During SHM, a paricle has displacement x form mean position. If accreleration. Kinetic energy and potential energy are represented by a K and U respectively, the choose the appropriate graph

Options

A.
B.
C.
D.

Answer

To solve the question regarding the graphs of kinetic energy (K), potential energy (U), and acceleration (A) during Simple Harmonic Motion (SHM), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Motion**: In SHM, the displacement \( x \) of the particle from the mean position varies sinusoidally with time. The equations governing SHM are:
\[ x(t) = A \sin(\omega t) \]
where \( A \) is the amplitude and \( \omega \) is the angular frequency.

2. **Find Velocity**: The velocity \( v \) is the derivative of displacement with respect to time:
\[ v(t) = \frac{dx}{dt} = A \omega \cos(\omega t) \]

3. **Find Acceleration**: The acceleration \( a \) is the derivative of velocity:
\[ a(t) = \frac{dv}{dt} = -A \omega^2 \sin(\omega t) \]
This indicates that acceleration is maximum when the displacement is maximum and vice versa.

4. **Kinetic Energy (K)**: The kinetic energy of the particle is given by:
\[ K = \frac{1}{2} m v^2 = \frac{1}{2} m (A \omega \cos(\omega t))^2 = \frac{1}{2} m A^2 \omega^2 \cos^2(\omega t) \]

5. **Potential Energy (U)**: The potential energy in SHM is given by:
\[ U = \frac{1}{2} k x^2 = \frac{1}{2} m \omega^2 (A \sin(\omega t))^2 = \frac{1}{2} m A^2 \omega^2 \sin^2(\omega t) \]

6. **Total Energy**: The total mechanical energy \( E \) in SHM is constant and is the sum of kinetic and potential energy:
\[ E = K + U = \frac{1}{2} m A^2 \omega^2 \]

7. **Graphing the Energies**:
- Kinetic Energy \( K \) varies as \( \cos^2(\omega t) \) which means it reaches maximum when displacement is zero.
- Potential Energy \( U \) varies as \( \sin^2(\omega t) \) which means it reaches maximum when displacement is maximum.
- The total energy remains constant throughout the motion.

8. **Choosing the Correct Graph**: Based on the above analysis, we can conclude:
- The graph of kinetic energy will be a cosine squared function (peaks at \( t = 0, T/2, T, \ldots \)).
- The graph of potential energy will be a sine squared function (peaks at \( t = T/4, 3T/4, \ldots \)).
- The total energy graph will be a horizontal line since it is constant.

9. **Final Selection**: After analyzing the options provided, the correct graph will be the one that shows the kinetic energy and potential energy oscillating out of phase with each other, while the total energy remains constant.