Question

In a new system of units called star units, $1\mathrm{kg}*=10\mathrm{kg},1\mathrm{m}*=1\mathrm{km}$ and $1\mathrm{s}*=1\min$, what will be the value of $1\mathrm{J}$ in the new system?

• A. $2.4\times {10}^{-5}\mathrm{J}$
• B. $3.6\times {10}^{-4}\mathrm{J}$
• C. $4.2\times {10}^{-3}\mathrm{J}$
• D. $4.2\times {10}^{-2}\mathrm{J}$

Answer 1

The correct option is B

$3.6\times {10}^{-4}\mathrm{J}$

Explanation for correct option:

Option (b) $3.6\times {10}^{-4}\mathrm{J}$

Step 1: Given data

• According to the question, the new unit system is:

$$\begin{gathered} 1\mathrm{kg}=10\mathrm{kg} \\ 1\mathrm{m}=1\mathrm{km} \end{gathered}$$

$1\sec =1\min$

Step 2: Analyze the Dimensional formula of Joule.

• The dimensional formula of Joule is the same as that of the work done.
• The dimensional formula of Joule is ${\mathrm{ML}}^2{\mathrm{T}}^{-2}$
• Now, the units of Joule are ${\mathrm{ML}}^2{\mathrm{T}}^{-2}=\mathrm{kg}{\mathrm{m}}^2\mathrm{s}{\mathrm{ec}}^{-2}$

Step 3: Find the new value of $1$Joule

• By looking at the dimensional formula of Joule, we will use the formula given below to find the new value of one Joule
  ${\mathrm{n}}_2={\mathrm{n}}_1\left[\frac{{\mathrm{M}}_1}{{\mathrm{M}}_2}\right]{\left[\frac{{\mathrm{L}}_1}{{\mathrm{L}}_2}\right]}^2{\left[\frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}\right]}^{-2}$
• Now, substituting the values according to the new unit system.
  ${\mathrm{n}}_2=1\left[\frac{1\mathrm{kg}}{10\mathrm{kg}}\right]{\left[\frac{1\mathrm{m}}{1\mathrm{km}}\right]}^2{\left[\frac{1\mathrm{s}}{1\min }\right]}^{-2}$
  $=1\left[\frac{1}{10}\right]{\left[\frac{1}{1000}\right]}^2{\left[\frac{1}{60}\right]}^{-2}=3.6\times {10}^{-4}\text{units}$

Hence, option (b) is correct.