Question

Three friends go camping in the forest with their families. They start collecting wood to make a fire. They find a large, branchless log in the forest. One end of the log is very heavy, while the other end is quite light. They decide to cut the log in half to carry it on their backs to the campsite. To ensure that each piece weighs the same, Hasan suggests balancing the log on a rock and cutting it from the balance point. After balancing the log, the following discussion takes place between them: Alice: 'If we cut it from the balance point as I suggested, both pieces will weigh the same.' Bob: 'I think the piece on the right side of the balance point is heavier.' Mark: 'I also think the piece on the left side of the balance point is heavier.' Please mark the correct answer. Explain the rule or thoughts you used to come to this conclusion.

This is a question that our instructor asked us and I was very confused because the provided answer to that question was:

If a log is balanced on a pivot, the heavier side will be closer to the pivot, and the lighter side will be further away. The torques, which are the product of the weight and distance from the pivot, will be equal, but this does not imply that the weights of the two sides are equal unless the log has uniform density and the pivot is at the midpoint.

To summarize, if they cut the log at the balance point:

The torques (moments) about the balance point are equal.
The weights of the two sides of the log are such that the heavier side is shorter and the lighter side is longer.
The weights of the two resulting pieces, when measured separately, will be the same.

But, I believe the last two points contradict each other. If both sides have the same weight, how come they have the same torque as their distance to the point would be different, it's not a uniform log. Am I right to think that way?

Answer 1

If a log is balanced on a pivot, the heavier side will be closer to the pivot, and the lighter side will be further away. The torques, which are the product of the weight and distance from the pivot, will be equal, but this does not imply that the weights of the two sides are equal.

Step-by-step explanation:

If a log is balanced on a pivot, the heavier side will be closer to the pivot, and the lighter side will be further away.

The torques, which are the product of the weight and distance from the pivot, will be equal, but this does not imply that the weights of the two sides are equal unless the log has uniform density and the pivot is at the midpoint.

To summarize, if they cut the log at the balance point:

The torques (moments) about the balance point are equal.

The weights of the two sides of the log are such that the heavier side is shorter and the lighter side is longer.

The weights of the two resulting pieces, when measured separately, will be the same.

But, the heavier side will have a shorter distance to the pivot point, while the lighter side will have a longer distance.

This means that although the torques are equal, the weights of the two sides are not necessarily the same because the log is not of uniform density.