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title: "A block of mass \\(m\\) is attached to an ideal spring on a frictionless horizontal surface. The spring is modified such that its effective spring constant increases linearly with time according to \\(k(t) = k_0 + bt\\), where \\(k_0\\) and \\(b\\) are positive constants. Which of the following correctly pairs the instantaneous rate of change of the block’s angular frequency \\(\\left.\\dfrac{d\\omega}{dt}\\right|_{t=0}\\) at time \\(t = 0\\) with the qualitative trend of the oscillation period \\(T(t)\\) as time increases?"
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url: "https://nerd-notes.com/ubq/117619/"
date_modified: "2026-08-04T07:49:21+00:00"
---

# A block of mass \(m\) is attached to an ideal spring on a frictionless horizontal surface. The spring is modified such that its effective spring constant increases linearly with time according to \(k(t) = k_0 + bt\), where \(k_0\) and \(b\) are positive constants. Which of the following correctly pairs the instantaneous rate of change of the block’s angular frequency \(\left.\dfrac{d\omega}{dt}\right|_{t=0}\) at time \(t = 0\) with the qualitative trend of the oscillation period \(T(t)\) as time increases?

A block of mass \(m\) is attached to an ideal spring on a frictionless horizontal surface. The spring is modified such that its effective spring constant increases linearly with time according to \(k(t) = k_0 + bt\), where \(k_0\) and \(b\) are positive constants. Which of the following correctly pairs the instantaneous rate of change of the block's angular frequency \(\left.\dfrac{d\omega}{dt}\right|_{t=0}\) at time \(t = 0\) with the qualitative trend of the oscillation period \(T(t)\) as time increases?

- **A.** \(\left.\dfrac{d\omega}{dt}\right|_{t=0} = \dfrac{b}{\sqrt{mk_0}}\); Period \(T(t)\) increases
- **B.** \(\left.\dfrac{d\omega}{dt}\right|_{t=0} = \dfrac{b}{2\sqrt{mk_0}}\); Period \(T(t)\) decreases
- **C.** \(\left.\dfrac{d\omega}{dt}\right|_{t=0} = \dfrac{b}{\sqrt{mk_0}}\); Period \(T(t)\) decreases
- **D.** \(\left.\dfrac{d\omega}{dt}\right|_{t=0} = \dfrac{b}{4\sqrt{mk_0}}\); Period \(T(t)\) increases

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117619/*