- #1
Chrono G. Xay
- 92
- 3
How would one go about finding the center of mass of a non-uniform rod when its cross-section has been modeling using a piecewise-defined function?
For a specific case let's use a drumstick, noting its volume as a sum of half of a sphere (the 'butt' end), a cylinder (the 'shaft'), the majority of half of an ellipsoid (the 'shoulder'), and the majority of either a smaller sphere or smaller ellipsoid.
With an ellipsoidal tip, the piecewise-defined function for its cross-section is given by:
f(x) = sqrt( (3)^2 - ( x + 97 )^2 ), x<-97;
f(x) = 3, -97 < x < 0;
f(x) = (3)*sqrt( 1 - ( x / 36 )^2 ), 0<x<32.8456;
f(x) = (2)*sqrt( 1 - ( ( x - 36 ) / 4 )^2 ), 32.8456<x=<40
(The value "32.8456" was obtained using equations related to the 'vesica pisces')
Now... If I were to take the integral of each piece... could I not easily compute the volume of the drumstick by squaring my sum and multiplying the result by π?
That might get me the volume, but I need to construct an equation for the object's linear density... I have read about one function called 'unit step' and another called 'heaviside', but am not sure which one would be appropriate, let alone how to input my pieces into the process...
The goal I am shooting for, assuming I can acquire an equation for the 'linear density' and 'center of mass' is to find the 'center of percussion'...
For a specific case let's use a drumstick, noting its volume as a sum of half of a sphere (the 'butt' end), a cylinder (the 'shaft'), the majority of half of an ellipsoid (the 'shoulder'), and the majority of either a smaller sphere or smaller ellipsoid.
With an ellipsoidal tip, the piecewise-defined function for its cross-section is given by:
f(x) = sqrt( (3)^2 - ( x + 97 )^2 ), x<-97;
f(x) = 3, -97 < x < 0;
f(x) = (3)*sqrt( 1 - ( x / 36 )^2 ), 0<x<32.8456;
f(x) = (2)*sqrt( 1 - ( ( x - 36 ) / 4 )^2 ), 32.8456<x=<40
(The value "32.8456" was obtained using equations related to the 'vesica pisces')
Now... If I were to take the integral of each piece... could I not easily compute the volume of the drumstick by squaring my sum and multiplying the result by π?
That might get me the volume, but I need to construct an equation for the object's linear density... I have read about one function called 'unit step' and another called 'heaviside', but am not sure which one would be appropriate, let alone how to input my pieces into the process...
The goal I am shooting for, assuming I can acquire an equation for the 'linear density' and 'center of mass' is to find the 'center of percussion'...