Torque, Force and Cross Product

September 21, 2016Mechanical designPhilippe Lucidarme

From force to torque

Let’s consider the body C, rotating around the point O. The force $\vec{F}$ is applied on body C at point A.

The torque around O is given by the cross product between the vectors $\vec{OA}$ and $\vec{F}$ :

$\vec{\Gamma} = \vec{OA} \times \vec{F}$

From torque to force

Assuming the torque is known, the force at point A is given by:

$\vec{F} = \frac {\vec{\Gamma} \times \vec{OA} } { \| \vec{OA} \| ^2 }$

Example

close all;
clear all;
clc;

%% Parameters

% Coordinates of point O
O=[0;0;0];
% Coordinates of point A
A=[1;2;0];
% Coordinates of point B
B=[3;1;0];

% Force applied at point A
FA=[0;2;0];


%% Compute torque

% Compute vector OA
vOA=A-O;
% Compute torque at point A
TO=cross (vOA,FA);


%% Compute force at point B

vOB=B-O;
FB=cross(TO,vOB)/(norm(vOB)*norm(vOB));



%% Draw system

plot3(O(1),O(2),O(3));
hold on;
line ([O(1) A(1)],[O(2) A(2)],[O(3) A(3)]);
line ([O(1) B(1)],[O(2) B(2)],[O(3) B(3)]);
text(O(1)+0.2,O(2)+0.2,O(3)+0.2,'O','FontSize',100);
text(A(1)+0.2,A(2)+0.2,A(3)+0.2,'A','FontSize',100);
text(B(1)+0.2,B(2)+0.2,B(3)+0.2,'B','FontSize',100);


% Draw force at point A
hFA=quiver3 (A(1),A(2),A(3),FA(1),FA(2),FA(3),'Color','k','LineWidth',3,'MaxHeadSize',1.5);

% Draw torque at point O
hTO=quiver3 (O(1),O(2),O(3),TO(1),TO(2),TO(3),'Color','g','LineWidth',3,'MaxHeadSize',1.5);

% Draw force at point B
hFB=quiver3 (B(1),B(2),B(3),FB(1),FB(2),FB(3),'Color','r','LineWidth',3,'MaxHeadSize',1.5);

% Display axis and legends
grid on;
axis square equal;
xlabel('X');
ylabel('Y');
zlabel('Z');
legend ([hFA,hTO,hFB],'Force at point A','Torque at point O','Force at point B');

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