This problem explores the relationship between the period of a pendulum and the length of the rope, which, in this case, corresponds to the height of the tree.

For a pendulum of length $L$ (the length of the rope), the period $T$ — the time for a complete cycle — can be approximated by: $T = 2\pi \sqrt{\frac{L}{g}}$ where $g$ is the acceleration due to gravity.

Step-by-Step Analysis

1. Period Proportionality: The period $T$ is proportional to the square root of the length $L$: $T \propto \sqrt{L}$

2. Comparing the Periods: The problem states that your friend’s pendulum has a period that is half of yours, $T_{\text{friend}} = \frac{1}{2} T_{\text{you}}$.

3. Relating Periods to Lengths: If we square both sides of the period relation, we get: $\frac{T_{\text{friend}}^2}{T_{\text{you}}^2} = \frac{L_{\text{friend}}}{L_{\text{you}}}$ Plugging in $T_{\text{friend}} = \frac{1}{2} T_{\text{you}}$: $\left(\frac{1}{2} T_{\text{you}}\right)^2 = \frac{L_{\text{friend}}}{L_{\text{you}}}$ $\frac{1}{4} T_{\text{you}}^2 = \frac{L_{\text{friend}}}{L_{\text{you}}}$ Thus: $L_{\text{friend}} = \frac{1}{4} L_{\text{you}}$

Conclusion

The height of your tree (represented by $L_{\text{you}}$) is four times the height of your friend's tree ($L_{\text{friend}}$). Therefore, the height of your tree is greater than the height of your friend’s tree.

Would you like further details, or do you have any questions?

Here are some related questions that could expand your understanding:

1. How would the period change if the pendulum were released from a much larger angle?
2. What assumptions are we making by using the small-angle approximation for a pendulum?
3. How would air resistance affect the period of the pendulum?
4. What would happen to the period if this experiment were conducted on the moon?
5. How would the period be affected if the rope were made of a stretchy material?

Tip: The period of a pendulum is largely independent of the amplitude (angle of release) as long as the angle remains small, due to the small-angle approximation.