Question

A mass attached to a spring performs S.H.M. whose displacement is \(\mathrm{x}=3 \times 10^{-3} \cos 2 \pi \mathrm{t}\) metre. The time taken to obtain maximum speed for the first time is

• A. \(\frac{1}{12}\text{ }\mathrm{s}\)
• B. \(\frac{1}{8}\text{ }\mathrm{s}\)
• C. \(\frac{1}{4}\text{ }\mathrm{s}\)
• D. \(\frac{1}{2}\text{ }\mathrm{s}\)

Hint:

Speed is maximum at the mean position, which a cosine motion reaches a quarter of a period after the extreme position.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Given \(x=3\times10^{-3}\cos 2\pi t\), so \(\omega=2\pi\ \text{rad/s}\).
Period \(T=\dfrac{2\pi}{\omega}=1\text{ s}\).

Step 2: Where is speed maximum
Speed is maximum at the mean position \(x=0\).

Step 3: Calculation
At \(t=0\), \(x\) is at the extreme (amplitude).
\(\cos 2\pi t=0\Rightarrow 2\pi t=\dfrac{\pi}{2}\Rightarrow t=\dfrac{1}{4}\text{ s}\).

Step 4: Compare the options
\(\dfrac{1}{4}\text{ s}\) matches the third option.

Final Answer:
The mass first attains maximum speed after one quarter of a second. \[ \boxed{\text{(C) }\dfrac{1}{4}\text{ s}} \]