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

Options

A. 4s
B. 2s
C. 6s
D. 3s

Answer

To solve the problem, we need to determine the shortest time taken for a body in Simple Harmonic Motion (S.H.M.) to move from a displacement of +6.5 cm to -6.5 cm. Given the parameters of the motion, we can follow these steps:

### Step 1: Understand the parameters of S.H.M.
- Mass of the body, \( m = 36 \, \text{g} = 0.036 \, \text{kg} \)
- Amplitude, \( A = 13 \, \text{cm} = 0.13 \, \text{m} \)
- Time period, \( T = 12 \, \text{s} \)
- Initial displacement at \( t = 0 \), \( x(0) = +13 \, \text{cm} \)

### Step 2: Write the equation of motion for S.H.M.
Since the body starts from the positive extreme, we can express the displacement \( x(t) \) as:
\[ x(t) = A \cos(\omega t) \]
where \( \omega \) is the angular frequency given by:
\[ \omega = \frac{2\pi}{T} = \frac{2\pi}{12} = \frac{\pi}{6} \, \text{rad/s} \]

### Step 3: Find the time when \( x = +6.5 \, \text{cm} \)
Set \( x(t) = 6.5 \, \text{cm} = 0.065 \, \text{m} \):
\[ 0.065 = 0.13 \cos\left(\frac{\pi}{6} t\right) \]
\[ \cos\left(\frac{\pi}{6} t\right) = \frac{0.065}{0.13} = \frac{1}{2} \]
The angle whose cosine is \( \frac{1}{2} \) is \( \frac{\pi}{3} \) radians. Thus:
\[ \frac{\pi}{6} t_1 = \frac{\pi}{3} \implies t_1 = 2 \, \text{s} \]

### Step 4: Find the time when \( x = -6.5 \, \text{cm} \)
Set \( x(t) = -6.5 \, \text{cm} = -0.065 \, \text{m} \):
\[ -0.065 = 0.13 \cos\left(\frac{\pi}{6} t\right) \]
\[ \cos\left(\frac{\pi}{6} t\right) = -\frac{1}{2} \]
The angle whose cosine is \( -\frac{1}{2} \) is \( \frac{2\pi}{3} \) radians. Thus:
\[ \frac{\pi}{6} t_2 = \frac{2\pi}{3} \implies t_2 = 4 \, \text{s} \]

### Step 5: Calculate the time difference
The time taken to move from \( +6.5 \, \text{cm} \) to \( -6.5 \, \text{cm} \) 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**.