For a particle executing `SHM`, the kinetic energy `(K)` is given by `K = K_(0)cos^(2) omegat`. The equation for its displacement is

Options

A. `((k_(0))/(momega^(2)))^(1//2) sin omega t`
B. `((2k_(0))/(m omega^(2)))^(1//2)sin omegat`
C. `((2omega^(2))/(m k_(0)))^(1//2)sin omegat`
D. `((2k_(0))/(m omega))^(1//2)sin omegat`

Answer

To solve the problem, we need to find the equation for the displacement of a particle executing Simple Harmonic Motion (SHM) given that its kinetic energy \( K \) is expressed as \( K = K_0 \cos^2(\omega t) \).

### Step-by-Step Solution:

1. **Assume the Displacement Equation**:
We start by assuming the standard form of the displacement in SHM:
\[ y(t) = A \sin(\omega t) \]
where \( A \) is the amplitude.

2. **Find the Velocity**:
To find the velocity, we differentiate the displacement with respect to time \( t \):
\[ v(t) = \frac{dy}{dt} = \frac{d}{dt}(A \sin(\omega t)) = A \omega \cos(\omega t) \]

3. **Express Kinetic Energy**:
The kinetic energy \( K \) of the particle can be expressed in terms of its velocity:
\[ K = \frac{1}{2} m v^2 \]
Substituting the expression for \( v(t) \):
\[ K = \frac{1}{2} m (A \omega \cos(\omega t))^2 = \frac{1}{2} m A^2 \omega^2 \cos^2(\omega t) \]

4. **Compare with Given Kinetic Energy**:
We are given that:
\[ K = K_0 \cos^2(\omega t) \]
By comparing the two expressions for kinetic energy, we have:
\[ \frac{1}{2} m A^2 \omega^2 \cos^2(\omega t) = K_0 \cos^2(\omega t) \]
This implies:
\[ \frac{1}{2} m A^2 \omega^2 = K_0 \]

5. **Solve for Amplitude \( A \)**:
Rearranging the equation gives:
\[ A^2 = \frac{2 K_0}{m \omega^2} \]
Taking the square root:
\[ A = \sqrt{\frac{2 K_0}{m \omega^2}} \]

6. **Substitute Amplitude into Displacement Equation**:
Now substitute the value of \( A \) back into the displacement equation:
\[ y(t) = A \sin(\omega t) = \sqrt{\frac{2 K_0}{m \omega^2}} \sin(\omega t) \]

7. **Final Displacement Equation**:
Therefore, the equation for the displacement of the particle is:
\[ y(t) = \sqrt{\frac{2 K_0}{m \omega^2}} \sin(\omega t) \]