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title: "A block of mass \\(m\\) attached to an ideal spring of spring constant \\(k\\) undergoes weakly damped simple harmonic motion. Due to a resistive force, the amplitude of oscillation as a function of time \\(t\\) is given by \\(A(t) = A_0 e^{-\\frac{b}{2m} t}\\), where \\(A_0\\) is the initial amplitude and \\(b\\) is a positive damping constant. Which of the following expressions represents the time \\(t_{1/2}\\) required for the total mechanical energy of the block-spring system to decrease to half of its initial value?"
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url: "https://nerd-notes.com/ubq/117614/"
date_modified: "2026-08-04T07:49:19+00:00"
---

# A block of mass \(m\) attached to an ideal spring of spring constant \(k\) undergoes weakly damped simple harmonic motion. Due to a resistive force, the amplitude of oscillation as a function of time \(t\) is given by \(A(t) = A_0 e^{-\frac{b}{2m} t}\), where \(A_0\) is the initial amplitude and \(b\) is a positive damping constant. Which of the following expressions represents the time \(t_{1/2}\) required for the total mechanical energy of the block-spring system to decrease to half of its initial value?

A block of mass \(m\) attached to an ideal spring of spring constant \(k\) undergoes weakly damped simple harmonic motion. Due to a resistive force, the amplitude of oscillation as a function of time \(t\) is given by \(A(t) = A_0 e^{-\frac{b}{2m} t}\), where \(A_0\) is the initial amplitude and \(b\) is a positive damping constant. Which of the following expressions represents the time \(t_{1/2}\) required for the total mechanical energy of the block-spring system to decrease to half of its initial value?

- **A.** \(\dfrac{2m \ln 2}{b}\)
- **B.** \(\dfrac{m \ln 2}{b}\)
- **C.** \(\dfrac{m \ln 2}{2b}\)
- **D.** \(\dfrac{m}{b \ln 2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117614/*