A mass `m=100 ` gram is attached at the end of a light spring which oscillates on a frictionless horizontal table with an amplitude equalt to `0.16m` and time period equal to 2 second. Initially the mass is released from rest at `t=0` and displacement `x=-0.16` metre. The expression for the displacement of mass at any time `(t)` is

Options

A. `x-0.16cos(pit)`
B. `x=-0.16 cos (pit)`
C. `x=0.16 cos (pit+pi)`
D. `x=-0.16 cos (pit+pi)`

Explanation

To find the expression for the displacement of the mass attached to the spring at any time \( t \), we can follow these steps:

### Step 1: Identify the parameters
We have the following parameters:
- Mass \( m = 100 \) grams (which is \( 0.1 \) kg)
- Amplitude \( A = 0.16 \) m
- Time period \( T = 2 \) seconds
- Initial displacement \( x(0) = -0.16 \) m (released from rest)

### Step 2: Calculate the angular frequency
The angular frequency \( \omega \) is given by the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the time period:
\[ \omega = \frac{2\pi}{2} = \pi \, \text{rad/s} \]

### Step 3: Write the general equation for SHM
The general equation for simple harmonic motion (SHM) can be expressed as:
\[ x(t) = A \sin(\omega t + \phi_0) \]
where \( \phi_0 \) is the phase constant.

### Step 4: Determine the phase constant \( \phi_0 \)
At \( t = 0 \), the displacement is given as \( x(0) = -0.16 \) m. Thus, we can write:
\[ -0.16 = 0.16 \sin(\phi_0) \]
This simplifies to:
\[ \sin(\phi_0) = -1 \]
The angle for which \( \sin(\phi_0) = -1 \) is:
\[ \phi_0 = \frac{3\pi}{2} \, \text{rad} \]

### Step 5: Substitute \( \omega \) and \( \phi_0 \) into the equation
Now we can substitute \( \omega \) and \( \phi_0 \) back into the general equation:
\[ x(t) = 0.16 \sin(\pi t + \frac{3\pi}{2}) \]

### Step 6: Simplify the equation
Using the sine addition formula, we can express \( \sin(\pi t + \frac{3\pi}{2}) \) as:
\[ \sin(\pi t + \frac{3\pi}{2}) = -\cos(\pi t) \]
Thus, the equation becomes:
\[ x(t) = 0.16 \cdot (-\cos(\pi t)) = -0.16 \cos(\pi t) \]

### Final Expression
The expression for the displacement of the mass at any time \( t \) is:
\[ x(t) = -0.16 \cos(\pi t) \]

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