EGR 280

Mechanics

2 – Moments, equivalent systems of forces

Statics of Rigid Bodies

Principle of Transmissibility: Forces can be moved along their lines of action without changing the overall effect on any rigid body. Any deformations or local changes in the body are, of course, dependent on where the forces are applied.

F F

Moment of a force about a point: Given a point O in space and a force F, the moment of F about O is defined as:

MO = r × F

where r is the position vector

from O to anywhere on the

line of action of F

MO = rFsinθ = Fd

where d = rsinθ

Rectangular components:

i j k (ryFz – rzFy) i

MO = r × F = rx ry rz = + (rzFx – rxFz) j

Fx Fy Fz + (rxFy – ryFx) k

O

r

F

d

θ

Moment of a force about a given axis:

Given a point O, force F and an axis defined by the unit vector λL, the magnitude of the component of the moment of F about O in the λL direction is:

MOL = (rO × F) • λL

And the moment of F about axis is

MOL = [(rO × F) • λL)] λL

Note: rO is from anywhere on axis λL to

anywhere on the line of action of F

O

λL rO

F

The magnitude of a moment is a measure of the tendency of the force to rotate the rigid body about point O. The direction of the moment give the direction of the rotation, by the right-hand rule.

OF

MO = rO × FrO

Moment of a couple: Two forces F and –F which have the same magnitude, parallel lines of action and opposite sense are said to form a couple.

|M| = |r×F|

= rFsinθ

= Fd

Equivalent couples – couples that have the same magnitude and sense are equivalent.

Since couples are moments, they can be written as vectors and all vector operations applied to them.

F -F

r

d

We have seen that we can move forces freely along their lines of action. Moving a force parallel to its line of action produces a couple:

Equivalent systems of forces: Two force systems are equivalent if they can be reduced to the same force-couple system at any point.

O O O

FF F

F

-F

M