Respuesta :
Answer:
1747.9 J
Step-by-step explanation:
The work done to lift a book by a certain height [tex]\Delta h[/tex] is
[tex]W=mg\Delta h[/tex]
where
m = 1.40 kg is the mass of one book
[tex]g=9.8 m/s^2[/tex] is the acceleration due to gravity
The first shelf is 15.0 cm (0.15 m) from the ground, and it can be filled with 28 books. So, the work done to fill the first shelf is
[tex]W_1=28mg\Delta h=28(1.40)(9.8)(0.15)=57.6 J[/tex]
The 2nd shelf is 38.0 cm above the first one, so its height from the ground is
[tex]\Delta h =15.0+38.0 =53.0 cm=0.53 m[/tex]
It can also contain 28 books, so the work done to fill this shelf is
[tex]W_2=28 mg\Delta h=28(1.40)(9.8)(0.53)=203.6 J[/tex]
The height of the third shelf is
[tex]\Delta h =15.0+38.0 +38.0=91.0 cm=0.91 m[/tex]
It can also contain 28 books, so the work done to fill this shelf is
[tex]W_3=28 mg\Delta h=28(1.40)(9.8)(0.91)=349.6 J[/tex]
The height of the 4th shelf is
[tex]\Delta h =15.0+38.0 +38.0+38.0=129.0 cm=1.29 m[/tex]
It can also contain 28 books, so the work done to fill this shelf is
[tex]W_4=28 mg\Delta h=28(1.40)(9.8)(1.29)=495.6 J[/tex]
The height of the 5th shelf is
[tex]\Delta h =15.0+38.0 +38.0+38.0+38.0=167.0 cm=1.67 m[/tex]
It can also contain 28 books, so the work done to fill this shelf is
[tex]W_5=28 mg\Delta h=28(1.40)(9.8)(1.67)=641.5 J[/tex]
Therefore, the total work done is:
[tex]W=W_1+W_2+W_3+W_4+W_5=57.6+203.6+349.6+495.6+641.5=1747.9 J[/tex]
In the given scenario, we have to calculate the game's energy stored if all of the books were placed on the shelves, and the following are the calculation of the work:
The number of books on every shelf [tex]=28[/tex]
weight of each book =[tex]1.4\ kg\\[/tex]
height of each book= [tex]22 \ cm[/tex]
Height of every book's center of mass in relation to its bottom: [tex]= \frac{22}{2}=11\ cm=0.11\ m[/tex]
Following are the calculation of books energy:
In [tex]1^{st}[/tex] shelf [tex]= 28\times 1.4 \times 9.8 \times (0.15+0.11) =99.88\ Joules[/tex]
In [tex]2^{nd}[/tex] shelf[tex]= 28\times 1.4\times 9.8\times (0.38+0.15+0.11)=245.86 \ Joules[/tex]
In [tex]3^{rd}[/tex] shelf[tex]= 28\times 1.4\times 9.8\times (0.38+0.38+0.15+0.11)=391.84 \ Joules[/tex]
In [tex]4^{th}[/tex] shelf [tex]= 28\times 1.4\times 9.8\times (0.38+0.38+0.38+0.15+0.11)=536.82\ Joules[/tex]
In [tex]5^{th}[/tex] shelf[tex]= 28\times 1.4 \times 9.8\times (0.38+0.38+0.38+0.38+0.15+0.11)=683.80 \ Joules[/tex]
Calculating the Total potential energy:
[tex]= 99.88+245.86+391.84+536.82+683.80\\\\=1958.20 \ Joules\\\\ = 1.96 \times 10^3 \ Joules[/tex]
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