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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 3.10 A*R*C*S

At this point, you have learned a wide range of skills that you can apply to a physics context, including:
While each of these skills is useful on its own, they are most powerful when used together to help you examine complicated contexts that often draw on real-world experiences. Activities where you should carry out all of the steps above are labeled A*R*C*S: Analyze-Represent-Calculate-Sensemake. The figure below shows all the individual steps to help you keep track of them.
Figure 3.10.1. Steps to follow when you consider an A*R*C*S activity.
You manage parks for Corvallis, and you are responsible for putting up fences around two new parks that have the same area. Park 1 is a square, while park 2 is a rectangle whose length is \(110 \mathrm{~m}\text{.}\) If park 2 requires twice as much fencing as park 1, what is the area of each park?
1. Analyze and Represent
  1. List quantities. The only known here is \(l = 110 \mathrm{~m}\text{,}\) the length of the rectangular park. There are many unknowns: \(s\) (the side length of the square park), \(w\) (the width of the rectangle), \(P_{square}\text{,}\) \(A_{rectangle}\text{,}\) and \(P_{rectangle}\text{,}\) and \(A_{square}\) (which we are trying to find).
  2. Identify assumptions. There is no physics in this activity, but the problem stated that the parks are rectangular, which is probably a simplification given that real parks are often strange shapes.
  3. Represent the situation physically. Below is a diagram of the square and the rectangle, with all distances labeled.
Figure 3.10.3. A rectangular park and a square park.
2. Calculate
  1. Represent principles symbolically. The area of a square is \(A_{square} = s^2\) and the perimeter (the length of the fence) is \(P_{square} = 4s\text{.}\) The area of a rectangle is \(A_{rectangle} = lw\) and the perimeter (the length of the fence) is \(P_{rectangle} = 2l + 2w\text{.}\)
  2. Solve unknown(s) symbolically. Start by setting the two areas equal and relating the perimeters
    \begin{equation*} A_{square} = A_{rectangle} \end{equation*}
    \begin{equation*} s^2 = lw \end{equation*}
    \begin{equation*} P_{rectangle} = 2P_{square} \end{equation*}
    \begin{equation*} 2l + 2w = 8s \end{equation*}
    Combine these expressions algebraically by solving the first equation for \(w\) and plugging it into the second equation:
    \begin{equation*} w = s^2/l \end{equation*}
    \begin{equation*} 2l + 2s^2/l = 8s \end{equation*}
    \begin{equation*} s^2 - 4sl + l^2 = 0 \end{equation*}
    Use the quadratic formula to solve this last equation for \(s\text{,}\) then square \(s\) to get the area of the square!
    \begin{equation*} s = \frac{4l \pm \sqrt{16l^2 - 4l^2}}{2} = \left(2 \pm \sqrt{3}\right)l \end{equation*}
    \begin{equation*} A = s^2 = \left(7 \pm 4\sqrt{3}\right)l^2 \end{equation*}
  3. Plug in numerical values. The two possible approximate values for \(A\) are \(13.9l^2 \approx 168190 \mathrm{~m^2}\) or \(0.07l^2 \approx 847 \mathrm{~m^2}\text{.}\)
3. Sensemake
  1. Are the units correct? The units for area should be \(\mathrm{m^2}\text{.}\) Each factor of \(l\) in our answer has units of \(\mathrm{m}\text{,}\) so squaring it gives \(\mathrm{m^2}\text{,}\) which is what we expect!
  2. Is your numerical answer reasonable? Something a little weird happened. You got two> numbers. What does that mean? In this case you were told that the length of the park was \(l\text{,}\) but you were not told whether the length was bigger or smaller than the width! The two cases correspond to those two possibilities. The units \(\mathrm{m^2}\) are not great for making sense of something as big as a park, so you can look up how many \(\mathrm{m^2}\) are in an acre (4047), which is a more reasonable measure of land area for a park. This gives us 41.6 acres (this would be a pretty big park!) and 0.2 acres (on the other hand, very small).
  3. Does your symbolic answer make physical sense? A quick way to check if our answer makes sense is to evaluate how it depends on the given quantities. In our case, the area varies like \(l^2\text{.}\) First of all, it depends on the square of \(l\text{,}\) which is something you have seen before in many other expressions for area, such as for squares, circles, polygons, etc. More specifically, this means that increasing \(l\) (making the given parameter bigger) corresponds to an increase in the area. Lastly, we can check special values of \(l\) and see what your answer is. The most special value of \(l\) is \(l = 0\text{,}\) which corresponds to no park at all! In this case, the area of the park should of course be 0, which is what your symbolic answer says as well:
    \begin{equation*} A = s^2 = \left(7 \pm 4\sqrt{3}\right) 0^2 = 0 \end{equation*}

Exercises Exercises

A*R*C*S 3.10.1. The Brick against the Rail.

A small brick of mass \(0.8 \mathrm{~kg}\) is positioned against a horizontal wooden railing. You pull on the rope at the \(60^o\) angle shown so that the acceleration of the brick is \(2 \mathrm{~m/s^2}\) to the left. Find the magnitudes of all forces on the brick.
Figure 3.10.4. A brick is pulled against a wooden railing.
1. Analyze and Represent
  1. Identify known and unknown quantities with both a symbol and a number.
  2. Identify and justify any assumptions.
  3. Choose an appropriate system and draw a-free-body diagram for the system.
2. Calculate
  1. Represent physics principles that will help you solve for the tensions.
  2. Determine a symbolic equation for each force in terms of known variables.
  3. Plug numbers into your symbolic answer.
3. Sensemake
  1. Check the units of your symbolic answer.
  2. Compare your numerical answers to other numerical forces with which you are familiar.
  3. Use covariational reasoning: how should your symbolic equations depend on \(m\text{,}\) \(a\text{,}\) and \(\theta\text{?}\)

A*R*C*S 3.10.2. The Block upon the Ice.

You are pulling a small, heavy block across an icy lake. You pull on the rope at the angle shown in such a way that the normal force on the block by the ice is zero and the acceleration of the block is \(2.4 \mathrm{~m/s^2}\) to the left. Find the magnitude of the tension and the mass of the block. Assume the gravitational force on the block points downward with a magnitude equal to \(mg\text{,}\) where \(m\) is the block’s mass and \(g = 10 \mathrm{~m/s^2}\text{.}\)
Figure 3.10.5. A block is pulled across an icy pond.
1. Analyze and Represent
  1. Identify known and unknown quantities with both a symbol and a number.
  2. Identify and justify any assumptions.
  3. Choose an appropriate system and draw a-free-body diagram for the system.
2. Calculate
  1. Represent physics principles that will help you solve for the forces.
  2. Determine a symbolic equation for each unknown quantity in terms of known variables.
  3. Plug numbers into your symbolic answer.
3. Sensemake
  1. Check the units of your symbolic answer.
  2. Compare your numerical answers to appropriate numerical quantities with the same units.
  3. Use covariational reasoning: how should your symbolic equations depend on \(a\text{,}\) and \(\theta\text{?}\)
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