A body performs S.H.M. Its kinetic energy K varies with time t as indicated by graph

Options

A.
B.
C.
D.

Answer

To solve the problem regarding the variation of kinetic energy (K) of a body performing simple harmonic motion (SHM) with time (t), we can follow these steps:

### Step 1: Understand the Kinetic Energy Formula
The kinetic energy (K) of a body is given by the formula:
\[ K = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the body and \( v \) is its velocity.

### Step 2: Write the Velocity Equation for SHM
In simple harmonic motion, the velocity \( v \) can be expressed as:
\[ v = A \omega \cos(\omega t + \alpha) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( \alpha \) is the phase constant.

### Step 3: Substitute Velocity into the Kinetic Energy Formula
Now, substitute the expression for velocity into the kinetic energy formula:
\[ K = \frac{1}{2} m (A \omega \cos(\omega t + \alpha))^2 \]
This simplifies to:
\[ K = \frac{1}{2} m A^2 \omega^2 \cos^2(\omega t + \alpha) \]

### Step 4: Analyze the Kinetic Energy Expression
From the expression \( K = \frac{1}{2} m A^2 \omega^2 \cos^2(\omega t + \alpha) \), we can observe:
- The term \( \cos^2(\omega t + \alpha) \) is always non-negative (i.e., it is always greater than or equal to zero).
- Therefore, the kinetic energy \( K \) will always be a positive quantity or zero.

### Step 5: Determine the Graph of Kinetic Energy
Since \( K \) varies with \( \cos^2(\omega t + \alpha) \), the graph of kinetic energy versus time will resemble the graph of \( \cos^2 \) function, which oscillates between 0 and a maximum value (when \( \cos^2 \) is 1).

### Step 6: Choose the Correct Option
From the options provided, we need to identify the graph that represents the positive oscillation of kinetic energy, which is consistent with the behavior of \( \cos^2 \) function. The correct option will be the one that shows a graph oscillating between 0 and a maximum positive value.

### Final Conclusion
Thus, based on the analysis, the correct option is the one that represents the kinetic energy graph as a function of time that is always non-negative.

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