Science:Math Exam Resources/Courses/MATH101/April 2015/Question 09 (b)

MATH101 April 2015

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Question 09 (b)

A sculpture, shaped like a pyramid 3 m high sitting on the ground, has been made by stacking smaller and smaller iron plates on top of one another. The iron plate at height $z$ m above ground level is a square whose side length is $(3-z)$ m. All of the iron plates started on the floor of a basement 2 m below ground level. Write down an integral that represents the work, in Joules, it took to move all of the iron from its starting position to its present position. Do not evaluate the integral. (You can use $9.8m/s^{2}$ for the force of gravity and $8000kg/m^{3}$ for the density of iron.)

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Let $W$ be the work that a constant force $F$ does, when moving an object over a distance of $d$. Then, $W=Fd$. If we move the object against the gravity, the required force has the formula: $F=mg=({\text{mass of the object}})\cdot ({\text{gravity acceleration}})$. For the constant density, we have $m=\rho V=({\text{density}})\cdot ({\text{volume}}).$

Hint 2
Calculate the work required to move the iron plate at the height $z$ m.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. First, calculate the work required to move the iron plate at the height $zm$ with the width $dz$. Since we move from the 2m basement below ground level, the distance that the object move is $\color {OrangeRed}d=(z+2)m$. On the other hand, since the plate at the height $z$ is a square with the side length $(3-z)m$, its volume is $V=(3-z)^{2}dzm^{3}$. Using the formula for the mass, this gives the mass of the plate: $m=\rho \cdot V=(3-z)^{2}dzm^{3}\cdot 8000kg/m^{3}=8000\cdot (3-z)^{2}dzkg$. Then, from the formula for the force, we have ${\color {OliveGreen}{F}}=mg=8000\cdot (3-z)^{2}dzkg\cdot 9.8m/s^{2}={\color {OliveGreen}9.8\cdot 8000\cdot (3-z)^{2}dz(kg\cdot m/s^{2})}$. Therefore, the required work to move the iron plate at the height $zm$ is $W={\color {OliveGreen}F}{\color {OrangeRed}d}=9.8\cdot 8000\cdot (3-z)^{2}dz(kg\cdot m/s^{2})\cdot (z+2)m=9.8\cdot 8000\cdot (3-z)^{2}(z+2)dz(kg\cdot m^{2}/s^{2})$ Finally, since the range of $z$ is from $0$ to $3$, we integrate in this range and get the total required work to move all of the iron from its starting position to its present position: $W=\color {blue}\int _{0}^{3}9.8\cdot 8000\cdot (3-z)^{2}(z+2)dz(kg\cdot m^{2}/s^{2})$.