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Topic 2.8: Practice Sheet — Multi-Spring Systems

Name: _______________________________ Date: _______________ Period: ______

Directions: For every problem, start from Hooke's Law ($F_s = k\Delta x$, magnitude only) and solve symbolically before substituting numbers.

  1. Two Springs, One Block. A block on a horizontal surface is pushed by two horizontal springs, both anchored to the same wall and both compressed, so both push the block in the same direction. Spring 1 ($k_1=170\text{ N/m}$) is compressed $0.12\text{ m}$; Spring 2 ($k_2=210\text{ N/m}$) is also compressed $0.12\text{ m}$. The block stays at rest, held by friction. Find the friction force needed.
  2. Springs Sharing a Load. A single spring ($k=180\text{ N/m}$) holds a $5.0\text{ kg}$ mass at rest. In a second setup, TWO identical springs (each $k=180\text{ N/m}$) share a $10.0\text{ kg}$ mass equally. Find the stretch in each case, and explain whether they're equal.
  3. The Buoyancy Twist. A $3.0\text{ kg}$ block hangs at rest from a vertical spring ($k=120\text{ N/m}$) in air. The block is then fully submerged in water and comes to rest again, with the spring now stretched only $0.10\text{ m}$. Find (a) the original stretch in air, (b) the spring force while submerged, (c) the buoyant force from the water.
  4. Conceptual: Direction of Acceleration. A block attached to a horizontal spring oscillates back and forth on a frictionless surface. At the instant the block is to the RIGHT of equilibrium and moving further right (away from equilibrium), which direction does its acceleration point? Explain using the restoring-force idea and Newton's second law.

Progress Check Style

  1. Two identical springs (each $k=100\text{ N/m}$) share a $6.0\text{ kg}$ hanging mass equally. Which of the following correctly gives the stretch in EACH spring? (A) $\dfrac{(6.0)(9.8)}{100}$ (B) $\dfrac{(6.0)(9.8)}{2(100)}$ (C) $\dfrac{2(6.0)(9.8)}{100}$ (D) $\dfrac{(6.0)(9.8)}{100^2}$

AP Classroom-Style Questions

(Modeled directly on the real Topic 2.8 AP Classroom quiz question styles — multi-spring reasoning, ranking across scenarios, and conceptual "why" justifications.)

  1. Two Springs, One Block. A block on a horizontal surface is attached to two horizontal springs (spring constants $k_1$ and $k_2$), one on each side. Both springs are compressed a distance $\Delta x$ from their unstretched length, and the block remains at rest due to static friction between the block and the surface. Write a symbolic expression for the magnitude of the static friction force acting on the block.
  2. Ranking Across Scenarios. Three identical vertical springs each hold up a hanging block at equilibrium: Scenario A holds mass $m$, Scenario B holds mass $2m$, and Scenario C uses TWO of these identical springs side by side (each sharing the load equally) to hold up mass $2m$. Rank the stretch distances $\Delta x_A$, $\Delta x_B$, and $\Delta x_C$ from smallest to largest, and justify your ranking.
  3. Conceptual: The Submerged Spring. A block hangs at rest from a vertical spring, stretching it to length $L$. The block is then completely submerged in a tank of water and comes to rest again — and the new spring length is LESS than $L$. A student claims the spring constant $k$ must have decreased because of the water. Explain why this reasoning is incorrect, and give the actual explanation for why the spring's length decreased.

Keep this sheet — it's part of your semester study materials, same as your Concept Sheets.