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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.

You find a very tall tree one day and decide to measure the height of the tree using a pendulum. You tie a small rock to a rope that is also tied to a branch at the top of the tree, so that when the rope is vertical the rock is very close to the ground. You then release the rock from rest at an initial angle of 5°. You measure the amount of time it takes for the rock to return to your hand. Your friend performs the same experiment with a different tree, but uses a smaller initial angle of only 2.5°. Your friend measures the amount of time it takes for the rock to return to her hand to be half as big as the time you found. Is the height of your tree greater than, less than, or equal to the height of your friend’s tree?

Explore a physics problem involving a pendulum to determine the height of a tree by comparing the period of pendulums on different trees. Learn how the period relates to pendulum length through proportionality and the small-angle approximation. This problem is suitable for high school students in Grades 11-12.

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