Science:Math Exam Resources/Courses/MATH101/April 2013/Question 03 (b)

MATH101 April 2013

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Question 03 (b)

Short-Answer Questions. Question 1-3 are short-answer questions. Put your answer in the box provided. Simplify your answer as much as possible. Full Marks will be awarded for a correct answer placed in the box. Show your work, for part marks. A force of 10 N (newtons) is required to hold a spring stretched 5 cm beyond its natural length. How much work, in joules (J), is done in stretching the spring from its natural length to 50 cm beyond its natural length?

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
Remember that force is $kx$ for a spring. But what is $k$ here?

Hint 2
Work is force multiplied by distance, for constant force. In this case, force is not constant - what would the formula for work be?

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. Recall for springs, $\displaystyle f(x)=kx$ We are given that force is 10 N when $x$ is 5 cm = 0.05 m, so $\displaystyle 10=k(0.05)$ giving $\displaystyle k=200$. Then, the work done in stretching the spring from its natural length to 50 cm = 0.5 m beyond its natural length is given by ${\begin{aligned}W&=\int _{0}^{0.5}200x\ dx\\&=200{\frac {x^{2}}{2}}{\big |}_{0}^{0.5}\\&=25\end{aligned}}$ So the work done in stretching the spring is 25J.

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