A simple pendulum 4 m long swings with an amplitude of 0.2 m. What is its acceleration at the ends of its path? (g = `10m//s^(2)`)

Options

A. Zero
B. `10m//s^(2)`
C. `0.5m//s^(2)`
D. `2.5m//s^(2)`

Answer

To find the acceleration of a simple pendulum at the ends of its path, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Length of the pendulum (L) = 4 m
- Amplitude (A) = 0.2 m
- Acceleration due to gravity (g) = 10 m/s²

2. **Understand the Concept:**
- The acceleration of a simple pendulum is maximum at the extreme positions (the ends of its path).
- The formula for the maximum acceleration (a_max) at the extreme position is given by:
\[ a_{\text{max}} = \omega^2 \cdot A \]
- Where \( \omega \) is the angular frequency.

3. **Calculate the Angular Frequency (ω):**
- The angular frequency \( \omega \) can be calculated using the formula:
\[ \omega = \sqrt{\frac{g}{L}} \]
- Substituting the values:
\[ \omega = \sqrt{\frac{10 \, \text{m/s}^2}{4 \, \text{m}}} = \sqrt{2.5} \, \text{rad/s} \]

4. **Calculate \( \omega^2 \):**
- Now, square \( \omega \):
\[ \omega^2 = 2.5 \, \text{rad}^2/\text{s}^4 \]

5. **Calculate Maximum Acceleration (a_max):**
- Substitute \( \omega^2 \) and amplitude (A) into the maximum acceleration formula:
\[ a_{\text{max}} = \omega^2 \cdot A = 2.5 \cdot 0.2 \]
- Calculate:
\[ a_{\text{max}} = 0.5 \, \text{m/s}^2 \]

6. **Final Result:**
- The maximum acceleration at the ends of the path is:
\[ a_{\text{max}} = 0.5 \, \text{m/s}^2 \]

### Summary:
The acceleration at the ends of the path for the simple pendulum is **0.5 m/s²**.