A particle executes S.H.M. with a time period of 3s. The time taken by the particle to go directly from its mean position to half of its amplitude is-

Options

A. 1s
B. 3/4 s
C. 1/3 s
D. 1/4 s

Answer

To solve the problem step by step, we will analyze the motion of the particle executing Simple Harmonic Motion (SHM) and find the time taken to go from the mean position to half of its amplitude.

### Step 1: Understand the given information
The time period (T) of the particle is given as 3 seconds.

### Step 2: Calculate the angular frequency (ω)
The angular frequency (ω) can be calculated using the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the value of T:
\[ \omega = \frac{2\pi}{3} \text{ rad/s} \]

### Step 3: Write the equation for displacement in SHM
The displacement (x) of a particle in SHM can be expressed as:
\[ x = A \sin(\omega t) \]
where A is the amplitude and t is the time.

### Step 4: Set up the equation for half of the amplitude
We need to find the time taken to reach half of the amplitude (A/2). Therefore, we set:
\[ \frac{A}{2} = A \sin(\omega t) \]
Dividing both sides by A (assuming A ≠ 0):
\[ \frac{1}{2} = \sin(\omega t) \]

### Step 5: Solve for ωt
We know that:
\[ \sin(\theta) = \frac{1}{2} \implies \theta = \frac{\pi}{6} \text{ (30 degrees)} \]
Thus, we have:
\[ \omega t = \frac{\pi}{6} \]

### Step 6: Solve for time (t)
Now, substituting the value of ω:
\[ \frac{2\pi}{3} t = \frac{\pi}{6} \]
To find t, we rearrange the equation:
\[ t = \frac{\pi/6}{2\pi/3} \]
This simplifies to:
\[ t = \frac{\pi}{6} \cdot \frac{3}{2\pi} = \frac{3}{12} = \frac{1}{4} \text{ seconds} \]

### Final Answer
The time taken by the particle to go directly from its mean position to half of its amplitude is:
\[ \frac{1}{4} \text{ seconds} \]

---