A simple pendulum of length 'L' has mass 'M' and it oscillates freely with amplitude A, Energy is
(g = acceleration due to gravity)

Options

A. `(MgA^2)/(2L)`
B. `(MgA)/(2L)`
C. `(MgA^2)/L`
D. `(2MgA^2)/L`

Answer

To find the total energy of a simple pendulum of length \( L \), mass \( M \), and amplitude \( A \), we can follow these steps:

### Step 1: Understand the Energy in Simple Harmonic Motion
In simple harmonic motion (SHM), the total mechanical energy \( E \) is the sum of kinetic energy \( K \) and potential energy \( U \). The total energy remains constant throughout the motion.

### Step 2: Identify Energy at Extreme Position
At the extreme position of the pendulum's swing (where it momentarily comes to rest), all the energy is potential energy. Therefore, we can express the total energy as:
\[ E = U = \text{Potential Energy at the extreme position} \]

### Step 3: Calculate Potential Energy
The potential energy \( U \) at the maximum height (extreme position) can be calculated using the formula:
\[ U = Mgh \]
where \( h \) is the height raised from the lowest point.

### Step 4: Determine Height \( h \)
The height \( h \) can be determined using the geometry of the pendulum. When the pendulum is at its maximum amplitude \( A \), the height \( h \) can be expressed as:
\[ h = L - L\cos(\theta) = L(1 - \cos(\theta)) \]
For small angles, we can approximate \( \cos(\theta) \) using the small angle approximation \( \cos(\theta) \approx 1 - \frac{A^2}{2L} \). Thus,
\[ h \approx \frac{A^2}{2L} \]

### Step 5: Substitute \( h \) into Potential Energy Formula
Substituting \( h \) into the potential energy formula gives:
\[ U = Mg\left(\frac{A^2}{2L}\right) \]

### Step 6: Total Energy Expression
Thus, the total energy \( E \) of the pendulum can be expressed as:
\[ E = U = \frac{MgA^2}{2L} \]

### Final Answer
The total energy of the simple pendulum is:
\[ E = \frac{MgA^2}{2L} \]

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