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Topic 3.3: Practice Sheet — Spring Potential Energy

Name: _______________________________ Date: _______________ Period: ______

Directions: Measure $\Delta x$ from the spring's relaxed length. Write $U_s=\tfrac12k(\Delta x)^2$ before substituting, and use $g=10\text{ m/s}^2$. Show every step.

Warm-Up Level (straightforward, one-step)

  1. A spring with $k=400\text{ N/m}$ is compressed $0.050\text{ m}$. Find its potential energy.
  2. A spring stores $2.0\text{ J}$ when stretched $0.10\text{ m}$. How much does it store when stretched $0.30\text{ m}$?
  3. The same spring is (a) stretched $0.10\text{ m}$ and (b) compressed $0.10\text{ m}$. Compare the stored energies and explain.

Standard Level (multi-step, matches typical AP Classroom depth)

  1. A spring ($k=100\text{ N/m}$) is compressed $0.10\text{ m}$, then compressed an additional $0.10\text{ m}$. Find $\Delta U_1$ and $\Delta U_2$, and compare them.
  2. A block on a horizontal spring ($k=80\text{ N/m}$) is held at $x=-0.30\text{ m}$, then moved slowly at constant speed to $x=+0.30\text{ m}$. Find $U_s$ at $-0.30\text{ m}$, $0$, and $+0.30\text{ m}$, and describe how $U_s$ changes.
  3. A $0.20\text{ kg}$ egg is placed on a vertical spring ($k=40\text{ N/m}$) at its relaxed length and lowered slowly to equilibrium. Find the compression, $\Delta U_g$, and $\Delta U_s$.
  4. A spring ($k=50\text{ N/m}$) is stretched from $x=0$ to $x=0.40\text{ m}$. (a) Sketch the $F$-vs-$x$ graph and find the area under it. (b) Confirm the area matches $\tfrac12kx^2$. (c) Find the work needed to stretch it from $x=0.20\text{ m}$ to $x=0.40\text{ m}$ using the area of the region between those positions.

AP-Level (multi-part: calculate + justify/represent/predict-a-change)

  1. A spring is already compressed by $x_0$. It is compressed a further $d$ (change $\Delta U_1$), then another $d$ (change $\Delta U_2$). a) Write $\Delta U_1$ and $\Delta U_2$ in terms of $k$, $x_0$, and $d$. b) Show that $\Delta U_2-\Delta U_1=kd^2$. c) A student argues, "The changes can't be compared without knowing the original compression $x_0$." Use part (b) to explain why the student is wrong.
  2. A person compresses a spring slowly at constant speed. Kinetic energy doesn't change, so net work on the block is zero. Explain how that's consistent with the person doing positive work and the spring storing energy.

Progress Check Style (evaluate the method — AP Classroom format)

  1. A block is attached to a compressed horizontal spring and held at $x=-A$. A person slowly moves the block at constant speed through $x=0$ (spring relaxed) to $x=+A$. How does the potential energy of the spring–block system change? (A) It decreases throughout the motion. (B) It increases throughout the motion. (C) It decreases, then increases. (D) It increases, then decreases.
  2. An egg is placed gently on a vertical spring at its unstretched length, then slowly lowered until it rests in equilibrium. Which correctly gives the signs of $\Delta U_g$ (Earth–egg) and $\Delta U_s$ (spring)? (A) $\Delta U_g>0$, $\Delta U_s>0$ (B) $\Delta U_g>0$, $\Delta U_s<0$ (C) $\Delta U_g<0$, $\Delta U_s>0$ (D) $\Delta U_g<0$, $\Delta U_s<0$

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