The acceleration of a particle, varies with time according to the relation a=-mw^2sinwt.The displacement of this particle at a time t will be 1) m sin(wt) 2) mwcos(wt) 3) mwsin(wt) 4) -1/2(mw^2sinwt)t^2

To determine the displacement of the particle given the acceleration function \( a = -mw^2 \sin(wt) \), we need to integrate the acceleration to find the velocity and then integrate the velocity to find the displacement. Let's break this down step by step.

Understanding the Relationship Between Acceleration, Velocity, and Displacement

Acceleration is the rate of change of velocity, and velocity is the rate of change of displacement. Therefore, we can find displacement by integrating the acceleration function twice.

Step 1: Integrate Acceleration to Find Velocity

The acceleration function is given as:

a(t) = -mw^2 \sin(wt)

To find the velocity \( v(t) \), we integrate the acceleration with respect to time:

v(t) = ∫a(t) dt = ∫(-mw^2 \sin(wt)) dt

Using the integral of \( \sin(wt) \), which is \( -\frac{1}{w} \cos(wt) \), we have:

v(t) = mw \cos(wt) + C_1

Here, \( C_1 \) is the constant of integration, which can be determined if initial conditions are provided. For now, we can assume \( C_1 = 0 \) for simplicity, leading to:

v(t) = mw \cos(wt)

Step 2: Integrate Velocity to Find Displacement

Now, we integrate the velocity function to find the displacement \( s(t) \):

s(t) = ∫v(t) dt = ∫(mw \cos(wt)) dt

The integral of \( \cos(wt) \) is \( \frac{1}{w} \sin(wt) \), so we have:

s(t) = mw \cdot \frac{1}{w} \sin(wt) + C_2

This simplifies to:

s(t) = m \sin(wt) + C_2

Again, \( C_2 \) is a constant of integration that depends on initial conditions. If we assume the particle starts from the origin, we can set \( C_2 = 0 \), leading to:

s(t) = m \sin(wt)

Final Result

Thus, the displacement of the particle at time \( t \) is:

s(t) = m \sin(wt)

Looking at the options provided, the correct answer is:

• 1) m sin(wt)

This approach illustrates how we can derive displacement from acceleration through integration, emphasizing the fundamental relationships in kinematics. If you have any further questions or need clarification on any steps, feel free to ask!