Aim for FivePhysics 1

Mathematical routines (MR)

Weighing a submerged aluminum block

  • Units 2 and 8
  • 10 points
  • About 22 minutes

You can use a calculator on this question, just like on exam day.

A multipart problem where you use math to analyze a physical situation. You draw or sketch a representation such as a free-body diagram or a graph, derive an equation using letters, calculate a numerical answer with units, and then make a claim or prediction and justify it with physics reasoning. On the exam: Question 1 of 4. Section II has 4 questions in 95 minutes (50% of the exam score), one of each type in this order; calculator allowed. The CED suggests 20–25 minutes.

The question

A solid aluminum block of volume V = 2.0 × 10⁻⁴ m³ and density ρ_Al = 2700 kg/m³ hangs at rest from a spring scale by a light string. The block is completely submerged in water (density ρ_w = 1000 kg/m³) in a beaker and does not touch the beaker's sides or bottom. The beaker sits on a digital balance (a platform scale). Use g = 9.8 m/s² (answers that use g = 10 m/s² are also accepted).

Suggested time: 22 minutes

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Part (a)

2 points

Describe the free-body diagram of the submerged block. List each force, name the object that exerts it, give its direction, and compare the lengths of the arrows.

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Part (b)

2 points

Derive an expression for the reading on the spring scale in terms of ρ_Al, ρ_w, V, and g.

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Part (c)

2 points

Calculate the reading on the spring scale.

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Part (d)

2 points

Before the block was lowered into the water, the balance under the beaker read 12.0 N. Predict the balance reading while the block hangs submerged as described. Justify your prediction using Newton's third law.

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Part (e)

2 points

The block is lowered farther, so it is deeper in the water but still not touching the bottom. Does the spring scale reading increase, decrease, or stay the same? Justify your answer.

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