Q062 - Physics For Engineers And Scientists - Solutions

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Question 62 - Problems - Chapter 1

Problem:

The table below shows the masses and the diameters of the different planets in our solar system.

In [1]:
import numpy as np
import pandas as pd

PlanetName = ['Mercury','Venus', 'Earth', 'Mars', 'Jupiter', 'Saturnus', 'Uranus', 'Neptunus', 'Pluto']
DistanceSun = [57.9e6, 108.0e6, 150.0e6, 228.0e6, 778.0e6, 1430.0e6, 2870.0e6, 4500.0e6, 5890.0e6]
OrbitalPeriod = [0.241, 0.615, 1, 1.88, 11.9, 29.5, 84, 165, 248]
Mass = [3.3e23, 4.87e24, 5.98e24, 6.42e23, 1.9e27, 5.67e26, 8.7e25, 1.03e26, 1.5e22]
Radius = [2439, 6052, 6378, 3393, 71399, 60000, 25400, 24300, 1500]
Gravity = [0.38, 0.91, 1, 0.38, 2.53, 1.07, 0.92, 1.19, 0.045]
RotationPeriod = [58.6, 243, 0.997, 1.026, 0.41, 0.43, 0.65, 0.77, 6.39]
PlanetDataSet = list(zip(PlanetName,DistanceSun,OrbitalPeriod,Mass,Radius,Gravity,RotationPeriod))
df = pd.DataFrame(data = PlanetDataSet, columns=['PlanetName','DistanceSun','OrbitalPeriod','Mass','Radius','Gravity','RotationPeriod'])
df.set_index('PlanetName', inplace=True)
df
Out[1]:
DistanceSun OrbitalPeriod Mass Radius Gravity RotationPeriod
PlanetName
Mercury 5.790000e+07 0.241 3.300000e+23 2439 0.380 58.600
Venus 1.080000e+08 0.615 4.870000e+24 6052 0.910 243.000
Earth 1.500000e+08 1.000 5.980000e+24 6378 1.000 0.997
Mars 2.280000e+08 1.880 6.420000e+23 3393 0.380 1.026
Jupiter 7.780000e+08 11.900 1.900000e+27 71399 2.530 0.410
Saturnus 1.430000e+09 29.500 5.670000e+26 60000 1.070 0.430
Uranus 2.870000e+09 84.000 8.700000e+25 25400 0.920 0.650
Neptunus 4.500000e+09 165.000 1.030000e+26 24300 1.190 0.770
Pluto 5.890000e+09 248.000 1.500000e+22 1500 0.045 6.390

Question:

a) Calculate the average density of each plannet and make a list of the planets in order of decreasing densities.
b) Is there a corelation between the density of the planet and the distance to the sun?


Solution:

To calcculate density Rho, we use the formula:

$$ \textrm{Density}\;\varrho =\;\frac{\textrm{Weight}\;\left(\textrm{in}\;\textrm{kg}\right)}{\textrm{Volume}\;\left(\textrm{in}\;m^3 \right)} $$

Therfore, we need the volume of each planet

$$ \textrm{Volume}\;v\;=\frac{4}{3}*\pi \;r^3 $$

In [2]:
# Add two columns
df['Volume'] = 0
df['Density'] = 0

#iterate the dataset and calculate volume and density 
for index, row in df.iterrows():
    Volume = (4/3) * np.pi * (row["Radius"]*1000)**3
    df.loc[index,['Volume']] = Volume            # add values to the table
    Density = (row['Mass']*1e3) / (Volume * 1e6) # in g/cm3
    df.loc[index,['Density']]  = Density
    print(f'For planet {index} the density is {Density:.2g} g/cm3')
    
df.sort_values(by=['Density'], ascending = False)
For planet Mercury the density is 5.4 g/cm3
For planet Venus the density is 5.2 g/cm3
For planet Earth the density is 5.5 g/cm3
For planet Mars the density is 3.9 g/cm3
For planet Jupiter the density is 1.2 g/cm3
For planet Saturnus the density is 0.63 g/cm3
For planet Uranus the density is 1.3 g/cm3
For planet Neptunus the density is 1.7 g/cm3
For planet Pluto the density is 1.1 g/cm3
Out[2]:
DistanceSun OrbitalPeriod Mass Radius Gravity RotationPeriod Volume Density
PlanetName
Earth 1.500000e+08 1.000 5.980000e+24 6378 1.000 0.997 1.086781e+21 5.502487
Mercury 5.790000e+07 0.241 3.300000e+23 2439 0.380 58.600 6.077487e+19 5.429876
Venus 1.080000e+08 0.615 4.870000e+24 6052 0.910 243.000 9.285074e+20 5.244977
Mars 2.280000e+08 1.880 6.420000e+23 3393 0.380 1.026 1.636214e+20 3.923691
Neptunus 4.500000e+09 165.000 1.030000e+26 24300 1.190 0.770 6.010456e+22 1.713680
Uranus 2.870000e+09 84.000 8.700000e+25 25400 0.920 0.650 6.864197e+22 1.267446
Jupiter 7.780000e+08 11.900 1.900000e+27 71399 2.530 0.410 1.524632e+24 1.246202
Pluto 5.890000e+09 248.000 1.500000e+22 1500 0.045 6.390 1.413717e+19 1.061033
Saturnus 1.430000e+09 29.500 5.670000e+26 60000 1.070 0.430 9.047787e+23 0.626673

b) Density is decresing if planets are further from the sun but not in a linear way so this can be a coincidence.


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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