Q52P - Fundamentals Of Physics

Question

As a contrast between the old and the modern and between the large and the small, consider the following: In old rural England 1 hide (between 100 and 120 acres) was the area of land needed to sustain one family with a single plough for one year. (An area of 1 acre is equal to 4047 m2.) Also, 1 wapentake was the area of land needed by 100 such families. In quantum physics, the cross-sectional area of a nucleus (defined in terms of the chance of a particle hitting and being absorbed by it) is measured in units of barns, where 1 barn is ${}^{\mathbf{1}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{28}}\mathbf{ }{\mathbf{m}}^{\mathbf{2}}}$ . (In nuclear physics jargon, if a nucleus is “large,” then shooting a particle at it is like shooting a bullet at a barn door, which can hardly be missed.) What is the ratio of ${}^{\mathbf{15}}$ wapentakes to ${}^{\mathbf{11}}$ barns?

Step-by-Step Solution

Answer

The ratio of 25 wapentakes to 11 barns is ${}^{1\times {10}^{36}}$ .

Step 1: Given data

The conversion factors are,

$$\begin{gathered} 1 \mathrm{hide} = 110 \mathrm{acres} \\ 1 \mathrm{wapentakes} =100 \mathrm{hide} \\ 1 \mathrm{acre} = 4047 \mathrm{square} \mathrm{meters} \\ 1 \mathrm{barn} =1\times {10}^{-28} \mathrm{square} \mathrm{meters} \end{gathered}$$

Step 2: Understanding the conversion factors

By using the proper conversion given in the problem, convert both ${}^{\mathbf{25}\mathbf{ }\mathbf{wapentakes}}$ and ${}^{\mathbf{11}\mathbf{ }\mathbf{barns}}$ into square meters. After that, find the ratio between them.

Step 3: Determination of the ratio of wapentakes to barns

After writing all conversion factors, it seems that you need to convert both 25 wapentakes and 11 barns in square meters. 

Convert 25 wapentakes into square meters.

$$\begin{gathered} 25 \mathrm{wapentakes}=25 \mathrm{wapentakes}\times \left(\frac{100 \mathrm{hide}}{1 \mathrm{wapentake}}\right)\times \left(\frac{110 \mathrm{acres}}{1 \mathrm{hide}}\right)\times \left(\frac{4047 {\mathrm{m}}^2}{1 \mathrm{acre}}\right) \\ =1.1\times {10}^9 {\mathrm{m}}^2 \end{gathered}$$

Now, convert 11 barns into square meters.

$$\begin{gathered} 11 \mathrm{barns}=11 \mathrm{barns}.\left(\frac{1\times {10}^{-28} {\mathrm{m}}^2}{1 \mathrm{barn}}\right) \\ =11\times {10}^{26} {\mathrm{m}}^2 \end{gathered}$$

Therefore, the required ratio is,

$$\begin{gathered} \frac{25 \mathrm{wapentakes}}{11 \mathrm{barns}}=\frac{1.1\times {10}^9 {\mathrm{m}}^2}{11\times {10}^{-28} {\mathrm{m}}^2} \\ =1\times {10}^{36} \end{gathered}$$ 

 

Thus, the ratio of 25 wapentakes to 11 barns is $1\times {10}^{36}$.