The acceleration of a particle starting from rest, varies with time according to the relation`A=-aomegasinmegat.`. The displacement of this particle at a time t will be

Options

A. `-(1)/(2)(aomega^(2)sinomegat)t^(2)`
B. `aomegasinomegat`
C. `aomegacosomegat`
D. `a(sinomegat)/omega -at`

Answer

To find the displacement of a particle whose acceleration varies with time according to the relation \( A = -a \omega \sin(\omega t) \), we can follow these steps:

### Step 1: Relate acceleration to velocity
The acceleration \( A \) of the particle is the rate of change of velocity \( v \) with respect to time \( t \). Thus, we can write:
\[ A = \frac{dv}{dt} = -a \omega \sin(\omega t) \]

### Step 2: Rearrange the equation
We can rearrange this equation to separate variables:
\[ dv = -a \omega \sin(\omega t) \, dt \]

### Step 3: Integrate to find velocity
Now, we will integrate both sides. The limits for velocity will be from 0 to \( v \) (since the particle starts from rest) and for time from 0 to \( t \):
\[ \int_0^v dv = -a \omega \int_0^t \sin(\omega t) \, dt \]
The left side integrates to \( v \). For the right side, we need to integrate \( \sin(\omega t) \):
\[ v = -a \omega \left[ -\frac{1}{\omega} \cos(\omega t) \right]_0^t \]
This simplifies to:
\[ v = a \left( \cos(\omega t) - \cos(0) \right) \]
Since \( \cos(0) = 1 \):
\[ v = a \cos(\omega t) - a \]

### Step 4: Relate velocity to displacement
Now, we relate the velocity \( v \) to displacement \( x \):
\[ v = \frac{dx}{dt} \]
Substituting for \( v \):
\[ \frac{dx}{dt} = a \cos(\omega t) - a \]

### Step 5: Rearrange and integrate to find displacement
Rearranging gives:
\[ dx = \left( a \cos(\omega t) - a \right) dt \]
Now, we integrate both sides. The limits for \( x \) will be from 0 to \( x \) and for \( t \) from 0 to \( t \):
\[ \int_0^x dx = \int_0^t \left( a \cos(\omega t) - a \right) dt \]
The left side integrates to \( x \). For the right side, we need to integrate:
\[ x = a \left[ \frac{1}{\omega} \sin(\omega t) - at \right]_0^t \]
This simplifies to:
\[ x = a \left( \frac{1}{\omega} \sin(\omega t) - at \right) \]
Evaluating the limits gives:
\[ x = \frac{a}{\omega} \sin(\omega t) - at \]

### Final Answer
Thus, the displacement of the particle at time \( t \) is:
\[ x = \frac{a}{\omega} \sin(\omega t) - at \]

---