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Learning Introductory Physics with Activities

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

Section 3.9 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 3.9.1. Steps to follow when you consider an A*R*C*S activity.
Below is an example of using all the A*R*C*S steps.

Exercises Exercises

1. The Brick against the Rail.

A small, heavy brick of mass \(40 \mathrm{~kg}\) is positioned against a horizontal metal railing. You pull on the rope at the \(60^o\) angle shown so that the acceleration of the brick is \(1.5 \mathrm{~m/s^2}\) to the left. Find the magnitudes of all forces on the brick.
Figure 3.9.2. A brick is pulled against a metal 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. Identify relevant laws 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 \(\theta\text{?}\)

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 3.9.3. 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. Identify relevant laws that will help you solve for the tensions.
  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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