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The SI units of mass, length and time are kg, m and s and the corresponding CGS units are g, cm and s.
Numerical Problems on Unit Conversion (Dimensional Analysis)
SQ
Unit Conversion Practice Set
Apply the formula $n_2 = n_1 \left[\frac{M_1}{M_2}\right]^a \left[\frac{L_1}{L_2}\right]^b \left[\frac{T_1}{T_2}\right]^c$1
Convert one dyne into newton. [Himachal 09]
Answer
$10^{-5}$ newton
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Detailed Solution
Dyne is the CGS unit of force, and Newton is the SI unit. The dimensional formula for force is $[\text{MLT}^{-2}]$.
Here $a = 1, b = 1, c = -2$.
We need to convert from CGS to SI. Thus, $n_1 = 1$.
$$n_2 = n_1 \left[\frac{M_1}{M_2}\right]^a \left[\frac{L_1}{L_2}\right]^b \left[\frac{T_1}{T_2}\right]^c$$
$$n_2 = 1 \left[\frac{1 \text{ g}}{1 \text{ kg}}\right]^1 \left[\frac{1 \text{ cm}}{1 \text{ m}}\right]^1 \left[\frac{1 \text{ s}}{1 \text{ s}}\right]^{-2}$$
$$n_2 = 1 \left[\frac{1}{1000}\right]^1 \left[\frac{1}{100}\right]^1 [1]^{-2} = 10^{-3} \times 10^{-2} = 10^{-5}$$
Therefore, $1 \text{ dyne} = 10^{-5} \text{ N}$.
2
If the value of universal gravitational constant in SI is $6.6 \times 10^{-11} \text{ Nm}^2 \text{ kg}^{-2}$, then find its value in CGS system. [Himachal 09]
Answer
$6.6 \times 10^{-8} \text{ dyne cm}^2 \text{ g}^{-2}$
📝
Detailed Solution
The dimensional formula for the universal gravitational constant ($G$) is $[\text{M}^{-1}\text{L}^3\text{T}^{-2}]$.
Here $a = -1, b = 3, c = -2$.
We are converting from SI to CGS. $n_1 = 6.6 \times 10^{-11}$.
$$n_2 = 6.6 \times 10^{-11} \left[\frac{1 \text{ kg}}{1 \text{ g}}\right]^{-1} \left[\frac{1 \text{ m}}{1 \text{ cm}}\right]^3 \left[\frac{1 \text{ s}}{1 \text{ s}}\right]^{-2}$$
$$n_2 = 6.6 \times 10^{-11} \left[\frac{10^3 \text{ g}}{1 \text{ g}}\right]^{-1} \left[\frac{10^2 \text{ cm}}{1 \text{ cm}}\right]^3 [1]^{-2}$$
$$n_2 = 6.6 \times 10^{-11} \times 10^{-3} \times (10^2)^3 = 6.6 \times 10^{-11} \times 10^{-3} \times 10^6$$
$$n_2 = 6.6 \times 10^{-8} \text{ CGS units}$$
3
The density of mercury is $13.6 \text{ g cm}^{-3}$ in CGS system. Find its value in SI units.
Answer
$13.6 \times 10^3 \text{ kg m}^{-3}$
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Detailed Solution
The dimensional formula for density is $[\text{ML}^{-3}\text{T}^0]$.
Here $a = 1, b = -3, c = 0$.
We are converting from CGS to SI. $n_1 = 13.6$.
$$n_2 = 13.6 \left[\frac{1 \text{ g}}{1 \text{ kg}}\right]^1 \left[\frac{1 \text{ cm}}{1 \text{ m}}\right]^{-3} \left[\frac{1 \text{ s}}{1 \text{ s}}\right]^0$$
$$n_2 = 13.6 \left[\frac{1}{1000}\right]^1 \left[\frac{1}{100}\right]^{-3} = 13.6 \times 10^{-3} \times (10^{-2})^{-3}$$
$$n_2 = 13.6 \times 10^{-3} \times 10^6 = 13.6 \times 10^3 \text{ kg m}^{-3}$$
4
The surface tension of water is $72 \text{ dyne cm}^{-1}$. Express it in SI units.
Answer
$0.072 \text{ Nm}^{-1}$
📝
Detailed Solution
Surface Tension = Force / Length. Its dimensional formula is $[\text{ML}^0\text{T}^{-2}]$.
Here $a = 1, b = 0, c = -2$.
We are converting from CGS to SI. $n_1 = 72$.
$$n_2 = 72 \left[\frac{1 \text{ g}}{1 \text{ kg}}\right]^1 \left[\frac{1 \text{ cm}}{1 \text{ m}}\right]^0 \left[\frac{1 \text{ s}}{1 \text{ s}}\right]^{-2}$$
$$n_2 = 72 \left[\frac{1}{1000}\right]^1 \times 1 \times 1 = 72 \times 10^{-3} = 0.072 \text{ Nm}^{-1}$$
5
An electric bulb has a power of $500 \text{ W}$. Express it in CGS units.
Answer
$5 \times 10^9 \text{ erg s}^{-1}$
📝
Detailed Solution
The dimensional formula for Power is $[\text{ML}^2\text{T}^{-3}]$.
Here $a = 1, b = 2, c = -3$.
We are converting from SI to CGS. $n_1 = 500$.
$$n_2 = 500 \left[\frac{1 \text{ kg}}{1 \text{ g}}\right]^1 \left[\frac{1 \text{ m}}{1 \text{ cm}}\right]^2 \left[\frac{1 \text{ s}}{1 \text{ s}}\right]^{-3}$$
$$n_2 = 500 \left[\frac{1000}{1}\right]^1 \left[\frac{100}{1}\right]^2 [1]^{-3}$$
$$n_2 = 500 \times 10^3 \times (10^2)^2 = 500 \times 10^3 \times 10^4 = 500 \times 10^7$$
$$n_2 = 5 \times 10^9 \text{ erg s}^{-1}$$
6
If the value of atmospheric pressure is $10^6 \text{ dyne cm}^{-2}$, find its value in SI units.
Answer
$10^5 \text{ Nm}^{-2}$
📝
Detailed Solution
The dimensional formula for Pressure is $[\text{ML}^{-1}\text{T}^{-2}]$.
Here $a = 1, b = -1, c = -2$.
We are converting from CGS to SI. $n_1 = 10^6$.
$$n_2 = 10^6 \left[\frac{1 \text{ g}}{1 \text{ kg}}\right]^1 \left[\frac{1 \text{ cm}}{1 \text{ m}}\right]^{-1} \left[\frac{1 \text{ s}}{1 \text{ s}}\right]^{-2}$$
$$n_2 = 10^6 \left[10^{-3}\right]^1 \left[10^{-2}\right]^{-1} [1]^{-2}$$
$$n_2 = 10^6 \times 10^{-3} \times 10^2 = 10^5 \text{ Nm}^{-2}$$
7
In SI units, the value of Stefan’s constant is $\sigma = 5.67 \times 10^{-8} \text{ Js}^{-1} \text{m}^{-2} \text{K}^{-4}$. Find its value in CGS system.
Answer
$5.67 \times 10^{-5} \text{ erg s}^{-1} \text{ cm}^{-2} \text{ K}^{-4}$
📝
Detailed Solution
First, find the dimensional formula of Stefan’s constant. The unit $\text{Js}^{-1}\text{m}^{-2}$ is $\text{Watts/m}^2$, which is Power/Area.
Since Power = $[\text{ML}^2\text{T}^{-3}]$ and Area = $[\text{L}^2]$, Power/Area = $[\text{MT}^{-3}]$.
With temperature included, $\sigma = [\text{M}^1\text{L}^0\text{T}^{-3}\text{K}^{-4}]$.
Here $a = 1, b = 0, c = -3, d = -4$.
We convert from SI to CGS. $n_1 = 5.67 \times 10^{-8}$.
$$n_2 = 5.67 \times 10^{-8} \left[\frac{1 \text{ kg}}{1 \text{ g}}\right]^1 \left[\frac{1 \text{ m}}{1 \text{ cm}}\right]^0 \left[\frac{1 \text{ s}}{1 \text{ s}}\right]^{-3} \left[\frac{1 \text{ K}}{1 \text{ K}}\right]^{-4}$$
$$n_2 = 5.67 \times 10^{-8} \times [10^3]^1 \times 1 \times 1 \times 1$$
$$n_2 = 5.67 \times 10^{-8} \times 10^3 = 5.67 \times 10^{-5}$$
8
Find the value of $100 \text{ J}$ on a system which has $20 \text{ cm}$, $250 \text{ g}$ and half minute as fundamental units of length, mass and time.
Answer
$9 \times 10^6$ new units
📝
Detailed Solution
Joules (J) is the SI unit of energy. The dimensional formula for energy is $[\text{ML}^2\text{T}^{-2}]$.
Here $a = 1, b = 2, c = -2$.
We are converting from the SI system ($M_1=1\text{ kg}$, $L_1=1\text{ m}$, $T_1=1\text{ s}$) to the New System ($M_2=250\text{ g}$, $L_2=20\text{ cm}$, $T_2=30\text{ s}$). $n_1 = 100$.
$$n_2 = 100 \left[\frac{1 \text{ kg}}{250 \text{ g}}\right]^1 \left[\frac{1 \text{ m}}{20 \text{ cm}}\right]^2 \left[\frac{1 \text{ s}}{30 \text{ s}}\right]^{-2}$$
$$n_2 = 100 \left[\frac{1000 \text{ g}}{250 \text{ g}}\right]^1 \left[\frac{100 \text{ cm}}{20 \text{ cm}}\right]^2 \left[\frac{1}{30}\right]^{-2}$$
$$n_2 = 100 \times [4]^1 \times [5]^2 \times^2$$
$$n_2 = 100 \times 4 \times 25 \times 900$$
$$n_2 = 10000 \times 900 = 9 \times 10^6 \text{ new units}$$
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