A body of mass 36 g moves with S.H.M. of amplitude A = 13 cm and time period T = 12 s. At time t = 0, the displacement x is + 13 cm. The shortest time of passage from x = + 6.5 cm to x = - 6.5 cm is

Explanation

To solve the problem step by step, let's break it down:

### Given Data:
- Mass of the body, \( m = 36 \, \text{g} = 0.036 \, \text{kg} \) (not needed for this calculation)
- Amplitude, \( A = 13 \, \text{cm} = 0.13 \, \text{m} \)
- Time period, \( T = 12 \, \text{s} \)
- Initial displacement at \( t = 0 \), \( x(0) = +13 \, \text{cm} = 0.13 \, \text{m} \)

### Step 1: Calculate Angular Frequency
The angular frequency \( \omega \) is given by the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the value of \( T \):
\[ \omega = \frac{2\pi}{12} = \frac{\pi}{6} \, \text{rad/s} \]

### Step 2: Write the Equation of Motion
The displacement \( x(t) \) in simple harmonic motion can be expressed as:
\[ x(t) = A \cos(\omega t) \]
Substituting the values of \( A \) and \( \omega \):
\[ x(t) = 0.13 \cos\left(\frac{\pi}{6} t\right) \]

### Step 3: Find Time \( t_1 \) when \( x = +6.5 \, \text{cm} \)
To find the time \( t_1 \) when the displacement is \( +6.5 \, \text{cm} = 0.065 \, \text{m} \):
\[ 0.065 = 0.13 \cos\left(\frac{\pi}{6} t_1\right) \]
Dividing both sides by \( 0.13 \):
\[ \cos\left(\frac{\pi}{6} t_1\right) = \frac{0.065}{0.13} = \frac{1}{2} \]
Taking the inverse cosine:
\[ \frac{\pi}{6} t_1 = \cos^{-1}\left(\frac{1}{2}\right) \]
Since \( \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \):
\[ \frac{\pi}{6} t_1 = \frac{\pi}{3} \]
Solving for \( t_1 \):
\[ t_1 = \frac{\pi/3}{\pi/6} = 2 \, \text{s} \]

### Step 4: Find Time \( t_2 \) when \( x = -6.5 \, \text{cm} \)
Now, find the time \( t_2 \) when the displacement is \( -6.5 \, \text{cm} = -0.065 \, \text{m} \):
\[ -0.065 = 0.13 \cos\left(\frac{\pi}{6} t_2\right) \]
Dividing both sides by \( 0.13 \):
\[ \cos\left(\frac{\pi}{6} t_2\right) = -\frac{0.065}{0.13} = -\frac{1}{2} \]
Taking the inverse cosine:
\[ \frac{\pi}{6} t_2 = \cos^{-1}\left(-\frac{1}{2}\right) \]
Since \( \cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3} \):
\[ \frac{\pi}{6} t_2 = \frac{2\pi}{3} \]
Solving for \( t_2 \):
\[ t_2 = \frac{2\pi/3}{\pi/6} = 4 \, \text{s} \]

### Step 5: Find the Shortest Time of Passage
The shortest time of passage from \( x = +6.5 \, \text{cm} \) to \( x = -6.5 \, \text{cm} \) is:
\[ \Delta t = t_2 - t_1 = 4 \, \text{s} - 2 \, \text{s} = 2 \, \text{s} \]

### Final Answer
The shortest time of passage from \( x = +6.5 \, \text{cm} \) to \( x = -6.5 \, \text{cm} \) is **2 seconds**.

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