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Math  /  Algebra

Question(II) Eight books, each 4.0 cm thick with mass 1.6 kg , lie flat on a table. How much work is required to stack them one on top of another?

Studdy Solution

STEP 1

1. Each book is 4.0 cm thick.
2. Each book has a mass of 1.6 kg.
3. The books are initially lying flat on the table.
4. We need to calculate the work done to stack the books one on top of another.
5. Work is calculated using the formula W=F⋅d W = F \cdot d , where F F is the force and d d is the distance over which the force is applied.
6. The force required to lift each book is equal to its weight, which can be calculated using F=m⋅g F = m \cdot g , where m m is the mass and g g is the acceleration due to gravity (approximately 9.8 m/s2 9.8 \, \text{m/s}^2 ).

STEP 2

1. Calculate the force required to lift one book.
2. Determine the distance each book is lifted.
3. Calculate the work done to lift each book to its new position.
4. Sum the work done for all books to find the total work required.

STEP 3

Calculate the force required to lift one book.
The force F F required to lift one book is equal to its weight: F=m⋅g=1.6 kg×9.8 m/s2 F = m \cdot g = 1.6 \, \text{kg} \times 9.8 \, \text{m/s}^2 F=15.68 N F = 15.68 \, \text{N}

STEP 4

Determine the distance each book is lifted.
Since each book is 4.0 cm thick, convert this to meters: d=4.0 cm=0.04 m d = 4.0 \, \text{cm} = 0.04 \, \text{m}
The first book is not lifted, the second book is lifted 0.04 m, the third book is lifted 0.08 m, and so on. The n n -th book is lifted (n−1)×0.04 m(n-1) \times 0.04 \, \text{m}.

STEP 5

Calculate the work done to lift each book to its new position.
The work done W W to lift the n n -th book is: Wn=F⋅dn=15.68 N×(n−1)×0.04 m W_n = F \cdot d_n = 15.68 \, \text{N} \times (n-1) \times 0.04 \, \text{m}
Calculate the work for each book: - First book: W1=15.68×0×0.04=0 J W_1 = 15.68 \times 0 \times 0.04 = 0 \, \text{J} - Second book: W2=15.68×1×0.04=0.6272 J W_2 = 15.68 \times 1 \times 0.04 = 0.6272 \, \text{J} - Third book: W3=15.68×2×0.04=1.2544 J W_3 = 15.68 \times 2 \times 0.04 = 1.2544 \, \text{J} - Fourth book: W4=15.68×3×0.04=1.8816 J W_4 = 15.68 \times 3 \times 0.04 = 1.8816 \, \text{J} - Fifth book: W5=15.68×4×0.04=2.5088 J W_5 = 15.68 \times 4 \times 0.04 = 2.5088 \, \text{J} - Sixth book: W6=15.68×5×0.04=3.136 J W_6 = 15.68 \times 5 \times 0.04 = 3.136 \, \text{J} - Seventh book: W7=15.68×6×0.04=3.7632 J W_7 = 15.68 \times 6 \times 0.04 = 3.7632 \, \text{J} - Eighth book: W8=15.68×7×0.04=4.3904 J W_8 = 15.68 \times 7 \times 0.04 = 4.3904 \, \text{J}

STEP 6

Sum the work done for all books to find the total work required.
Total work Wtotal W_{\text{total}} is the sum of the work done for each book: Wtotal=W1+W2+W3+W4+W5+W6+W7+W8 W_{\text{total}} = W_1 + W_2 + W_3 + W_4 + W_5 + W_6 + W_7 + W_8 Wtotal=0+0.6272+1.2544+1.8816+2.5088+3.136+3.7632+4.3904 W_{\text{total}} = 0 + 0.6272 + 1.2544 + 1.8816 + 2.5088 + 3.136 + 3.7632 + 4.3904 Wtotal=17.5616 J W_{\text{total}} = 17.5616 \, \text{J}
The total work required to stack the books is approximately 17.56 J \boxed{17.56} \, \text{J} .

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