---
title: "A block of mass \\(M\\) is attached to an ideal horizontal spring of force constant \\(k\\) and oscillates on a frictionless horizontal surface with an initial amplitude \\(A_0\\) and total mechanical energy \\(E_0\\). At the exact instant the block passes through its equilibrium position (\\(x = 0\\)), a small lump of sticky clay of mass \\(m\\) is dropped vertically onto the block from a negligible height and immediately sticks to it. Which of the following correctly gives the new amplitude of oscillation \\(A_f\\) and the ratio of the new total mechanical energy to the original mechanical energy \\(\\dfrac{E_f}{E_0}\\) of the oscillating system?"
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url: "https://nerd-notes.com/ubq/120992/"
date_modified: "2026-08-23T04:44:58+00:00"
---

# A block of mass \(M\) is attached to an ideal horizontal spring of force constant \(k\) and oscillates on a frictionless horizontal surface with an initial amplitude \(A_0\) and total mechanical energy \(E_0\). At the exact instant the block passes through its equilibrium position (\(x = 0\)), a small lump of sticky clay of mass \(m\) is dropped vertically onto the block from a negligible height and immediately sticks to it. Which of the following correctly gives the new amplitude of oscillation \(A_f\) and the ratio of the new total mechanical energy to the original mechanical energy \(\dfrac{E_f}{E_0}\) of the oscillating system?

A block of mass \(M\) is attached to an ideal horizontal spring of force constant \(k\) and oscillates on a frictionless horizontal surface with an initial amplitude \(A_0\) and total mechanical energy \(E_0\). At the exact instant the block passes through its equilibrium position (\(x = 0\)), a small lump of sticky clay of mass \(m\) is dropped vertically onto the block from a negligible height and immediately sticks to it. Which of the following correctly gives the new amplitude of oscillation \(A_f\) and the ratio of the new total mechanical energy to the original mechanical energy \(\dfrac{E_f}{E_0}\) of the oscillating system?

![A schematic diagram showing a horizontal mass-spring oscillator on a smooth horizontal surface. On the left, a vertical wall is fixed. An ideal horizontal coil spring of spring constant \(k\) is attached to the wall and extends horizontally to the right, connected to a rectangular block of mass \(M\). The block rests on a horizontal line representing a frictionless surface. A vertical dashed line passes through the center of the block, labeled below as \(x = 0\). Directly above the block, a small rounded lump of clay of mass \(m\) is depicted with a downward vertical arrow indicating it is dropped onto the top surface of the block. A horizontal double-sided arrow below the surface indicates the original amplitude of oscillation between \(-A_0\) and \(+A_0\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460298-LssqbJ.jpg)

- **A.** \(A_f = A_0\) and \(\dfrac{E_f}{E_0} = 1\)
- **B.** \(A_f = A_0 \left(\dfrac{M}{M+m}\right)\) and \(\dfrac{E_f}{E_0} = \left(\dfrac{M}{M+m}\right)^2\)
- **C.** \(A_f = A_0 \sqrt{\dfrac{M+m}{M}}\) and \(\dfrac{E_f}{E_0} = \dfrac{M+m}{M}\)
- **D.** \(A_f = A_0 \sqrt{\dfrac{M}{M+m}}\) and \(\dfrac{E_f}{E_0} = \dfrac{M}{M+m}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120992/*