The pendulum of a grandfather clock is 74.0 cm long. When the pendulum swings from one side to the other side, it travels a horizontal distance of 19.0 cm. Determine the angle through which the pendulum swings. Round your answer to the nearest tenth of a degree

To determine the angle through which the pendulum swings, we need to use some trigonometry.

Given:
- Length of the pendulum \( L = 74.0 \) cm
- Horizontal distance traveled by the pendulum \( d = 19.0 \) cm

1. **Determine the half-angle \(\theta\):**
   Since the pendulum swings from one side to the other, the horizontal distance of 19.0 cm is the total distance. Therefore, the distance from the center to one side (half of the total distance) is \( \frac{19.0}{2} = 9.5 \) cm.

2. **Form a right triangle:**
   The pendulum length forms the hypotenuse of a right triangle, and half of the horizontal distance forms one leg of this triangle. We need to find the angle \(\theta\) between the pendulum and the vertical line.

3. **Use the cosine function:**
   \[
   \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{L - x}{L}
   \]
   where \(x\) is the vertical distance from the pivot to the point where the pendulum reaches 9.5 cm horizontally. This distance \(x\) can be found using:
   \[
   x = \sqrt{L^2 - (9.5)^2}
   \]

4. **Solve for \(x\):**
   \[
   x = \sqrt{74.0^2 - 9.5^2}
   \]
   \[
   x = \sqrt{5476 - 90.25}
   \]
   \[
   x = \sqrt{5385.75}
   \]
   \[
   x \approx 73.4 \text{ cm}
   \]

5. **Find the angle \(\theta\):**
   \[
   \cos(\theta) = \frac{x}{L} = \frac{73.4}{74.0}
   \]
   \[
   \theta = \cos^{-1}\left(\frac{73.4}{74.0}\right)
   \]
   \[
   \theta \approx \cos^{-1}(0.9919)
   \]
   \[
   \theta \approx 7.9^\circ
   \]

6. **Determine the total angle through which the pendulum swings:**
   Since the pendulum swings equally on both sides of the vertical, the total angle is:
   \[
   2\theta \approx 2 \times 7.9^\circ = 15.8^\circ
   \]

So, the angle through which the pendulum swings is approximately \( 15.8^\circ \).

Would you like more details or have any questions?

Here are 8 related questions for further exploration:
1. How does the length of a pendulum affect its period of oscillation?
2. What is the relationship between the angle of swing and the period of a pendulum?
3. How can the period of a pendulum be calculated?
4. What assumptions are made when analyzing pendulum motion?
5. How does air resistance affect the motion of a pendulum?
6. What is simple harmonic motion, and how is it related to pendulums?
7. How would the angle change if the pendulum length was doubled?
8. What other factors, besides length and angle, influence the pendulum’s motion?

**Tip:** When working with trigonometric functions, always ensure your calculator is set to the correct mode (degrees or radians) based on the context of your problem.

Learn how to calculate the angle through which a pendulum swings using trigonometry. Given a pendulum length of 74.0 cm and a horizontal swing distance of 19.0 cm, find the precise angle of swing rounded to the nearest tenth of a degree. Explore step-by-step solution and trigonometric formula application.

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