MTH6P2B: MOMENT OF A FORCE

Equivalent Forces
• We defined equivalent forces as being forces with the same magnitude acting in the same direction and acting along the same line of action (this is through the Principle of Transmissibility), but why do the forces need to act along the same line?

Introduction to Moments
The tendency of a force to rotate a rigid body about any defined axis is called the Moment of the force about the axis.

The moment, M, of a force about a point provides a measure of the tendency for rotation (sometimes called a torque).
M = F * d

Moment caused by a Force

The Moment of Force (F) about an axis through Point (A) or for short, the Moment of F about A, is the product of the magnitude of the force and the perpendicular distance between Point (A) and the line of action of Force (F)

Units of a Moment
The units of a Moment are:
N·m in the SI system
ft·lbs or in·lbs in the US Customary system

Beams are often used to bridge gaps in walls. We have to know what the effect of the force on the beam will have on the beam supports.
What do you think those impacts are at points A and B?

Carpenters often use a hammer in this way to pull a stubborn nail. Through what sort of action does the force at the handle pull the nail? How can you mathematically model the effect of force at point O?

Properties of a Moment

Moments not only have a magnitude, they also have a sense to them.

The sense of a moment is clockwise or counterclockwise depending on which way it will tend to make the object rotate.

Properties of a Moment
The sense of a Moment is defined by the direction it is acting on the Axis and can be found using Right Hand Rule.

Varignon’s Theorem
The moment of a Force about any axis is equal to the sum of the moments of its components about that axis
This means that resolving or replacing forces with their resultant force will not affect the moment on the object being analyzed

In the 2-D case, the magnitude of the moment is

As shown, d is the perpendicular distance from point O to the line of action of the force.
In 2-D, the direction of is either clockwise or counter-clockwise, depending on the tendency for rotation.

A 100-lb vertical force is applied to the end of a lever which is attached to a shaft at O.
Determine:
a) Moment about O,
b) Horizontal force at A which creates the same moment,
c) Smallest force at A which produces the same moment,
d) Location for a 240-lb vertical force to produce the same moment,
e) Whether any of the forces from b, c, and d is equivalent to the original force.

Example #1

a) Moment about O is equal to the product of the force and the perpendicular distance between the line of action of the force and O. Since the force tends to rotate the lever clockwise, the moment vector is into the plane of the paper.

Example #1

b) Horizontal force at A that produces the same moment,

Example #1

Example #1

Example #1

Principle of Moments
Varignon’s Theorem: The moment of a force about a point is equal to the sum of moments of the components of the force about the point:

Often it is easier to determine  by using the components of F as shown (Varignon’s Theorem).

Note the different signs on the terms! The typical sign convention for a moment in 2-D is that counter-clockwise is considered positive. We can determine the direction of rotation by imagining the body pinned at O and deciding which way the body would rotate because of the force.

Given: A 20 lb force is applied to the hammer.
Find: The moment of the force at A.
Plan:

Since this is a 2-D problem:
1) Resolve the 20 lb force along the handle’s x and y axes.
2) Determine  using a scalar analysis.

Moments in 3D 4.5 Moment of a Force about a Specific Axis

In 2D bodies the moment is due to a force contained in the plane of action perpendicular to the axis it is acting around. This makes the analysis very easy.
In 3D situations, this is very seldom found to be the case.

Moments in 3D

The moment about an axis is still calculated the same way (by a force in the plane perpendicular to the axis) but most forces are acting in abstract angles.

By resolving the abstract force into its rectangular components (or into its components perpendicular to the axis of concern) the moment about the axis can then be found the same way it was found in 2D – M = Fd (where d is the distance between the force and the axis of concern)

Notation for Moments
In simpler terms the Moment of a Force about the y-axis  can be found by using the
projection of the Force on the x-z Plane
The Notation used to denote Moments about the Cartesian Axes are

3D Moment:

3D Moments Example:

•Given the tension in cable BC is 700 N, find Mx, My, and Mz about point A.