A particle is executing simple harmonic motion with a time period `T`. At time `t=0,` it is at its position of equilibium. The kinetice energy -time graph of the particle will look like

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Explanation

To solve the problem of determining the kinetic energy-time graph of a particle executing simple harmonic motion (SHM) with a time period \( T \), we can follow these steps:

### Step 1: Understand the Motion
The particle is executing SHM, which can be described by the equation:
\[ y(t) = A \sin(\omega t + \phi) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.

### Step 2: Determine the Phase Constant
At time \( t = 0 \), the particle is at its position of equilibrium, which means:
\[ y(0) = 0 \implies A \sin(\phi) = 0 \]
Since \( A \neq 0 \), we have \( \sin(\phi) = 0 \). This implies that \( \phi = 0 \) (since we consider the simplest case). Thus, the equation simplifies to:
\[ y(t) = A \sin(\omega t) \]

### Step 3: Find the Velocity
The velocity \( v(t) \) of the particle is the derivative of the displacement \( y(t) \) with respect to time:
\[ v(t) = \frac{dy}{dt} = A \omega \cos(\omega t) \]

### Step 4: Calculate the Kinetic Energy
The kinetic energy \( KE \) of the particle is given by:
\[ KE = \frac{1}{2} m v^2 = \frac{1}{2} m (A \omega \cos(\omega t))^2 \]
This simplifies to:
\[ KE = \frac{1}{2} m A^2 \omega^2 \cos^2(\omega t) \]

### Step 5: Analyze the Kinetic Energy Function
The kinetic energy is proportional to \( \cos^2(\omega t) \). The maximum value of \( KE \) occurs when \( \cos^2(\omega t) = 1 \) (i.e., when \( \omega t = 0, \pi, 2\pi, \ldots \)), and the minimum value occurs when \( \cos^2(\omega t) = 0 \) (i.e., when \( \omega t = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \)).

### Step 6: Determine Key Points in the Cycle
- At \( t = 0 \): \( KE \) is maximum.
- At \( t = \frac{T}{4} \): \( KE = 0 \) (since \( \cos^2(\frac{\pi}{2}) = 0 \)).
- At \( t = \frac{T}{2} \): \( KE \) is maximum again.
- At \( t = \frac{3T}{4} \): \( KE = 0 \).
- At \( t = T \): \( KE \) is maximum again.

### Step 7: Sketch the Kinetic Energy-Time Graph
The graph of \( KE \) versus time will be a cosine squared function, oscillating between 0 and a maximum value, with maxima at \( t = 0, \frac{T}{2}, T \) and minima at \( t = \frac{T}{4}, \frac{3T}{4} \).

### Conclusion
The kinetic energy-time graph will look like a cosine squared wave, oscillating between 0 and its maximum value.