I found centre of mass of a juggling club is ~20cm along, by analysing an image from the internet. I wanted to find the centre of mass of a juggling club, but was unable to access an real club at the time.
Using a search engine I quickly found an image of a juggling club. The image was of a red club with a white background, oriented straight on with the handle pointing up. I reduced the number of colours in the image down to two colours using Irfan View. The image was very clean and with the exception of a reflection on the handle, reducing the colours produced a single solid shape that indicated the area occupied by the club. I filled the reflection, cut the image in half down the centre of the club using MS Paint, before writing a program to analyse it.
The reduced image represents a map of the area occupied by half of the juggling club. Each row of the image, a horizontal slice of the club. I found the height of a club on Wikipedia is 500mm. The type of club I was looking at are made of moulded plastic and are hollow in the centre. I assumed that the width of the moulded plastic in the club ~2mm. It is not necessary to know the exact thickness or density of the plastic, only the relative weight of each layer. The formula for centre of gravity, Centre = sum [ layer weight * layer height ] / sum [ layer weight ]
The analysis program loaded the image, scanned over it one row at a time counting the number of pixels in each row, the number of rows, and calculated the centre of mass. The size of each image pixel is equal to, club height (500mm) / count of occupied slices, which came out at 1.75mm per pixel. The radius of each slice of the club is equal to, count of occupied pixels in the slice * 1.75mm. Each layer has an outer radius from the measured value, and inner radius equal to, measured value – plastic width. The weight of each layer approximated by evaluating, the area of the outer radius – the area of the inner radius. Finally the position and weight of each layer feed into the centre of mass formula which tells me that the centre of a juggling club is 307mm down the handle towards the large end.
Inspecting a real club later confirmed my calculations.
Left, the image used for analysis, right, mockup club cross section.