Question

A mass attached to a spring performs S.H.M. whose displacement is \(x = 3\times 10^{-3}cos2πt\) m. The time taken to obtain maximum speed for the first time is (in s)

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

Hint:

Maximum speed occurs at the mean position, where x = 0.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In SHM the speed is maximum when the mass passes through the mean position (\(x=0\)).

Step 2: Set x = 0:
\(3\times10^{-3}\cos2\pi t=0\), so \(\cos2\pi t=0\). The smallest positive solution is \(2\pi t=\dfrac\pi2\), so \(t=\dfrac14\) s.

Step 3: Check with the period:
\(\omega=2\pi\), so \(T=1\) s. The body starts at the extreme position and reaches the mean position after \(\dfrac T4=\dfrac14\) s. Option C.

Step 4: Why the other options are wrong.
1 s is a full period, back at the extreme. 1/2 s is the opposite extreme. 1/8 s is not a special point of the motion.

Final Answer:
The time is 1/4 s. \[ \boxed{\text{(C) }\dfrac14\ \text{s}} \]