Question

You have a meter stick that balances at the \(55-\mathrm{cm}\) mark. Is your meter stick homogeneous?

Short answer

Answer: No, the meter stick is not homogeneous as the balance point (center of mass) is located at the 55 cm mark, which is different from its geometrical center (50 cm mark).

Step-by-step solution

Understand the Problem

First, we need to determine the center of mass of a homogeneous object. In this case, the object is a meter stick. If the meter stick is homogeneous, its center of mass should be located exactly at its geometrical center.

Find the Balance Point

The given information states that the balance point (i.e., the center of mass) is at the 55 cm mark on the meter stick.

Compare the Balance Point to the Geometrical Center

Since the length of the meter stick is 100 cm, its geometrical center is located at the 50 cm mark. To determine if the meter stick is homogeneous, we need to compare the balance point (center of mass) to the geometrical center.

Conclusion

As the balance point is located at the 55 cm mark, which is different from its geometrical center (50 cm mark), we can conclude that the meter stick is not homogeneous.

Key concepts

Homogeneous Object

When we refer to a homogeneous object, we're talking about an object with a uniform composition throughout. This means that the material making up the object is distributed evenly, resulting in no variation in density from one part of the object to another.

For instance, consider a perfectly made metal rod. If it's homogeneous, every segment of the rod — no matter how small — has precisely the same density and weight per unit length. In the classroom, we might use a meter stick as an example. If the meter stick were indeed homogeneous, the material that composes the stick would be spread out in such a way that each centimeter of the stick has the same mass as the next.

In our exercise, the expectation was that if the meter stick was homogeneous, its center of mass should align with its geometrical center, assuming uniform density.

Balance Point

The balance point of an object, often referred to as the center of mass, is a crucial concept in physics. This point represents the location where the mass of an object is considered to be concentrated. For practical purposes, if you were to support the object at its balance point, the object would remain in equilibrium and not tip over.

To find the balance point, various methods can be employed depending on the object's shape and density distribution. With our meter stick example, the balance point is naturally where the stick rests horizontally without tilting, which is at the 55 cm mark as per the information given.

This balance point is a telltale indicator of how the mass is distributed along the stick. If the balance point is not at the center, it implies that there is an uneven distribution of mass — meaning, the object isn't homogeneous.

Geometrical Center

The geometrical center of an object is also known as its centroid. It's the 'average' position of all the points of an object. Imagine folding a shape along every possible axis; the place where it balances perfectly each time is its geometrical center.

For two-dimensional figures like squares or circles, it's quite simple to find this point because of their symmetry. But even in three dimensions, like cubes or spheres, the concept remains intact. A meter stick, being a long, thin rectangle, has its geometrical center at the midway point — at the 50 cm mark, as it's 100 cm long.

The geometrical center assumes an important requirement: that the material of the object is uniformly distributed. However, in real-life applications, finding the true geometrical center may not always indicate the balance point of an object as external factors or material inconsistencies can shift the actual center of mass away from the geometrical center.