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

Options

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

Answer

To solve the problem of how long it takes for a particle oscillating according to the equation \( X = 7 \cos(0.5 \pi t) \) to move from the position of equilibrium to maximum displacement, we can follow these steps:

### Step 1: Identify the parameters from the equation
The equation of motion is given as:
\[ X = 7 \cos(0.5 \pi t) \]

From this equation, we can identify:
- The amplitude \( a = 7 \) (maximum displacement)
- The angular frequency \( \omega = 0.5 \pi \)

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

Substituting the value of \( \omega \):
\[ T = \frac{2\pi}{0.5 \pi} \]
\[ T = \frac{2\pi}{\frac{1}{2} \pi} = 4 \text{ seconds} \]

### Step 3: Determine the time to move from equilibrium to maximum displacement
In one complete oscillation (from maximum displacement to maximum displacement), the time is divided into four equal parts:
- From equilibrium (0) to maximum displacement (+7)
- From maximum displacement to equilibrium (0)
- From equilibrium (0) to minimum displacement (-7)
- From minimum displacement back to equilibrium (0)

Since the time period \( T \) is 4 seconds, the time taken to move from equilibrium to maximum displacement is:
\[ \text{Time from equilibrium to maximum displacement} = \frac{T}{4} = \frac{4 \text{ seconds}}{4} = 1 \text{ second} \]

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

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