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title: "A block of mass \\(M\\) is attached to an ideal horizontal spring of spring constant \\(k\\) and undergoes simple harmonic motion on a frictionless surface with amplitude \\(A_0\\) and frequency \\(f_0\\). A lump of sticky clay of mass \\(m\\) is dropped vertically from a negligible height onto the block and immediately sticks to it without slipping. In Scenario 1, the clay sticks to the block at the instant the block is at a turnaround point (\\(x = A_0\\)); in Scenario 2, an identical lump of clay sticks to the block at the instant the block passes through the equilibrium position (\\(x = 0\\)). Which of the following correctly expresses the resulting amplitude and frequency of oscillation for each scenario?"
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url: "https://nerd-notes.com/ubq/124319/"
date_modified: "2026-09-28T14:04:48+00:00"
---

# A block of mass \(M\) is attached to an ideal horizontal spring of spring constant \(k\) and undergoes simple harmonic motion on a frictionless surface with amplitude \(A_0\) and frequency \(f_0\). A lump of sticky clay of mass \(m\) is dropped vertically from a negligible height onto the block and immediately sticks to it without slipping. In Scenario 1, the clay sticks to the block at the instant the block is at a turnaround point (\(x = A_0\)); in Scenario 2, an identical lump of clay sticks to the block at the instant the block passes through the equilibrium position (\(x = 0\)). Which of the following correctly expresses the resulting amplitude and frequency of oscillation for each scenario?

A block of mass \(M\) is attached to an ideal horizontal spring of spring constant \(k\) and undergoes simple harmonic motion on a frictionless surface with amplitude \(A_0\) and frequency \(f_0\). A lump of sticky clay of mass \(m\) is dropped vertically from a negligible height onto the block and immediately sticks to it without slipping. In Scenario 1, the clay sticks to the block at the instant the block is at a turnaround point (\(x = A_0\)); in Scenario 2, an identical lump of clay sticks to the block at the instant the block passes through the equilibrium position (\(x = 0\)). Which of the following correctly expresses the resulting amplitude and frequency of oscillation for each scenario?

![A horizontal surface with a vertical wall on the far left. An ideal horizontal coiled spring extends horizontally to the right from the wall to a rectangular block of mass \(M\). The block rests on the horizontal surface. A vertical dashed line passes through the center of the block's neutral position and is labeled \(x = 0\). A second vertical dashed line to the right is labeled \(x = A_0\). Directly above the position \(x = A_0\), a small circular lump of clay of mass \(m\) is shown with a single downward straight arrow pointing toward the surface of the block. A double-headed horizontal arrow below the surface spans from \(-A_0\) to \(A_0\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604287-tKihbP.jpg)

- **A.** \(A_1 = A_0\), \(A_2 = A_0\sqrt{\dfrac{M}{M+m}}\), and \(f_1 = f_2 = f_0\sqrt{\dfrac{M}{M+m}}\)
- **B.** \(A_1 = A_0\sqrt{\dfrac{M}{M+m}}\), \(A_2 = A_0\sqrt{\dfrac{M}{M+m}}\), and \(f_1 = f_2 = f_0\sqrt{\dfrac{M}{M+m}}\)
- **C.** \(A_1 = A_0\), \(A_2 = A_0\left(\dfrac{M}{M+m}\right)\), and \(f_1 = f_2 = f_0\sqrt{\dfrac{M}{M+m}}\)
- **D.** \(A_1 = A_0\), \(A_2 = A_0\sqrt{\dfrac{M}{M+m}}\), and \(f_1 = f_0, \; f_2 = f_0\sqrt{\dfrac{M}{M+m}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124319/*