A body osicllates with SHM according to the equation `x=6cos(2pit+(pi)/(4))`. Its instantaneous displacement at t = 1 sec is

Options

A. `(sqrt(2))/(5)`
B. `(1)/(sqrt(3))`
C. `(1)/(sqrt(2))`
D. `(6)/(sqrt(2))`

Explanation

To find the instantaneous displacement of a body oscillating with simple harmonic motion (SHM) described by the equation \( x = 6 \cos(2\pi t + \frac{\pi}{4}) \) at \( t = 1 \) second, we can follow these steps:

### Step-by-Step Solution:

1. **Write down the equation of motion**:
The equation given is:
\[ x = 6 \cos(2\pi t + \frac{\pi}{4}) \]

2. **Substitute \( t = 1 \) second into the equation**:
We need to find \( x \) when \( t = 1 \):
\[ x = 6 \cos(2\pi \cdot 1 + \frac{\pi}{4}) \]
Simplifying this, we get:
\[ x = 6 \cos(2\pi + \frac{\pi}{4}) \]

3. **Use the periodic property of the cosine function**:
The cosine function is periodic with a period of \( 2\pi \), which means:
\[ \cos(2\pi + \theta) = \cos(\theta) \]
Therefore:
\[ x = 6 \cos(\frac{\pi}{4}) \]

4. **Calculate \( \cos(\frac{\pi}{4}) \)**:
The value of \( \cos(\frac{\pi}{4}) \) is:
\[ \cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}} \]

5. **Substitute this value back into the equation**:
Now substituting this back, we have:
\[ x = 6 \cdot \frac{1}{\sqrt{2}} = \frac{6}{\sqrt{2}} \]

6. **Rationalize the denominator** (if necessary):
To express it in a more standard form:
\[ x = \frac{6 \sqrt{2}}{2} = 3\sqrt{2} \]

### Final Answer:
Thus, the instantaneous displacement at \( t = 1 \) second is:
\[ x = \frac{6}{\sqrt{2}} \text{ units} \quad \text{(or equivalently } 3\sqrt{2} \text{ units)} \]