A particle performs linear S.H.M. of period 4 seconds and amplitude 4 cm. Find the time taken by it to travel a distance of 1 cm from the positive extreme position.

Answer

To solve the problem step by step, we will follow the principles of simple harmonic motion (SHM).

### Given Data:
- Period (T) = 4 seconds
- Amplitude (A) = 4 cm
- Distance to travel from the positive extreme position = 1 cm

### Step 1: Calculate Angular Frequency (ω)
The angular frequency (ω) can be calculated using the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the value of T:
\[ \omega = \frac{2\pi}{4} = \frac{\pi}{2} \text{ rad/s} \]

**Hint:** Remember that the angular frequency is related to the period of the motion.

### Step 2: Determine the Displacement from Mean Position
Since the particle is moving from the positive extreme position (which is at +A), the displacement (x) when the particle has traveled 1 cm from the positive extreme position is:
\[ x = A - 1 \text{ cm} = 4 \text{ cm} - 1 \text{ cm} = 3 \text{ cm} \]

**Hint:** The displacement from the mean position is calculated by subtracting the distance traveled from the amplitude.

### Step 3: Write the Equation of SHM
The equation of motion for SHM is given by:
\[ x = A \cos(\omega t) \]
Substituting the values we have:
\[ 3 = 4 \cos\left(\frac{\pi}{2} t\right) \]

**Hint:** The equation of SHM relates displacement to time through the cosine function.

### Step 4: Solve for Cosine
Rearranging the equation:
\[ \cos\left(\frac{\pi}{2} t\right) = \frac{3}{4} \]

**Hint:** Isolate the cosine term to find the angle.

### Step 5: Find the Angle
Now, we need to find the angle whose cosine is \( \frac{3}{4} \):
\[ \frac{\pi}{2} t = \cos^{-1}\left(\frac{3}{4}\right) \]

Using a calculator, we find:
\[ \cos^{-1}\left(\frac{3}{4}\right) \approx 41.41^\circ \text{ or } 0.722 \text{ radians} \]

**Hint:** Use a scientific calculator to find the inverse cosine value.

### Step 6: Solve for Time (t)
Now we can solve for t:
\[ t = \frac{2 \cdot \cos^{-1}\left(\frac{3}{4}\right)}{\pi} \]
Substituting the value:
\[ t \approx \frac{2 \cdot 0.722}{\pi} \approx 0.459 \text{ seconds} \]

**Hint:** Remember to convert the angle from degrees to radians if necessary.

### Final Answer
The time taken by the particle to travel a distance of 1 cm from the positive extreme position is approximately:
\[ t \approx 0.459 \text{ seconds} \]