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# Units and Measurements

Some Physical Quantities

 S. No. Basic Physical Quantities Fundamental Unit Symbol 1 Mass (M) kilogram kg 2 Length (L) meter m 3 Time (T) second s 4 Temperature ($\theta$) kelvin k 5 Electric Current (I) amphere A 6 Luminous Intensity (cd) candela cd 7 Quantity of matter (mol) mole mol

Some Derived Units on SI

 S. No. Physical Quantity Derived Unit Symbol 1 Force Newton N 2 Work, Energy Joule J 3 Power Watt W 4 Electric Potential Volt V 5 Electric capacity Farad F 6 Magnetic Flux Weber wb  etc.

Dimensional Formula and S.I. Units of Some Mechanical Quantities

 S. No. Physical Quantity Relation with Other Quantity Dimensional Formula S.I. Unit 1 Area Length $\times$ Breadth L $\times$ L = L2 = [M0L2T0] m2 2 Volume Length $\times$ Breadth $\times$ Height L $\times$ L $\times$ L = L3 = [M0L3T0] m3 3 Density $\frac{\text{Mass}} {\text{Volume}}$ $\frac{M}{L^3}$ = [M1L-3T0] kgm-3 4 Velocity $\frac{\text{Displacement}} {\text{Time}}$ $\frac{L}{T}$ = [M0L1T-1] ms-1 5 Linear Momentum Mass $\times$ Velocity M $\times$ LT-1 = [M1L1T-1] kgms-1 6 Acceleration $\frac{\text{Change in Velocity}}{\text{Time Taken}}$ $\frac{\frac{L}{T}}{T}$ = [M0L1T-2] ms-2 7 Acc. Due to gravity $\frac{\text{Velocity}}{\text{Time}}$ $\frac{\frac{L}{T}}{T}$ = [M0L1T-2] ms-2 8 Force Mass $\times$ Acceleration M $\times$ LT-2 = [M1L1T-2] N 9 Pressure $\frac{\text{Force}}{\text{Area}}$ $\frac{MLT^{-2}}{L^2}$ = [M1L-1T-2] Nm-2 10 Work Force $\times$ Distance MLT-2 $\times$ L = [M1L2T-2] J 11 Energy Force $\times$ Distance MLT-2 $\times$ L = [M1L2T-2] J 12 Power $\frac{\text{Work}}{\text{Time}}$ $\frac{ML^2T^{-2}} {T}$ = [M1L2T-3] W 13 Surface Tension $\frac{\text{Force}}{\text{Length}}$ $\frac{MLT^{-2}}{L}$ = [M1L0T-2] Nm-1 14 Stress, Pressure $\frac{\text{Force}}{\text{Area}}$ $\frac{MLT^{-2}}{L^2}$ = [M1L-1T-2] Nm-2 15 Strain $\frac{\text{Change in Dimension}} {\text{Original Dimension}}$ $\frac{L}{L}$ = 1 [M0L0T0] No unit 16 Torque Moment of Inertia $\times$ Angular Acceleration ML2(T-2) = [M1L2T-2] Nm 17 Heat (Q) Energy [M1L2T-2] J 18 Latent Heat (L) $\frac{\text{Heat($$\theta$)}} {\text{Mass(m)}}$$ $\frac{ML^2T^{-2}}{M}$ = [M0L2T-2] Jkg-1

Dimensions of More Physical Quantities

• Moment of Inertia (I)
­­­­­­­­­I = mr2 where, m = mass and r = distance
[I] = [M1L2]
S.I. unit = kgm2

• Angular Momentum (L)
L = mvr where r = distance, v = velocity and m = mass
[L] = [M1L1T-1].[L1]
[L] = [M1L2T-1]
S.I. unit = $\frac{kgm^2}{s}$

• Young’s Modulus
Y = $\frac{P}{\epsilon}$ = $\frac{\text{stress}}{\text{strain}}$
[Y] = [$\frac{M^1L^{-1}T^{-2}}{M^0L^0T^0}$] = [M1L-1T-2]
[Y] = [p] = [$\tau$]

• Bulk Modulus (B)
B = $\frac{\text{Stress}}{\text{Volume Strain}}$
B = -$\frac{\Delta P}{\frac{\Delta v}{v}}$
B’s dimensional formula
B = $\frac{\text{Stress}}{\text{Volumetric strain}}$
B = $\frac{\frac{M^1L^1T^{-2}}{L^2}}{M^0L^0T^0}$
B = [M1L-1T-2]
S.I. unit = kg/ms2 = N/sec

• Compressibility
K = $\frac{1}{B}$ = $\frac{1}{M^1L^{-1}T^{-2}}$
K = [M-1L1T2]

• Current (I)
I = $\frac{Q}{t}$
Dimension = A1
S.I. unit Amphere
Unit = Coloumb
Charge C = It
Dimension = A1T1 = [M0L0T1A1]

• If E = F/q, where E = electric field, q = charge and F = force, then [E] = ?
E = F/q
[E] = $\frac{M^1L^1T^{-2}}{A^1T^1}$ = M1L1T-3A-1
S.I. unit =N/Columb

• Angular Displacement ($\theta$) S.I. unit of angular displacement is Radian and it is dimensionless.

• Angular Velocity ($\omega$)
$\omega$ = $\frac{\text{Angular Displacement}}{\text{Time}}$
$\omega$ = $\frac{\theta}{t}$
S.I. unit = $\frac{\text{Radian}}{S}$
Dimensional formula = T-1

• Angular Acceleration ($\alpha$)
$\alpha$ = $\frac{\text{Angular Velocity}}{\text{Time}}$
$\alpha$ = $\frac{\omega}{t}$
S.I. unit = $\frac{\frac{\text{Radian}}{S}}{S}$ = $\frac{\text{Radian}}{S^2}$
Dimensional formula = T-2

• Frequency ($\nu$)
$\frac{1}{T}$ where, T = time period
S.I. unit = $\frac{1}{s}$ = S-1 = 1 Hz = Hertz
Dimensional formula = T-1

• Thermodynamic Temperature
S.I. unit = Kelvin
Dimensional formula = K1

• Specific Heat (C)
Q = mc$\Delta$T Where $\Delta$ T = Change in Temperature, m = mass and Q = Heat
C = $\frac{Q}{m\Delta T}$
Dimensional formula [C] = $\frac{M^1L^2T^{-2}}{M^1K^1}$ = L2T-2K-1
S.I. unit = J/kgk

Dimension Analysis

• Two or more physical quantities can be added or subtracted if there dimensions are same.
Y = A + B – C + D
[A] = [B] = [C] = [D]
• If two or more physical quantities are equal than there dimension will be also equal.
Y = $\frac{AB}{C}$
Dimension of [Y] = Dimension of $[\frac{AB}{C}]$

Note
If any formula is dimensionally incorrect than the formula will be surely incorrect.
BUT
If a formula is dimensionally correct than it may be correct or incorrect.