Question

You have a meter stick that balances at the 55 -cm mark. Is your meter stick homogeneous?

Short answer

Answer: No, the meter stick is not homogeneous.

Step-by-step solution

Find the center of mass of a homogeneous meter stick

For a homogeneous meter stick, the center of mass is in the middle, which is at the 50 cm mark. So, the theoretical center of mass is at 50 cm.

Compare the balancing point to the theoretical center of mass

In the exercise, it is stated that the meter stick balances at 55 cm mark. This is the balancing point and it is an indication of the center of mass for the given meter stick.

Determine if the meter stick is homogeneous

Now, compare the balancing point (55 cm) to the theoretical center of mass (50 cm). Since they are not the same, it indicates that the meter stick is not homogeneous. In conclusion, the meter stick is not homogeneous, as its balancing point (55 cm) is different from the theoretical center of mass for a homogeneous meter stick (50 cm).

Key concepts

Homogeneous Meter Stick

Understanding the term 'homogeneous' when related to a meter stick is essential in the context of physics. A homogeneous meter stick is one that has a uniform density and composition throughout its entire length. This uniformity implies that if you were to cut the stick at any point along its length, both pieces would have exactly the same mass per unit length.

This characteristic results in the center of mass, which is the point at which the entire mass of the meter stick can be considered to be concentrated, being right at its midpoint. For a standard meter stick, which is typically 100 cm long, the center of mass would theoretically be located at the 50 cm mark if the stick were truly homogeneous.

An easy experiment to demonstrate homogeneity involves balancing the stick on a single point such as the edge of a table or a finger. If the stick balances perfectly at the 50 cm mark, it supports the claim of homogeneity.

Balancing Point

The balancing point of an object is closely tied to the concept of its center of mass. It is the exact spot along the object's length where it can be supported and remain horizontal without tipping over. This point is where the gravitational torque about any part of the object is equal and thus, the object is in a state of equilibrium.

For an idealized rod such as a meter stick, the balancing point and the center of mass should conceptually be the same if the object is homogeneous. However, external factors or irregularities in the material can cause these points to diverge. If the balancing point is not at the expected theoretical center of mass, this suggests that the mass distribution within the object is not even, leading to an unbalanced mass distribution.

The balancing point serves not only as a helpful tool for understanding an object's stability but also offers insights into the distribution of mass throughout the object. If a meter stick, which is assumed to be uniform, has its balancing point shifted from the center, it indicates an anomaly in its structure or density.

Theoretical Center of Mass

The term theoretical center of mass refers to the predicted point on an object where its mass is considered to be perfectly distributed and hence, the point about which the object would balance under the force of gravity. It is a fundamental concept in physics that helps us understand how forces act on a body and how that body will respond to those forces.

For objects with a well-defined geometry and uniform density, such as a homogeneous meter stick, the theoretical center of mass can be easily calculated. For example, for such a stick measuring 100 cm, it would be located at the 50 cm mark, assuming uniformity in its construction.

However, the theoretical center of mass is not always the actual balancing point, as demonstrated in our exercise. When discrepancies arise between the theoretical prediction and the observed balancing point, it indicates that the theory's assumptions about the object—such as being homogeneous—may not hold true. It's a valuable insight for both practical and educational purposes, as it encourages a deeper exploration into the composition and properties of the object in question.