Statics

Forces and Moments

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Contents:

  1. Forces
  2. Moments
  3. Introduction to Statics

Forces

Force can be thought of as a push or pull, causing the motion of an object. Force is measured in Newtons (N) and consists of:

  • The point of application
  • Magnitude (Newtons)
  • Direction of the force

Since force is defined by both a magnitude and direction it is a vector. Newton summed up the behaviour of forces in three laws:

  1. Law of Inertia: Unless acted upon by a net force, an object at rest will tend to stay at rest and an object in motion will tend to stay in motion.
  2. The net force applied to an object is equal to that objects mass multiplied by its acceleration: \bm{\overrightarrow{F}}=m\bm{\overrightarrow{a}}. Further extened to \bm{\overrightarrow{F}}=\dfrac{d}{dt}(m\bm{\overrightarrow{v}}).
  3. For every action there is an equal and opposite reaction.

Very common examples of forces are:

  • Force due to gravity: \bm{\overrightarrow{F}}=m\bm{\overrightarrow{g}}
  • Friction: F=\mu F_N
  • Tension
  • Reaction forces

Moments

Moment is mathematically and physically identical to torque. The term moment is typically used in applications where a force applied at a distance to a pivot point will cause rotation in a structure. The term torque is typically used for applications on an axle, such as a gear shaft. Moment is defined by taking the cross product of a force and the distance between the force and the pivot point of interest, \bm{\overrightarrow{r}}.

\bm{\overrightarrow{M}}=\bm{\overrightarrow{r}}\times\bm{\overrightarrow{F}}

Since moment is defined by cross product it is a vector. The direction of rotation is defined by the right-hand rule:

  1. Point your fingers in the direction of the force.
  2. Face your palm such that the pivot vector would enter through your palm and exit through the back of your hand.
  3. The direction of your thumb will be the direction of the moment vector.
  4. The direction that your fingers curl will be the direction the moment will tend to rotate the object.

When your thumb points in the direction of positive x, y, or z, then the moment is positive for that direction. Therefore, in terms of rotation, counter-clockwise is deemed the positive direction of rotation.

A couple is a special type of moment. Couples occur when two equal and opposite parallel forces applied to an object are separated by a distance. In the case of a couple a moment does not require a reference point as it will be equal at all points. For this reason couples are deemed free vectors, they can be moved anywhere on an object and not change the result of the analysis.

Introduction to Statics

Statics describes a subset of problems within the field of mechanics. Static systems are defined by having a net force and net moment of zero, therefore experiencing zero acceleration. Systems that fulfill such conditions are in static equilibrium, and fall within the field of statics. The static condition is met when every point in a system has a net force and moment of zero. Therefore, a static system has the following governing equations for every point in the system:

\displaystyle\sum \bm{\overrightarrow{F}}=0

\displaystyle\sum \bm{\overrightarrow{M}}=0

Where force and moment are vectors and can thus be expressed in terms of x, y, and z components. Therefore we may express the above equations in terms of these components.

\displaystyle\sum\begin{bmatrix}F_x\\F_y\\F_z\end{bmatrix}=\begin{bmatrix}\sum F_x\\\sum F_y\\\sum F_z\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}

\displaystyle\sum\begin{bmatrix}M_x\\M_y\\M_z\end{bmatrix}=\begin{bmatrix}\sum M_x\\\sum M_y\\\sum M_z\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}