Force systems

Part of the Statics course offered by the Division of Applied Mechanics, School of Engineering and the Engineering and Technology Portal

Lecture
Force is a component of dynamics, which describes the causes of motion. Dynamics is in turn part of Newtonian Mechanics in the large field of Physics. Force systems are also the starting point of engineering analysis.

Force Vectors
A force vector is a force defined in two or more dimensions with a component vector in each dimension which may all be summed to equal the force vector. Similarly, the magnitude of each component vector, which is a scalar quantity, may be multiplied by the unit vector in that dimension to equal the component vector.

$$\vec F = \vec F_x + \vec F_y + \vec F_z = F_x\hat{i} + F_y\hat{j} + F_z\hat{k}$$

Moment
For a system wherein a rigid body experiences a force F at a orthogonal distance L from a fixed point, the moment M is the quantity (oddly enough of the same units as energy) defined by the force multiplied by the length of distance between the fixed point and the point where the force is applied. The direction of the moment is perpendicular to the force ecotro and the length, using the right hand rule.

$$ \vec M \ = \vec F * L $$

In the event that a force impacts the rigid body at an angle other than a right angle $$\vec F = F\angle\alpha = F_x + F_y$$, the moment is determined by the component of the force vector $$\vec F$$ that is orthogonal to the length L.

The general case in three dimensions can be calculated with the cross product. Do note that the order of the distance vector $$\vec r$$ and the force $$F$$ does matter in cross products as opposite order will change signs. $$ \vec M \ = \vec r \times \vec F $$ The components of a moment vector is the moment around the respective axis, following the right hand rule.

Example:

M = Force * Length = 100 Newtons * 10 Meters = 1,000 Newton-meters (N-m)

Example: Force F is incident on the end of a rigid body of length L at an angle A degrees from the central axis of the body x (Hint: draw a free body diagram).

Then  $$F_y \ = \vec F \sin (A)$$   and   $$M \ = F_y \ * L $$

Couple
A couple is a pair of equal and opposite force vectors that are some distance apart and that act upon the same body, thus causing a rotation. Imagine that force $$ F_1 \ $$ and force $$ F_2 \ $$ are incident at two locations along a rigid body of total length $$ L \ $$ at positions $$ a \ $$ and $$ b \ $$, where $$ a \ + b = L $$. (Hint: draw a free body diagram)

Then  $$ \vec M = F(a+b) - F a$$

In 3D the same rule applies, using that $$ \vec M = \vec r_1 \times \vec F_1 + \vec r_2 \times \vec F_2 = \vec r_1 \times \vec F_1 - \vec r_2 \times \vec F_1 = (\vec r_1 - \vec r_2) \times \vec F_1$$ which means that the moment will be the same around any point in the system.

Resultant
Any system of forces may be reduced to a system of components and a resulting moment.

That is to say, $$\vec R = \sum \vec F $$  and   $$\vec M_o \ = \sum M $$ about the point $$ O \ $$

$$\vec R_x = \sum \vec F_x $$ and $$\vec R_y = \sum \vec F_y $$ and $$\vec R_z = \sum \vec F_z $$ and then $$R = \sqrt{\vec R_x^2 + \vec R_y^2 + \vec R_z^2}$$ with $$\theta = \arctan \frac{F_y}{F_x} $$

...then $$\vec R$$ is magnitude $$R \ $$ in the direction of $$\theta $$

Assignments
Activities:
 * Create an activity

Readings:
 * Peruse the appropriate sections of Statics

Study guide:
 * 1) Wikipedia article:Force System
 * 2) Wikipedia article:Vector
 * 3) Wikipedia article:Force Vector
 * 4) Wikipedia article:Moment
 * 5) Wikipedia article:Couple
 * 6) Wikipedia article:Resultant