Ratio of kinetic energy at mean position to potential energy at A/2 of a particle performing SHM

Options

A. `2:1`
B. `4:1`
C. `8:1`
D. `1:1`

Answer

To solve the problem of finding the ratio of kinetic energy at the mean position to potential energy at \( y = \frac{A}{2} \) for a particle performing Simple Harmonic Motion (SHM), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding SHM**:
The displacement of a particle in SHM can be expressed as:
\[ y = A \sin(\omega t) \]
where \( A \) is the amplitude and \( \omega \) is the angular frequency.

2. **Finding Velocity**:
The velocity \( v \) of the particle can be derived from the displacement:
\[ v = \frac{dy}{dt} = A \omega \cos(\omega t) \]
We can express \( v \) in terms of displacement \( y \):
\[ v = \omega \sqrt{A^2 - y^2} \]
This is derived using the identity \( \cos^2(\omega t) = 1 - \sin^2(\omega t) \).

3. **Kinetic Energy at Mean Position**:
At the mean position, \( y = 0 \). The kinetic energy \( KE \) is given by:
\[ KE = \frac{1}{2} m v^2 \]
Substituting \( v = \omega A \) (since \( y = 0 \)):
\[ KE = \frac{1}{2} m (\omega A)^2 = \frac{1}{2} m \omega^2 A^2 \]
Let’s denote this as Equation (1).

4. **Potential Energy at \( y = \frac{A}{2} \)**:
The potential energy \( PE \) in SHM is given by:
\[ PE = \frac{1}{2} k y^2 \]
where \( k = m \omega^2 \). At \( y = \frac{A}{2} \):
\[ PE = \frac{1}{2} m \omega^2 \left(\frac{A}{2}\right)^2 = \frac{1}{2} m \omega^2 \frac{A^2}{4} = \frac{1}{8} m \omega^2 A^2 \]
Let’s denote this as Equation (2).

5. **Finding the Ratio**:
Now, we need to find the ratio of kinetic energy at the mean position to potential energy at \( y = \frac{A}{2} \):
\[ \text{Ratio} = \frac{KE}{PE} = \frac{\frac{1}{2} m \omega^2 A^2}{\frac{1}{8} m \omega^2 A^2} \]
The \( m \), \( \omega^2 \), and \( A^2 \) terms cancel out:
\[ \text{Ratio} = \frac{\frac{1}{2}}{\frac{1}{8}} = \frac{1}{2} \times \frac{8}{1} = 4 \]

### Final Answer:
The ratio of kinetic energy at the mean position to potential energy at \( y = \frac{A}{2} \) is \( 4:1 \).