The acceleration of a particle at time is given by `A=-aomega^(2) sinomegat`. Its displacement at time t is

Options

A. `-a omega^(2)sinomegat`
B. `a sin omegat`
C. `a cosomegat`
D. `aomega sin((omegat)/(2))`

Answer

To find the displacement of a particle given its acceleration, we can follow these steps:

### Step 1: Understand the relationship between acceleration, velocity, and displacement
The acceleration \( A \) of the particle is given by:
\[ A = -a \omega^2 \sin(\omega t) \]
We know that acceleration is the rate of change of velocity:
\[ A = \frac{dv}{dt} \]

### Step 2: Set up the equation
From the acceleration equation, we can express it as:
\[ \frac{dv}{dt} = -a \omega^2 \sin(\omega t) \]

### Step 3: Integrate to find velocity
To find the velocity \( v \), we integrate both sides with respect to \( t \):
\[ dv = -a \omega^2 \sin(\omega t) \, dt \]
Integrating gives:
\[ v = \int -a \omega^2 \sin(\omega t) \, dt \]
The integral of \( \sin(\omega t) \) is \( -\frac{1}{\omega} \cos(\omega t) \), thus:
\[ v = -a \omega^2 \left(-\frac{1}{\omega} \cos(\omega t)\right) + C \]
\[ v = a \omega \cos(\omega t) + C \]
Here, \( C \) is the constant of integration which can be determined if initial conditions are known.

### Step 4: Relate velocity to displacement
Velocity is also the rate of change of displacement:
\[ v = \frac{ds}{dt} \]
Substituting for \( v \):
\[ \frac{ds}{dt} = a \omega \cos(\omega t) + C \]

### Step 5: Integrate to find displacement
Now we integrate to find the displacement \( s \):
\[ ds = \left(a \omega \cos(\omega t) + C\right) dt \]
Integrating gives:
\[ s = \int \left(a \omega \cos(\omega t) + C\right) dt \]
The integral of \( \cos(\omega t) \) is \( \frac{1}{\omega} \sin(\omega t) \), thus:
\[ s = a \omega \left(\frac{1}{\omega} \sin(\omega t)\right) + Ct + D \]
Where \( D \) is another constant of integration. Simplifying gives:
\[ s = a \sin(\omega t) + Ct + D \]

### Step 6: Final expression for displacement
If we assume that the particle starts from rest (i.e., \( C = 0 \) and \( D = 0 \)), the displacement simplifies to:
\[ s = a \sin(\omega t) \]

### Conclusion
Thus, the displacement of the particle at time \( t \) is:
\[ s = a \sin(\omega t) \]