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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Answer

To solve the problem of the kinetic energy-time graph of a particle executing simple harmonic motion (SHM), we will follow these steps:

### Step 1: Understand the Motion
A particle in simple harmonic motion can be described by the equation:
\[ x(t) = A \sin(\omega t) \]
where:
- \( A \) is the amplitude,
- \( \omega = \frac{2\pi}{T} \) is the angular frequency,
- \( T \) is the time period.

### Step 2: Find the Velocity
The velocity \( v(t) \) of the particle is the derivative of the displacement:
\[ v(t) = \frac{dx}{dt} = A \omega \cos(\omega t) \]

### Step 3: Calculate the Kinetic Energy
The kinetic energy \( KE \) of the particle is given by:
\[ KE = \frac{1}{2} m v^2 \]
Substituting the expression for velocity:
\[ KE = \frac{1}{2} m (A \omega \cos(\omega t))^2 \]
\[ KE = \frac{1}{2} m A^2 \omega^2 \cos^2(\omega t) \]

### Step 4: Analyze the Kinetic Energy Function
The expression for kinetic energy can be rewritten as:
\[ KE = \frac{1}{2} m A^2 \omega^2 \cos^2(\omega t) \]
This shows that the kinetic energy varies with time according to the function \( \cos^2(\omega t) \).

### Step 5: Determine the Graph Shape
- At \( t = 0 \), \( \cos^2(0) = 1 \), so \( KE \) is at its maximum value.
- As \( t \) increases to \( \frac{T}{4} \), \( \cos^2(\omega t) \) decreases to 0, making \( KE \) minimum.
- At \( t = \frac{T}{2} \), \( \cos^2(\omega t) \) returns to 1, making \( KE \) maximum again.

The kinetic energy will oscillate between a maximum and a minimum value, repeating this pattern every half period.

### Conclusion
The kinetic energy-time graph will be a waveform that starts at its maximum value, decreases to zero, returns to maximum, and repeats this cycle. This corresponds to the graph of \( \cos^2(\omega t) \).

### Final Answer
The kinetic energy-time graph of the particle executing simple harmonic motion will look like a cosine squared wave, starting from maximum kinetic energy at \( t = 0 \).

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