A particle is oscillating according to the equation `X=7 cos 0.5 pit`, where t is in second. The point moves from the position of equilibrium to maximum displacement in time

Options

A. 4.0 sec
B. 2.0 sec
C. 1.0 sec
D. 0.5 sec

Answer

To solve the problem of how long it takes for a particle to move from the position of equilibrium to maximum displacement, we will follow these steps:

### Step 1: Identify the given equation
The equation of motion for the particle is given as:
\[ X = 7 \cos(0.5 \pi t) \]
where \( t \) is in seconds.

### Step 2: Determine the amplitude and angular frequency
From the equation, we can identify:
- Amplitude \( A = 7 \) (the maximum displacement from the equilibrium position)
- Angular frequency \( \omega = 0.5 \pi \) (the coefficient of \( t \) in the cosine function)

### Step 3: Calculate the time period
The time period \( T \) of the oscillation can be calculated using the formula:
\[ T = \frac{2\pi}{\omega} \]
Substituting the value of \( \omega \):
\[ T = \frac{2\pi}{0.5\pi} = \frac{2}{0.5} = 4 \text{ seconds} \]

### Step 4: Determine the time to move from equilibrium to maximum displacement
In simple harmonic motion, the time taken to move from the equilibrium position to the maximum displacement is one-fourth of the time period:
\[ \text{Time from equilibrium to maximum displacement} = \frac{T}{4} \]
Substituting the value of \( T \):
\[ \text{Time} = \frac{4 \text{ seconds}}{4} = 1 \text{ second} \]

### Conclusion
Thus, the time taken for the particle to move from the position of equilibrium to maximum displacement is **1 second**.

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