The equation of motion of a body executing S.H.M. is ` x= acos . (pi)/(3) ( t +1)`. Find the time at which the body comes to rest for first time.

Answer

To find the time at which the body comes to rest for the first time, we can follow these steps:

### Step 1: Understand the Equation of Motion
The equation of motion given is:
\[ x = a \cos\left(\frac{\pi}{3}(t + 1)\right) \]
where \( a \) is the amplitude of the motion.

### Step 2: Find the Velocity
The velocity \( v \) of the body is given by the derivative of the displacement \( x \) with respect to time \( t \):
\[ v = \frac{dx}{dt} \]

Differentiating \( x \):
\[ v = \frac{d}{dt}\left(a \cos\left(\frac{\pi}{3}(t + 1)\right)\right) \]
Using the chain rule, we get:
\[ v = -a \sin\left(\frac{\pi}{3}(t + 1)\right) \cdot \frac{\pi}{3} \]
Thus, the velocity can be expressed as:
\[ v = -\frac{a\pi}{3} \sin\left(\frac{\pi}{3}(t + 1)\right) \]

### Step 3: Set the Velocity to Zero
To find when the body comes to rest, we set the velocity \( v \) to zero:
\[ 0 = -\frac{a\pi}{3} \sin\left(\frac{\pi}{3}(t + 1)\right) \]

Since \( -\frac{a\pi}{3} \) is a constant and cannot be zero, we can simplify this to:
\[ \sin\left(\frac{\pi}{3}(t + 1)\right) = 0 \]

### Step 4: Solve for Time
The sine function is zero at integer multiples of \( \pi \):
\[ \frac{\pi}{3}(t + 1) = n\pi \]
where \( n \) is an integer.

Rearranging this gives:
\[ t + 1 = 3n \]
Thus,
\[ t = 3n - 1 \]

### Step 5: Find the First Time the Body Comes to Rest
To find the first time the body comes to rest, we take the smallest non-negative integer value for \( n \). The smallest value for \( n \) is 0:
\[ t = 3(0) - 1 = -1 \]
This is not a valid time since time cannot be negative.

Next, we try \( n = 1 \):
\[ t = 3(1) - 1 = 2 \]

### Conclusion
The first time the body comes to rest is:
\[ t = 2 \text{ seconds} \]