the two seemingly big parallelograms actually are not parallelograms because of how I designed the lightning bolt (the equations I used are here https://www.desmos.com/calculator/g6crwsecp1). However, I could break the shape up into 2 big parallelograms, 2 very thin parallelograms, and a triangle. I'm assuming I could still then find the CM of the 5 point masses the usual way.kuruman said:
Yes, you can carve it up into simple non overlapping shapes, but since you have equations why not just integrate them and add the (signed) results?JorkThePork said:
Sorry, I’m a little confused how this would give me the center of mass. wouldn’t this just give the area?haruspex said:
So what do you need to multiply the expressions by before integrating?JorkThePork said:
The center of mass of a piecewise function is the point at which the function's mass distribution is balanced. It is essentially the weighted average position of the mass, taking into account the different segments of the piecewise function.
To find the center of mass of a piecewise function, you need to integrate the function over its entire domain, taking into account the different segments. You calculate the moment of the function about a point (usually the origin) and then divide by the total mass. Mathematically, this involves computing the integral of x*f(x) over the domain and dividing it by the integral of f(x) over the same domain.
The steps to calculate the center of mass for a piecewise function are: 1. Divide the function into its segments.2. Compute the integral of x*f(x) for each segment.3. Compute the integral of f(x) for each segment.4. Sum the integrals from steps 2 and 3 over all segments.5. Divide the total moment (from step 2) by the total mass (from step 3) to find the center of mass.
Common challenges include correctly setting up the integrals for each segment, ensuring continuity at the boundaries of the segments, and accurately performing the integration, especially if the function is complex or involves discontinuities. Additionally, errors can arise if the limits of integration are not properly managed.
Yes, software tools like MATLAB, Mathematica, and Python (with libraries like NumPy and SciPy) can significantly simplify the process of finding the center of mass of a piecewise function. These tools can handle complex integrations and provide accurate results quickly, reducing the risk of manual calculation errors.