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Step-by-Step Solution
Step 1: Write down the displacement equation
The displacement of the object undergoing simple harmonic motion (SHM) is given by:
$x = 2 \times 10^{-2} \cos(\pi t)$
This represents the position of the object at any time $t$ (in seconds).
Step 2: Differentiate to find the velocity
The velocity $v$ is the first derivative of displacement with respect to time:
$v = \frac{dx}{dt} = \frac{d}{dt}\bigl(2 \times 10^{-2} \cos(\pi t)\bigr)$
Using the derivative of $\cos(\pi t)$, we get:
$v = 2 \times 10^{-2} \times (-\sin(\pi t)) \times \pi$
Hence,
$v = 2 \times 10^{-2} \,\pi \sin(\pi t)$
(Note the negative sign from the derivative of cosine is canceled because in SHM, we often focus on the magnitude of velocity when discussing maximum speed, but here the sign only indicates direction. For maximum speed, we consider the absolute value.)
Step 3: Determine the condition for maximum speed
The speed (the magnitude of velocity) will be maximum when the magnitude of the factor multiplying $2 \times 10^{-2} \,\pi$ is at its greatest. The factor is $|\sin(\pi t)|$, which is maximum when $\sin(\pi t) = \pm 1.$ For the first instance when this happens positively:
$\sin(\pi t) = 1$
Step 4: Solve for time
We know $\sin(\pi t) = 1$ occurs first at:
$\pi t = \frac{\pi}{2}
\quad \Rightarrow \quad
t = \frac{1}{2} = 0.5 \text{ s}$
This is the first time the speed is maximum. Therefore, the correct answer is 0.5 s.
