The graph between kinetic energy and displacement of a particle performing S.H.M. is

Options

A. parabola
B. straight line
C. ellipse
D. circle

Answer

To determine the graph between kinetic energy (KE) and displacement (x) of a particle performing Simple Harmonic Motion (SHM), we can follow these steps:

### Step 1: Understand the Energy in SHM
In SHM, the total mechanical energy (E) of the system is constant and is the sum of kinetic energy (KE) and potential energy (PE). The potential energy in SHM is given by the formula:
\[ PE = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant and \( x \) is the displacement from the mean position.

### Step 2: Write the Expression for Kinetic Energy
The total energy in SHM can be expressed as:
\[ E = KE + PE \]
From this, we can express kinetic energy as:
\[ KE = E - PE \]
Substituting the expression for potential energy, we get:
\[ KE = E - \frac{1}{2} k x^2 \]

### Step 3: Analyze the Equation
Since the total energy \( E \) is constant, we can rewrite the equation as:
\[ KE = E - \frac{1}{2} k x^2 \]
This shows that kinetic energy is a function of displacement \( x \). As \( x \) increases, the term \( \frac{1}{2} k x^2 \) increases, thus reducing the kinetic energy.

### Step 4: Identify the Nature of the Graph
The equation \( KE = E - \frac{1}{2} k x^2 \) can be rearranged to:
\[ KE = -\frac{1}{2} k x^2 + E \]
This is a quadratic equation in terms of \( x \) and represents a downward-opening parabola. The maximum kinetic energy occurs when \( x = 0 \) (the mean position), and it decreases to zero as \( x \) approaches the amplitude of the motion.

### Step 5: Conclusion
Thus, the graph of kinetic energy versus displacement for a particle in SHM is a downward-opening parabola, with the maximum kinetic energy at the mean position and zero kinetic energy at the maximum displacement (amplitude).

### Final Answer
The graph between kinetic energy and displacement of a particle performing SHM is a downward-opening parabola.

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