How to Calculate Moments of Force

Written by james porter
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How to Calculate Moments of Force
If a moment of force vector points along this man's thumb, it indicates a rotational force turning from his palm along with his fingers. (thumbs up image by Kathy Burns from Fotolia.com)

When force is applied to an object at a certain point, it does two things: push the object, and rotate the object. The amount of that rotational tendency is described by the moment of force. A moment of force is a vector: it has both a magnitude (the strength of the rotational force) and a direction (the axis along which the rotation will take place). The direction can be determined using the right hand rule: with your thumb pointed along the moment of force, your fingers curl in the direction of rotation. Calculating the moment of force is simple vector math.

Skill level:
Easy

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Things you need

  • Vector of the force being applied to the object (Fx, Fy, Fz)
  • Vector of the position where the force is applied on the object (Rx, Ry, Rz)
  • Vector of the position of the object's point of rotation (Ax, Ay, Az)

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Instructions

  1. 1

    Subtract the position vector of the point of rotation from the position vector of the point where the force is applied. In other words, calculate the vector (Rx -- Axe, Ry -- Ay, Rz -- Az). For example, if a force is applied at coordinates (2, 3, 6) to an object whose centre of gravity (and thus position of rotation) is at coordinates (-2, 8, 0), you would get a vector of (2 -- (-2), 3 -- 8, 6 -- 0) = (4, -5, 6). This vector points from the point of rotation to the point of force application.

  2. 2

    Find the cross product of the vector from step 1 (which we will hereafter call B) and the force vector (F), as described in this and the next two steps. Firstly, find the x component of the cross product by subtracting the product of the y component of F and the z component of B from the product of the y component of B and the z component of F. To put it succinctly, calculate (B X F)x = ByFz -- BzFy.

  3. 3

    Find the y component of the cross product in a similar fashion, by subtracting the product of the z component of F and the x component of B from the product of the z component of B and the x component of F. In other words, calculate (B X F)y = BzFx -- BxFz.

  4. 4

    Find the z component of the cross product by subtracting the product of the x component of F and the y component of B from the product of the x component of B and the y component of F. In other words, calculate (B X F)z = BxFy -- ByFx.

  5. 5

    Write the moment of force as the vector with x, y, and z components as the results of steps 2, 3, and 4, respectively. To put it all into one formula, the moment M is (ByFz -- BzFy, BzFx -- BxFz, BxFy -- ByFx).

Tips and warnings

  • To simplify the calculation, use the point of rotation as the origin. Then, your vector for the position where the force is applied (R) is already the vector you would calculate in step 1 (B).
  • Make sure all of your starting vectors use the same origin. If the vectors for the position of the point of rotation and the position of the applied force refer to different origins, the result will be useless.

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