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Chapter 11: Problem 13

You have a meter stick that balances at the \(50-\mathrm{cm}\) mark. Is it possible for your meter stick to be inhomogeneous?

Short Answer

Expert verified
Answer: Yes, it is possible for a meter stick to be inhomogeneous even if it balances at the 50-cm mark because the inhomogeneous distribution of mass along the meter stick can still result in a center of mass at the 50-cm mark as long as the total moment about the balancing point is equal and opposite.

Step by step solution

01

1. Understanding the balancing point

The balancing point of an object is where its center of mass is located. The center of mass is the average position of the mass of the object. If an object is uniform and homogeneous, the center of mass will be at its geometrical center. In the case of a meter stick, a homogeneous meter stick would have a center of mass at the 50-cm mark, which is where the given meter stick balances.
02

2. Analyzing the possibility of inhomogeneity

The meter stick could be inhomogeneous in various ways, with more mass distributed on one side than the other. For it to balance at the 50-cm mark despite this inhomogeneity, the center of mass still needs to be at the 50-cm mark. For example, if the meter stick has more mass at the first 25 cm and less mass at the last 75 cm, the total mass on both sides must be equal for the center of mass to be at the 50-cm mark. Let's assume the mass distribution in the first 25 cm is \(m_1\) and the mass distribution in the last 75 cm is \(m_2\). For the center of mass to be at the 50-cm mark, the total moment about the balancing point (50-cm mark) must be equal and opposite.
03

3. Balancing the moments

The moments about the 50-cm mark are equal if \(m_1 \times (50 - x_1) = m_2 \times x_2\); here, \(x_1\) is the distance from the center of mass of \(m_1\) to the 50-cm mark, and \(x_2\) is the distance from the center of mass of \(m_2\) to the 50-cm mark. Since the distances (\(x_1\) and \(x_2\)) depend on the specific distribution of mass in the first 25 cm and the last 75 cm, it is possible that there exists a distribution of masses \(m_1\) and \(m_2\) such that the moments are equal and the meter stick is inhomogeneous.
04

4. Conclusion

Yes, it is possible for your meter stick to be inhomogeneous even if it balances at the 50-cm mark. The inhomogeneous distribution of mass along the meter stick can still result in a center of mass at the 50-cm mark as long as the total moment about the balancing point is equal and opposite.

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