A particle executes SHM with time period 8 s. Initially, it is at its mean position. The ratio of distance travelled by it in the 1st second to that in the 2nd second is

Options

A. `sqrt2:1`
B. `1:(sqrt2-1)`
C. `(sqrt2+1):sqrt2`
D. `(sqrt2-1):1`

Explanation

To solve the problem step by step, we will calculate the distance traveled by the particle in the first second and the second second during its Simple Harmonic Motion (SHM).

### Step 1: Determine the angular frequency (ω)
Given that the time period (T) is 8 seconds, we can calculate the angular frequency (ω) using the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the value of T:
\[ \omega = \frac{2\pi}{8} = \frac{\pi}{4} \text{ radians/second} \]

**Hint:** Remember that angular frequency is related to the time period by this formula.

### Step 2: Write the equation of motion for SHM
Since the particle starts from the mean position, the displacement (x) as a function of time (t) is given by:
\[ x(t) = a \sin(\omega t) \]
Substituting the value of ω:
\[ x(t) = a \sin\left(\frac{\pi}{4} t\right) \]

**Hint:** The sine function describes the position of the particle in SHM, starting from the mean position.

### Step 3: Calculate the distance traveled in the first second (t = 1s)
Substituting \(t = 1\) into the equation:
\[ x(1) = a \sin\left(\frac{\pi}{4} \cdot 1\right) = a \sin\left(\frac{\pi}{4}\right) = a \cdot \frac{1}{\sqrt{2}} = \frac{a}{\sqrt{2}} \]

**Hint:** Use the sine value for \(\frac{\pi}{4}\) to find the position at \(t = 1\) second.

### Step 4: Calculate the distance traveled in the second second (t = 2s)
First, we find the position at \(t = 2\):
\[ x(2) = a \sin\left(\frac{\pi}{4} \cdot 2\right) = a \sin\left(\frac{\pi}{2}\right) = a \]

**Hint:** Remember that \(\sin\left(\frac{\pi}{2}\right) = 1\).

### Step 5: Calculate the distance traveled in the second second
The distance traveled in the second second is the difference between the positions at \(t = 2\) seconds and \(t = 1\) second:
\[ \text{Distance in 2nd second} = x(2) - x(1) = a - \frac{a}{\sqrt{2}} = a\left(1 - \frac{1}{\sqrt{2}}\right) \]

**Hint:** The distance traveled in a time interval is the change in position.

### Step 6: Find the ratio of distances traveled in the first second to the second second
Now we can find the ratio:
\[ \text{Ratio} = \frac{\text{Distance in 1st second}}{\text{Distance in 2nd second}} = \frac{\frac{a}{\sqrt{2}}}{a\left(1 - \frac{1}{\sqrt{2}}\right)} \]
This simplifies to:
\[ \text{Ratio} = \frac{1/\sqrt{2}}{1 - \frac{1}{\sqrt{2}}} \]

### Step 7: Simplify the ratio
To simplify:
\[ \text{Ratio} = \frac{1}{\sqrt{2} \left(1 - \frac{1}{\sqrt{2}}\right)} = \frac{1}{\sqrt{2} - 1} \]

**Hint:** Make sure to simplify the fraction carefully.

### Final Answer
Thus, the ratio of the distance traveled by the particle in the first second to that in the second second is:
\[ \text{Ratio} = \frac{1}{\sqrt{2} - 1} \]