---
title: "A block of mass \\(m\\) attached to an ideal horizontal spring of spring constant \\(k\\) oscillates along the \\(x\\)-axis on a frictionless surface with amplitude \\(A\\) and kinematic period \\(T_0\\). The accompanying graph displays the kinetic energy \\(K\\) (dashed curve) and the elastic potential energy \\(U\\) (solid curve) of the block-spring system as functions of time \\(t\\) over the interval \\(0 \\le t \\le T_0\\).  Which of the following statements correctly interprets the graphical features of the energy curves in terms of the kinematic motion of the block?"
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url: "https://nerd-notes.com/ubq/124247/"
date_modified: "2026-09-28T14:04:29+00:00"
---

# A block of mass \(m\) attached to an ideal horizontal spring of spring constant \(k\) oscillates along the \(x\)-axis on a frictionless surface with amplitude \(A\) and kinematic period \(T_0\). The accompanying graph displays the kinetic energy \(K\) (dashed curve) and the elastic potential energy \(U\) (solid curve) of the block-spring system as functions of time \(t\) over the interval \(0 \le t \le T_0\).

Which of the following statements correctly interprets the graphical features of the energy curves in terms of the kinematic motion of the block?

A block of mass \(m\) attached to an ideal horizontal spring of spring constant \(k\) oscillates along the \(x\)-axis on a frictionless surface with amplitude \(A\) and kinematic period \(T_0\). The accompanying graph displays the kinetic energy \(K\) (dashed curve) and the elastic potential energy \(U\) (solid curve) of the block-spring system as functions of time \(t\) over the interval \(0 \le t \le T_0\).

Which of the following statements correctly interprets the graphical features of the energy curves in terms of the kinematic motion of the block?

![A 2D line graph plotted on a rectangular coordinate system. The horizontal axis is labeled with time t and features five evenly spaced major tick marks labeled 0, T_0/4, T_0/2, 3T_0/4, and T_0. The vertical axis is labeled Energy and features two tick marks labeled 0 at the origin and E_total at the top. Two curves are plotted from t = 0 to t = T_0. A solid curve, representing potential energy U, begins at (0, E_total), curves downward to touch 0 at t = T_0/4, rises to a peak at (T_0/2, E_total), drops to 0 at t = 3T_0/4, and rises back to (T_0, E_total). A dashed curve, representing kinetic energy K, begins at (0, 0), rises to a peak at (T_0/4, E_total), drops to 0 at (T_0/2, 0), rises to a peak at (3T_0/4, E_total), and returns to 0 at (T_0, 0). A rectangular legend in the upper right indicates that the solid line represents U and the dashed line represents K. No other labels, gridlines, text, or curves appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604269-609Zxo.jpg)

- **A.** The period of the energy oscillations is \(T_0\), because mechanical energy can complete a full cycle only when the block returns to both its initial position and initial direction of motion.
- **B.** The frequency of the energy oscillations is \(\dfrac{1}{2}f_0\), because complete conversion between kinetic and potential energy occurs only once every kinematic period \(T_0\).
- **C.** The period of the energy oscillations is \(\dfrac{1}{2}T_0\), because the restoring force vanishes twice per kinematic cycle, causing \(K\) and \(U\) to simultaneously reach minimum values at those times.
- **D.** The frequency of the energy oscillations is \(2f_0\), because the energy functions depend on the squares of velocity and displacement, causing each energy to reach its maximum twice per kinematic cycle.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124247/*