In a simple harmonic motion, when the displacement is one-half of the amplitude, what fraction of the total energy is kinetic?

Options

A. 44198
B. 44259
C. Zero
D. 44200

Explanation

To solve the problem of finding the fraction of total energy that is kinetic energy when the displacement is one-half of the amplitude in simple harmonic motion, we can follow these steps:

### Step 1: Understand the Definitions
In simple harmonic motion (SHM), the total mechanical energy (E) is constant and is given by the formula:
\[ E = \frac{1}{2} k A^2 \]
where \(k\) is the spring constant and \(A\) is the amplitude.

### Step 2: Identify the Given Displacement
We are given that the displacement \(x\) is half of the amplitude:
\[ x = \frac{A}{2} \]

### Step 3: Calculate Kinetic Energy
The kinetic energy (K.E.) at any displacement \(x\) in SHM can be calculated using the formula:
\[ K.E. = \frac{1}{2} k A^2 - \frac{1}{2} k x^2 \]
Substituting \(x = \frac{A}{2}\):
\[ K.E. = \frac{1}{2} k A^2 - \frac{1}{2} k \left(\frac{A}{2}\right)^2 \]
\[ = \frac{1}{2} k A^2 - \frac{1}{2} k \frac{A^2}{4} \]
\[ = \frac{1}{2} k A^2 - \frac{1}{8} k A^2 \]
\[ = \frac{4}{8} k A^2 - \frac{1}{8} k A^2 \]
\[ = \frac{3}{8} k A^2 \]

### Step 4: Relate Kinetic Energy to Total Energy
Now, we can express the kinetic energy in terms of total energy:
\[ K.E. = \frac{3}{8} k A^2 \]
Since the total energy \(E = \frac{1}{2} k A^2\), we can write:
\[ K.E. = \frac{3}{8} \cdot 2E = \frac{3}{4} E \]

### Step 5: Find the Fraction of Total Energy that is Kinetic Energy
To find the fraction of total energy that is kinetic energy, we take the ratio:
\[ \text{Fraction of K.E.} = \frac{K.E.}{E} = \frac{\frac{3}{4} E}{E} = \frac{3}{4} \]

### Conclusion
Thus, when the displacement is one-half of the amplitude, the fraction of the total energy that is kinetic energy is:
\[ \frac{3}{4} \]