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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 (\(\theta\))
     
    Angular  Displacement
    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.


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