Question

A soda can of mass \(m\) and height \(L\) is filled with soda of mass \(M .\) A hole is punched in the bottom of the can to drain out the soda. a) What is the center of mass of the system consisting of the can and the soda remaining in it when the level of soda in the can is \(h,\) where \(0

Short answer

Question: Calculate the minimum value of the center of mass of a system consisting of a soda can with mass m and height L, and the remaining soda in the can with height h, where 0 < h < L. Answer: To find the minimum value of the center of mass, differentiate the expression y_cm with respect to h and set it equal to zero. Then, find the value of h (h_min) that corresponds to the minimum value of the center of mass and substitute it back into the expression of y_cm. The minimum value of the center of mass y_cm_min is obtained after this substitution.

Step-by-step solution

Compute the center of mass of the soda can

Consider the soda can as a hollow cylinder with mass m and height L. The center of mass of a cylinder is located at its geometric center. So the center of mass of the soda can, denoted by y1, is at height L/2 from its base: y1 = L/2.

Compute the center of mass of the remaining soda

Assume the remaining soda forms a cylinder with height h and mass M'. The center of mass of the remaining soda, denoted by y2, is at height h/2 from its base: y2 = h/2.

Compute the mass of the remaining soda

The mass of the remaining soda, M', varies linearly with its volume (height). Therefore, we can express M' as a function of h as follows: M' = M * (h / L)

Compute the total center of mass of the system

The center of mass of the system consisting of the can and the remaining soda is given by the weighted average of their individual centers of mass: y_cm = (m * y1 + M' * y2) / (m + M') Substitute M', y1, and y2 as computed from steps 1, 2, and 3: y_cm = (m * (L/2) + M * (h / L) * (h/2)) / (m + M * (h / L)) #b) The minimum value of the center of mass as the soda drains out#

Compute the minimum value of the center of mass

To find the minimum value of the center of mass, we differentiate the expression of y_cm in Step 4 with respect to h and then set the result equal to zero: dy_cm/dh = 0 Differentiate and set the result equal to zero to find h_min, which corresponds to the minimum value of the center of mass: (h_min) = L * m / (2 * (M + m))

Compute the minimum center of mass

Substitute the value of h_min back into the expression of y_cm obtained in Step 4: y_cm_min = (m * (L/2) + M * ((L * m) / (2 * (M + m))) * (((L * m) / (2 * (M + m)))/2)) / (m + M * ((L * m) / (2 * (M + m))) / L)

Key concepts

Mass Distribution

Understanding mass distribution is fundamental when studying the center of mass in physics. The distribution of mass across a body determines how weight is balanced and how the object will behave in various situations, such as rotation or free fall. For instance, consider a soda can filled with soda—an everyday example. The mass is not just at a single point; rather, it's distributed throughout the volume of the can and the soda.

When we talk about the center of mass of the can and soda together, we're considering the entire mass distribution of this system. As the soda drains from the can, this distribution changes, which in turn alters the system's center of mass. The fact that the soda level, denoted by 'h', changes linearly as the soda is drained reflects a linear mass distribution between the bottom of the can and its current soda level.

Geometric Center

The geometric center of an object, also known as its centroid, is the point where the shape would balance perfectly if it were made of a uniformly dense material. When approaching problems in physics, especially those involving symmetry, the geometric center can often be assumed to coincide with the center of mass—if the object has uniform density.

For example, in the case of the soda can—which we can simplify as a cylinder—the geometric center is intuitively at the middle of its height, hence at a distance of L/2 from the base. This neatly coincides with its center of mass, assuming the can’s material is uniformly distributed along its height. However, when the soda begins to drain, this assumption no longer holds because the soda inside the can creates a variable distribution of mass.

Weighted Average

The concept of a weighted average is crucial when we calculate the center of mass for a system of particles or extended objects like our soda can and the soda inside it. Unlike a simple average that treats all values equally, a weighted average accounts for the significance, or 'weight', of each value. In physics, the weights are related to the masses of the objects in question.

When finding the center of mass for our can-soda system, we determine the weighted average of their individual centers of mass. Each center of mass is 'weighted' by the corresponding mass of that part of the system. If the soda level is halfway down the can, the mass of the soda remaining in the can and the mass of the can itself influence where the combined center of mass of the system lies. This aggregated point is a crucial concept for understanding how objects will behave in a physical context, especially in terms of stability and motion.