DIMENSIONS
CHARACTERISTICS OF DIMENSIONS
- Numerical quantities have no dimensions.
- Quantities that are expressed as the ratio of two quantities having similar units have dimensionless.
- The dimensions of a quantity will be the same in all systems of unit.
- Two Physical quantities having different dimensions cannot be added or subtracted.
Dimensions of Base Quantities
- Length [L]
- Mass [M]
- Time [T]
- Temperature [K]
- Amount of Substance [n]
- Electric Current [A]
- Luminous Intensity [J]
Dimensions of Derived Quantities
SPEED/VELOCITY
`\Rightarrow` `v=` `\frac(d)(t)`
`\Rightarrow` `v=` `\frac([L])([T])`
`\Rightarrow` `v=` [`LT^-1`]
`\Rightarrow` `v=` [`M^0LT^-1`]
ACCELERATION
`\Rightarrow` `a=\frac vt`
`\Rightarrow` `a=\frac{\left[LT^{-1}\right]}{\left[T\right]}`
`\Rightarrow` `a=[LT^{-1}T^{-1}]`
`\Rightarrow``a=[LT^{-2}]`
`\Rightarrow` `a=[M^0LT^{-2}]`
FORCE
`\Rightarrow` `F=ma`
`\Rightarrow` `F=\left[M\right]\left[LT^{-2}\right]`
`\Rightarrow` `F=\left[MLT^{-2}\right]`
WORK
`\Rightarrow` `W=Fd`
`\Rightarrow` `W=\left[MLT^{-2}\right]\left[L\right]`
`\Rightarrow` `W=\left[ML^2T^{-2}\right]`
POWER
`\Rightarrow` `P=\frac Wt`
`\Rightarrow` `P=\frac{\left[ML^2T^{-2}\right]}{\left[T\right]}`
`\Rightarrow` `P=\left[ML^2T^{-2}\right]\left[T^{-1}\right]`
`\Rightarrow` `P=\left[ML^2T^{-3}\right]`
AREA
`\Rightarrow` `A=L\times W`
`\Rightarrow` `A=\left[L\right]\left[L\right]`
`\Rightarrow` `A=\left[L^2\right]`
`\Rightarrow` `A=\left[M^0L^2T^0\right]`
VOLUME
`\Rightarrow` `V=l\times w\times h`
`\Rightarrow` `V=\left[L\right]\left[L\right]\left[L\right]`
`\Rightarrow` `V=\left[L^3\right]`
`\Rightarrow` `V=\left[M^0L^3T^0\right]`
PRESSURE
`\Rightarrow` `P=\frac FA`
`\Rightarrow` `P=\frac{\left[MLT^{-2}\right]}{\left[L^2\right]}`
`\Rightarrow` `P=\left[MLT^{-2}\right]\left[L^{-2}\right]`
`\Rightarrow` `P=\left[ML^{-1}T^{-2}\right]`
SURFACE TENSION
`\Rightarrow` `T=\frac FL`
`\Rightarrow` `T=\frac{\left[MLT^{-2}\right]}{\left[L\right]}`
`\Rightarrow` `T=\left[MT^{-2}\right]`
`\Rightarrow` `T=\left[ML^0T^{-2}\right]`
MOMENTUM
`\Rightarrow` `P=mv`
`\Rightarrow` `P=\left[M\right]\left[LT^{-1}\right]`
`\Rightarrow` `P=\left[MLT^{-1}\right]`
TORQUE
`\Rightarrow` `\tau=F\times r`
`\Rightarrow` `\tau=\left[MLT^{-2}\right]\left[L\right]`
`\Rightarrow` `\tau=\left[ML^2T^{-2}\right]`
FREQUENCY
`\Rightarrow` `f=\frac1T`
`\Rightarrow` `f=\frac1{\left[T\right]}`
`\Rightarrow` `f=\left[T^{-1}\right]`
`\Rightarrow` `f=\left[M^0L^0T^{-1}\right]`
STRESS
`\Rightarrow` `S=\frac FA`
`\Rightarrow` `S=\frac{\left[MLT^{-2}\right]}{\left[L^2\right]}`
`\Rightarrow` `S=\frac{\left[MT^{-2}\right]}{\left[L^1\right]}`
`\Rightarrow` `S=\left[MT^{-2}\right]\left[L^{-1}\right]`
`\Rightarrow` `S=\left[ML^{-1}T^{-2}\right]`
STRAIN
`\Rightarrow` `\sigma=\frac{\triangle L}L`
`\Rightarrow` `\sigma=\frac{\left[L\right]}{\left[L\right]}`
`\Rightarrow` `\sigma=\left[L^0\right]`
`\Rightarrow` `\sigma=\left[M^0L^0T^0\right]` (Dimensionless)
DENSITY
`\Rightarrow` `\rho=\frac mV`
`\Rightarrow` `\rho=\frac{\left[M\right]}{\left[L^3\right]}`
`\Rightarrow` `\rho=\left[M\right]\left[L^{-2}\right]`
`\Rightarrow` `\rho=\left[ML^{-2}T^0\right]`
LINEAR DENSITY
`\Rightarrow` `\mu=\frac ml`
`\Rightarrow` `\mu=\frac{\left[M\right]}{\left[L\right]}`
`\Rightarrow` `\mu=\left[M\right]\left[L^{-1}\right]`
`\Rightarrow` `\mu=\left[ML^{-1}\right]`
`\Rightarrow` `\mu=\left[ML^{-1}T^0\right]`
SPRING CONSTANT
`\Rightarrow` `K=\frac Fx`
`\Rightarrow` `K=\frac{\left[MLT^{-2}\right]}{\left[L\right]}`
`\Rightarrow` `K=\left[MT^{-2}\right]`
`\Rightarrow` `K=\left[ML^0T^{-2}\right]`
ANGULAR DISPLACEMENT
`\Rightarrow` `\theta=\frac sr`
`\Rightarrow` `\theta=\frac{\left[L\right]}{\left[L\right]}`
`\Rightarrow` `\theta=\left[L^0\right]`
`\Rightarrow` `\theta=\left[M^0L^0T^0\right]` (Dimensionless)
ANGULAR VELOCITY
`\Rightarrow` `\omega=\frac\theta t`
`\Rightarrow` `\omega=\frac1{\left[T\right]}`
`\Rightarrow` `\omega=\left[T^{-1}\right]`
`\Rightarrow` `\omega=\left[M^0L^0T^{-1}\right]`
ANGULAR ACCELERATION
`\Rightarrow` `a=\frac\omega t`
`\Rightarrow` `a=\frac{\left[T^{-1}\right]}{\left[T\right]}`
`\Rightarrow` `a=\left[T^{-1}\right]\left[T^{-1}\right]`
`\Rightarrow` `a=\left[T^{-2}\right]`
`\Rightarrow` `a=\left[M^0L^0T^{-2}\right]`
ANGULAR MOMENTUM
`\Rightarrow` `L=mvr`
`\Rightarrow` `L=\left[M\right]\left[LT^{-1}\right]\left[L\right]`
`\Rightarrow` `L=\left[ML^2T^{-1}\right]`
KINETIC ENERGY
`\Rightarrow` `K.E=\frac12mv^2`
`\Rightarrow` `K.E=\left[M\right]\left[LT^{-1}\right]^2`
`\Rightarrow` `K.E=\left[M\right]\left[L^2T^{-2}\right]`
`\Rightarrow` `K.E=\left[ML^2T^{-2}\right]`
POTENTIAL ENERGY
`\Rightarrow` `P.E=mgh`
`\Rightarrow` `P.E=\left[M\right]\left[LT^{-2}\right]\left[L\right]`
`\Rightarrow` `P.E=\left[ML^2T^{-2}\right]`
PLANK'S CONSTANT
`\Rightarrow` `h=\frac Ef`
`\Rightarrow` `h=\frac{\left[ML^2T^{-2}\right]}{\left[T^{-1}\right]}`
`\Rightarrow` `h=\left[ML^2T^{-1}\right]`
GRAVITATIONAL CONSTANT
`\Rightarrow` `G=\frac{Fr^2}{m_1m_2}`
`\Rightarrow` `G=\frac{\left[MLT^{-2}\right]\left[L^2\right]}{\left[M\right]\left[M\right]}`
`\Rightarrow` `G=\frac{\left[LT^{-2}\right]\left[L^2\right]}{\left[M\right]}`
`\Rightarrow` `G=\left[L^3T^{-2}\right]\left[M^{-1}\right]`
`\Rightarrow` `G=\left[M^{-1}L^3T^{-2}\right]`
SPECIFIC HEAT CAPACITY
`\Rightarrow` `C=\frac Q{m\triangle T}`
`\Rightarrow` `C=\frac{\left[ML^2T^{-2}\right]}{\left[M\right]\left[K\right]}`
`\Rightarrow` `C=\frac{\left[L^2T^{-2}\right]}{\left[K\right]}`
`\Rightarrow` `C=\left[L^2T^{-2}\right]\left[K^{-1}\right]`
`\Rightarrow` `C=\left[M^0L^2T^{-2}K^{-1}\right]`
LATENT HEAT
`\Rightarrow` `L=\frac Qm`
`\Rightarrow` `L=\frac{\left[ML^2T^{-2}\right]}{\left[M\right]}`
`\Rightarrow` `L=\left[L^2T^{-2}\right]`
`\Rightarrow` `L=\left[M^0L^2T^{-2}\right]`
ENTROPY
`\Rightarrow` `S=\frac{\triangle Q}{\triangle T}`
`\Rightarrow` `S=\frac{\left[ML^2T^{-2}\right]}{\left[K\right]}`
`\Rightarrow` `S=\left[ML^2T^{-2}\right]\left[K^{-1}\right]`
`\Rightarrow` `S=\left[ML^2T^{-2}K^{-1}\right]`
BOLTZMANN CONSTANT
`\Rightarrow` `K_B=\frac{PV}{T}`
`\Rightarrow` `K_B=\frac{\left[ML^{-1}T^{-2}\right]\left[L^3\right]}{\left[K\right]}`
`\Rightarrow` `K_B=\left[ML^{-2}T^{-2}\right]\left[K^{-1}\right]`
`\Rightarrow` `K_B=\left[ML^{-2}T^{-2}K^{-1}\right]`
Dimensional Correction of Equations.
(i) `v_f=v_i+at`where: `v_f=` Final Velocity `v_i=` Initial Velocity `a=` Acceleration `t=` Time
`\left[LT^{-1}\right]=\left[LT^{-1}\right]+\left[LT^{-2}\right]\left[T\right]`
`\left[LT^{-1}\right]=\left[LT^{-1}\right]+\left[LT^{-1}\right]`
`\left[LT^{-1}\right]=2\left[LT^{-1}\right]`
By neglecting `2`
`\left[LT^{-1}\right]=\left[LT^{-1}\right]` (Dimensionally Correct)
(ii) `S=v_it+\frac(1)(2)at^2`where: `S=` Distance `v_i=` Initial Velocity `a=` Acceleration `t=` Time
`\left[L\right]=\left[LT^{-1}\right]\left[T\right]+\frac(1)(2)\left[LT^{-2}\right]\left[T^2\right]`
`\left[L\right]=\left[L\right]+\frac(1)(2)\left[L\right]`
`\left[L\right]=\frac(3)(2)\left[L\right]`
By neglecting `\frac(3)(2)`
`\left[L\right]=\left[L\right]` (Dimensionally Correct)
(iii) `v_f^2=2aS-v_i^2`where: `S=` Distance `v_i=` Initial Velocity `a=` Acceleration `t=` Time `v_f=` Final Velocity
`\left[LT^{-1}\right]^2=2\left[LT^{-2}\right][L]+\left[LT^{-1}\right]^2`
`\left[L^2T^{-2}\right]=2\left[L^2T^{-2}\right]+\left[L^2T^{-2}\right]`
`\left[L^2T^{-2}\right]=3\left[L^2T^{-2}\right]`
By neglecting `3`
`\left[L^2T^{-2}\right]=\left[L^2T^{-2}\right]` (Dimensionally Correct)
(iv) `v=\sqrt{\frac{2GM}R}`
Where: `M=` mass of Earth `G=` Gravitational constant `v=` velocity `R=` Radius of Earth
`\left[LT^{-1}\right]=\sqrt{\frac{\left[M^{-1}L^3T^{-2}\right]\left[M\right]}{\left[L\right]}}`
`\left[LT^{-1}\right]=\sqrt{\left[L^2T^{-2}\right]}`
`\left[LT^{-1}\right]=\left[LT^{-1}\right]` (Dimensionally Correct)
(v) `T=2\pi\sqrt{\frac (l)(g)}`
Where: `T=` Time `l=` lenght `g=` gravitaional acceleration
`\left[T\right]=\sqrt{\frac{\left[L\right]}{\left[LT^{-2}\right]}}`
`\left[T\right]=\sqrt{\frac1{\left[T^{-2}\right]}}`
`\left[T\right]=\sqrt{\left[T^2\right]}`
`\left[T\right]=\left[T\right]` (Dimensionally Correct)
(vi) `v=\sqrt{\frac F\mu}`
Where: `v=` velocity `F=` Force `\mu=` linear density
`\left[LT^{-1}\right]=\sqrt{\frac{\left[MLT^{-2}\right]}{\left[ML^{-1}\right]}}`
`\left[LT^{-1}\right]=\sqrt{\frac{\left[LT^{-2}\right]}{\left[L^{-1}\right]}}`
`\left[LT^{-1}\right]=\sqrt{\left[L\right]\left[LT^{-2}\right]}`
`\left[LT^{-1}\right]=\sqrt{\left[L^2T^{-2}\right]}`
`\left[LT^{-1}\right]=\left[LT^{-1}\right]` (Dimensionally Correct)
(vii) `T=2\pi\sqrt{\frac{m}{K}}`
Where: `T=` Time `m=` Mass `K=`Spring constant
`[T]=\sqrt{\frac{\left[M\right]}{\left[MT^{-2}\right]}}`
`\left[T\right]=\sqrt{\frac1{\left[T^{-2}\right]}}`
`\left[T\right]=\sqrt{\left[T^2\right]}`
`\left[T\right]=\left[T\right]` (Dimensionally Correct)
(viii) `v=F\lambda`
Where: `v=` Velocity `f=` Frequency `\lambda=` Wave length
`\left[LT^{-1}\right]=\left[\frac1T\right]\left[L\right]`
`\left[LT^{-1}\right]=\left[LT^{-1}\right]` (Dimensionally Correct)
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