Physics For Engineers And Scientists - Solutions - Q069

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Question 69 - Problems - Chapter 1

Problem:

The diameter of the trunk of a large tree is roughly proportional to the $\frac{3}{2}$ power of the length of the tree.

The highest Sequoia in the Sequoia National Park in California has a height of 81m and a diameter of 7.6m. his mass is 6100 metric-ton.

In Nevada, a petrified 90 m sequoia was found.

Question:

Approximate the diameter to its base and its weight when it was still alive

Solution:

The referential tree: $\textrm{ratio}=\sqrt{{\textrm{length}}^3 }$

$\frac{\textrm{diameter}}{\textrm{ratio}}=\textrm{constant}\;\textrm{factor}$

In [1]:
import numpy as np
length = 81;
diameter = 7.6;
weight = 6.1e6;
ratio = np.sqrt(length**3);
factor = diameter / ratio
print(f'The ratio is {factor:.3}')
The ratio is 0.0104
In [2]:
length2 = 90;
ratio2 = np.sqrt(length2**3);
diameter2 = factor * ratio2;
print(f'Diameter petrified tree = {diameter2:.2} m.')
Diameter petrified tree = 8.9 m.

The density $\rho$ of the tree $=\frac{\textrm{weight}}{\textrm{height}*{\left(\frac{\textrm{diameter}}{2}\right)}^2 *\pi \;}\;\textrm{in}\;\frac{\textrm{kg}}{m^2 }$

In [3]:
rho = ((weight)/(length * (diameter / 2)**2 * np.pi))
print(f'The density Rho = {rho:.2g} kg/m3.')
The density Rho = 1.7e+03 kg/m3.

With the same specific weight, the petrified tree would have weighed $\textrm{height}*{\left(\frac{\textrm{diameter}}{2}\right)}^2 *\pi *\rho \;\textrm{in}\;\textrm{kg}$

In [4]:
weight2 = length2 * (diameter2/2)**2 * np.pi * rho;
print(f'The mass of the tree would have been {weight2/1000:.2e} metric-ton.')
The mass of the tree would have been 9.30e+03 metric-ton.

سُوۡرَةُ حٰمٓ السجدة / فُصّلَت
ثُمَّ اسْتَوَىٰ إِلَى السَّمَاءِ وَهِيَ دُخَانٌ فَقَالَ لَهَا وَلِلْأَرْضِ ائْتِيَا طَوْعًا أَوْ كَرْهًا قَالَتَا أَتَيْنَا طَائِعِينَ ﴿١١
فَقَضَاهُنَّ سَبْعَ سَمَاوَاتٍ فِي يَوْمَيْنِ وَأَوْحَىٰ فِي كُلِّ سَمَاءٍ أَمْرَهَا ۚ وَزَيَّنَّا السَّمَاءَ الدُّنْيَا بِمَصَابِيحَ وَحِفْظًا ۚ ذَٰلِكَ تَقْدِيرُ الْعَزِيزِ الْعَلِيمِ ﴿١٢


Surah Fussilat: Thereafter turned He to the heaven and it was as smoke, and said Unto it and Unto the earth: come ye twain, willingly or loth. They said: we come willingly. (11) Then He decreed them as seven heavens in two days, and revealed Unto each heaven the command thereof; and We bedecked the nether heaven with lamps and placed therein a guard. That is the ordinance of the Mighty, the Knower. (12)

De hemelen en aarde (waren) een samenhangende massa. Wij hebben ze toen van elkaar gescheiden… Wij hebben de hemel tot een beschermend dak gemaakt … en de dag en de nacht, de zon en de maan geschapen. (Koran 41 11:12)


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