Science:Math Exam Resources/Courses/MATH110/April 2013/Question 08

MATH110 April 2013

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Question 08

In mechanics, a force exerted on a steel rod is called stress, and denoted by σ. The deformation of the rod due to this stress is called strain, and denoted ε. Stress and strain are related by the Ramberg-Osgood equation $\displaystyle \epsilon =a\sigma +b\sigma ^{n}$ where a, b and n are constants. Suppose the stress applied to a steel rod increases at a constant rate of R. Determine the rate at which the strain is increasing when the stress is equal to σ0. Hint: your answer will be an expression involving several constants.

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Note the mention of rates of change - this will be a related rates problem. Before you differentiate, make sure you know which values in the given equation are constants and which are variables.

Checking a solution serves two purposes: helping you if, after having used the hint, 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. The question is asking us for the rate at which strain is increasing. Since strain is indicated by the variable $\epsilon$, this means we are seeking ${\frac {d\epsilon }{dt}}$. To find ${\frac {d\epsilon }{dt}}$ we differentiate the given formula with respect to t. Remember that a, b and n are constants, while $\epsilon$ and $\sigma$ are variables that depend on t. So $\displaystyle \epsilon (t)=a\sigma (t)+b(\sigma (t))^{n}$ becomes ${\frac {d\epsilon }{dt}}(t)=a{\frac {d\sigma }{dt}}(t)+bn(\sigma (t))^{n-1}{\frac {d\sigma }{dt}}(t)$ We were asked to find ${\frac {d\epsilon }{dt}}$ where $\sigma (t)=\sigma _{0}$, so we can plug that value into the expression above. Finally, recall we are also told earlier in the problem that the stress is increasing at a constant rate R. In the expression above, the rate of stress increasing is given by ${\frac {d\sigma }{dt}}(t)$, so we replace ${\frac {d\sigma }{dt}}(t)$ with R. This gives our final answer of: ${\frac {d\epsilon }{dt}}=aR+bn\sigma _{0}^{n-1}R$

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