Question

A force of 10 N of gravitational force in CGS units is represented as:
(A) \[10\]Dynes
(B) \[\mathop {10}\nolimits^2 \]Dynes
(C) \[\mathop {10}\nolimits^5 \]Dynes
(D) \[\mathop {10}\nolimits^6 \]Dynes

Answer 1

Hint:This type of given problem can be solved by the conversion of a quantity from one system of units into another system of units. Here in this given problem, a force that is measured in newton is converted into dynes.

Step-by-step solution:
Step 1: For solving this type of equation we can go firstly by defining the given quantity i.e. force.
We know that Force can be calculated by multiplying the mass of the body and the acceleration produced in it i.e.,
\[\overrightarrow F = m\overrightarrow a \] ………………... (1)
The dimensional formula of the force is \[F = \left[ {\mathop M\nolimits^1 \mathop L\nolimits^1 \mathop T\nolimits^{ - 2} } \right]\].
There are two types of units of force: Absolute units and Gravitational units.
Newton (in SI) and dyne (in C.G.S.) come under absolute units.
Step 2: Conversion from one system of units into another:
\[\mathop n\nolimits_2 = \mathop n\nolimits_1 \mathop {\left( {\dfrac{{\mathop M\nolimits_1 }}{{\mathop M\nolimits_2 }}} \right)}\nolimits^a \mathop {\left( {\dfrac{{\mathop L\nolimits_1 }}{{\mathop L\nolimits_2 }}} \right)}\nolimits^b \mathop {\left( {\dfrac{{\mathop T\nolimits_1 }}{{\mathop T\nolimits_2 }}} \right)}\nolimits^c \] ………………..(2)
Where \[\mathop M\nolimits_1 = \]1kg, \[\mathop L\nolimits_1 = \]1m, \[\mathop T\nolimits_1 = \]1s, \[\mathop M\nolimits_2 = \]1g, \[\mathop L\nolimits_2 = \]1cm, \[\mathop T\nolimits_2 = \]1s, \[\mathop n\nolimits_2 = \](number of dynes) ?, \[\mathop n\nolimits_1 = \]10N, \[a = \]1, \[b = \]1, and \[c = \]-2.
Keeping all the values in the above equation, we will get –
\[\mathop n\nolimits_2 = 10\mathop {\left( {\dfrac{{\mathop {10}\nolimits^3 g}}{{1g}}} \right)}\nolimits^1 \mathop {\left( {\dfrac{{\mathop {10}\nolimits^2 cm}}{{1cm}}} \right)}\nolimits^1 \mathop {\left( {\dfrac{{1s}}{{1s}}} \right)}\nolimits^{ - 2} \]
\[\mathop n\nolimits_2 = \mathop {10}\nolimits^6 \]Dynes.
So, 10N of gravitational force equals to \[\mathop {10}\nolimits^6 \]Dynes.

So, the option (D) is correct.

Note:
-It should be remembered that the equation (2) is to be applied only when the quantity is in its absolute units. This is because when M, L, T are in absolute units, the derived unit of the quantity involved will be in absolute units only. The gravitational or practical units of any quantity can be obtained by proper conversion.
-The conversion can be done based on the fact that the magnitude of a physical quantity remains the same, whatever be the system of its measurement.