To solve this problem, we need to determine how high the lower end of a pendulum rises when it swings through an angle of 20° on either side from its vertical rest position. We shall use the following approach:
1. Given Data:
- Length of the pendulum, [tex]\( L \)[/tex] = 30 cm
- Angle of swing, [tex]\( \theta \)[/tex] = 20° (since it swings 20° on either side, the total angle is 40°, but we will use 20° for calculations)
2. Convert the angle to radians:
- Recall that angles in trigonometric calculations are often required in radians.
- To convert degrees to radians: [tex]\( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \)[/tex]
- For [tex]\( \theta = 20° \)[/tex]:
[tex]\[
\theta_{\text{radians}} = 20 \times \frac{\pi}{180} \approx 0.3491
\][/tex]
3. Calculate the height the pendulum rises:
- In a pendulum swing, the rise in height, [tex]\( h \)[/tex], when the pendulum swings through an angle, can be calculated using the formula:
[tex]\[
h = L \times (1 - \cos(\theta))
\][/tex]
- Substituting the given values:
[tex]\[
h = 30 \times (1 - \cos(0.3491))
\][/tex]
- Using the cosine function:
[tex]\[
\cos(0.3491) \approx 0.9397
\][/tex]
- Thus:
[tex]\[
h = 30 \times (1 - 0.9397) = 30 \times 0.0603 \approx 1.8092 \text{ cm}
\][/tex]
4. Conclusion:
- The lower end of the pendulum rises approximately 1.8092 cm when it swings through an angle of 20° on either side.
Therefore, the height the lower end of the pendulum rises is about 1.8092 cm.
