A^2]}{[MT^{-2}A^{-1}]^2} )

Fundamental Quantities and Units

• Length, time, and velocity cannot all be fundamental quantities in a system of units because velocity can be expressed in terms of length and time.
• If 1 unit of time is 5 sec, 1 unit of mass is 20 kg, and 1 unit of length is 10 m, then one unit of power is 16 watts.
• Given ( p = p_0 e^{-\alpha t^2} ), the constant ( \alpha ) has dimensions ( T^{-2} ) because powers are dimensionless.

Dimensional Analysis of Equations

• For the force ( F = A\cos(Bx) + C\sin(DT) ), the dimensional formula of ( \frac{D}{B} ) is ( [M^0L^1T^{-1}] ).
• ( Bx ) and ( DT ) must be dimensionless, leading to ( x = L^{-1} ) and ( D = T^{-1} ).

Constants and Definitions

• The dimensions of ( \frac{e^2}{4\pi\epsilon_0 hc} ) are dimensionless, where ( e ) is electronic charge, ( \epsilon_0 ) is electric permittivity, ( h ) is Planck's constant, and ( c ) is the velocity of light.

Comparing Physical Quantities

• Dipole moment, electric flux, and electric field is the group with different dimensions.
• ( \phi = \frac{Q}{m} ) where ( Q ) = Heat, ( m ) = mass, heat is a form of Energy with dimensions ([M^1L^2T^{-2}]).

Unit Conversion

• If ( 1 , \text{g} \cdot \text{cm} \cdot \text{s}^{-1} = x ) newton-second, then ( x = 1 \times 10^{-5} ).
• If length is 10 cm, mass is 10 g, and time is 0.1 s, then the unit of force in this system is 0.1 N, where ( n_2 = 1 \times \frac{1 , \text{kg}}{10 , \text{g}} \cdot \frac{1 , \text{m}}{10 , \text{cm}} \cdot \frac{1 , \text{s}}{0.1 , \text{s}} ).
• If ( y ) is distance and ( x ) is time, the dimensions of ( \frac{d^2y}{dx^2} ) are ( LT^{-2} ).
• Planck’s constant has the same dimensions as angular momentum, both being ( ML^2T^{-1} ).
• If the units of M and L are quadrupled, the units of torque become 64 times the original, calculated from changing ( ML^2T^{-2} ) to ( (4M)(4L)^2T^{-2} ).

Radar Signals and Displacement

• Radar signal velocity = ( 3 \times 10^8 ) m/s, where distance = ( 6.3 \times 10^{10} ) m, and time = 7 minutes.
• For displacement ( x = A\sin^2(kt) ), the unit of ( k ) is hertz (Hz), since ( kt ) is dimensionless, ( k = 1/t = \text{sec}^{-1} ).

Ohm's Law and Stefan's Constant

• The dimensions of ohm are the same as ( \frac{h}{e^2} ) where ( h ) is Planck's constant and ( e ) is charge.
• Light year is not a unit of time; it measures distance.
• The dimensional formula for Stefan's constant is ( [ML^0T^{-3}\Theta^{-4}] ).
• A cube with a side of ( 1.2 \times 10^{-2} ) m has a volume recorded as ( 1.7 \times 10^{-6} ) m(^3) with two significant figures.

Definitions and Formulas

• Muscle times speed equals power, which is defined by force multiplied by velocity.
• Miscle represents Force, where the formula is ([MLT^{-2}]).
• If force, length, and time are fundamental units, mass's dimensional formula is ( FL^{-1}T^2 ).
• According to ( M!=!KF^{a} L^{b} T^{c} ) then ( [MLT^{-2}]^{a} [L]^{b} [T]^{c} ).
• Then ( a=1, b=-1, c=2 ) so the answer is ( FL^{-1}T^{2} ).

Waves and Density

• For a wave ( y = a\sin(At - Bx + C) ), the dimensions of ( A ), ( B ), and ( C ) are ( T^{-1} ), ( L^{-1} ), and dimensionless, respectively.
• A density of ( 0.5 , \text{g/cm}^3 ) in the CGS system corresponds to 500 in SI units.
• Planck's constant has dimensions of ( ML^2T^{-1} ).
• Given ( v = at - \frac{b}{t+c} ), the dimensions of ( a ), ( b ), and ( c ) are ( LT^{-2} ), ( L ), and ( T ), respectively.
• For ( x(t) = \frac{V_0}{\alpha}(1 - e^{-\alpha t}) ), the dimensions of ( V_0 ) and ( \alpha ) are ( M^0LT^{-1} ) and ( T^{-1} ), respectively.
• If ( P = \frac{a - t^2}{bx} ), the dimensions of ( \frac{a}{b} ) are ( MT^{-2} ).
• Given ( v \propto g^p h^q ), ( p = \frac{1}{2} ) and ( q = \frac{1}{2} ).
• If errors in mass and length measurements are 3% and 2%, the maximum error in density is 9%, based on the formula ( d = \frac{m}{V} \implies d = \frac{m}{L^3} ).