When a person of mass (m) stands at the left end of a uniform plank of length (L) and mass (M), which is placed on a frictionless ice surface, and walks to the opposite end, the plank slides. This intriguing scenario can be analyzed using the principles of center of mass and system physics. In this article, we explore the underlying physics and the mathematics involved in determining the distance the plank slides while the person walks.

Understanding the Basics

First, it is crucial to recognize that the ice surface is frictionless. This means that there are no external forces acting to affect the motion of the system. Additionally, the center of mass (COM) of a system is the point at which the entire mass of the system can be considered to be concentrated for the purpose of translational motion. Importantly, the center of mass of a system will not change unless an external force acts upon it.

The Comolina Analysis

Let's consider a person of mass (m) standing at the left end of a uniform plank of length (L) and mass (M). The plank is positioned on a frictionless ice surface. The person then starts to walk towards the right end of the plank. To maintain the center of mass of the system at a constant position, the plank will need to move in the opposite direction (i.e., to the left).

Maintaining the Center of Mass

Let’s represent the origin (COM) of the system as the center of the plank. We divide the plank into two parts: the left part with mass (m) and the right part with mass ( frac{M}{2} ) once the person reaches the right end. As the person walks to the right end of the plank, the COM remains at the origin (center of the plank).

Mathematical Analysis

When the person moves from the left end to the right end of the plank, the plank and the person as a system constitute a rigid body. To maintain the center of mass at the same position, the plank must move in the opposite direction. Let's denote the distance the person has walked as (x) and the distance the plank has slid as (d).

For the center of mass to remain at the center of the plank, the following equation must hold:

[m(L - x) M cdot 0 (m M)(x d)]

Here, (L - x) represents the distance to the center of the plank for (m), and (x d) represents the position of the center of the plank for the entire system. Simplifying this equation, we get:

[m(L - x) (m M)(x d)]

Rearranging terms and simplifying:

[mL - mx mx md Mx Md] [mL 2mx md Mx Md] [mL x(2m M) Md]

Since (x eq 0), solving for (d), we have:

[d frac{mL}{Md} - x]

Conclusion and Reflection

In conclusion, the distance the plank slides (d) when the person of mass (m) walks to the right end of the plank of length (L) and mass (M) can be derived from the above equation. This is a classic problem in the dynamics of systems with moving parts. Understanding these principles not only aids in the analysis of such physical systems but also in more complex scenarios involving multiple objects and forces. Whether in school, or in real-world engineering applications, the understanding of center of mass and its preservation in rigid body motion is invaluable.

By applying the principles of physics, we can determine the precise distance the plank will slide. This exercise serves as a captivating example of how basic physical laws can be applied to solve interesting and practical problems. Happy learning!