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Introduction to statics

What is statics?

Statics is the branch of mechanics that analyzes bodies at rest or in equilibrium. In statics, we analyze forces and moments acting on bodies to determine conditions for equilibrium, design structures, and solve engineering problems. It is also assumed that objects are rigid, meaning they do not deform under applied forces. Statics is distinct from dynamics, which deals with bodies in motion and solid mechanics, which deals with deformable bodies.

	Rigid	Deformable
No motion \( \sum \vec{F} = 0 \) \( \sum \vec{M} = 0 \)	Statics	Introductory solid mechanics
Motion \( \sum \vec{F} \neq 0 \) \( \sum \vec{M} \neq 0 \)	Introductory dynamics	Vibration, FEA, robotics, etc.

Rigid

Deformable

No motion

$ \sum \vec{F} = 0 $

$ \sum \vec{M} = 0 $

Statics

Introductory solid mechanics

Motion

$ \sum \vec{F} \neq 0 $

$ \sum \vec{M} \neq 0 $

Introductory dynamics

Vibration, FEA, robotics, etc.

Newton's laws

Newton's equations relate the acceleration $ \vec{a} $ of a point mass with mass $ m $ to the total applied force $ \vec{F} $ on the mass (sum of all applied forces).

Newton's first law

A particle at rest stays that way unless acted on by unbalanced forces.

Newton's second law

Newton's second law:

$$ \vec{F} = m\vec{a} $$

Newton's third law

The mutual forces of action and reaction between two particles are equal, opposite, and collinear.

Example

A point mass moving in the plane with an applied force. You can try to made the mass move in a circle and then see what happens when the force is suddenly removed, which will demonstrate Newton's first law (no net force implies motion at constant speed in a constant direction). Also observe which force directions cause the speed to increase or decrease.

Click and drag to impart a force on the particle.

Idealizations

Particles

In this course, we assume two things about particles:

The mass of the particle is not 0.
The radius of the mass is 0.

Through these assumptions, we are essentially concentrating all of the mass of an object at a single point in space. The particle has a mass but the size and shape of the particle is not taken into account.

Rigid bodies

For rigid bodies, we assume that the object has both mass (similar to particles) but also take its shape into account.

We call these bodies "rigid" because we assume that they do not deform under applied forces or moments.

A rigid body is an extended area of material that includes all the points inside it, and which moves so that the distances and angles between all its points remain constant. The location of a rigid body can be described by the position of one point $P$ inside it, together with the rotation angle of the body (one angle in 2D, three angles in 3D).

	location description
point mass	position vector \( \vec{r}_P \)
rigid body in 2D	position vector \( \vec{r}_P \) angle \( \theta \)
rigid body in 3D	position vector \( \vec{r}_P \) angles \( \theta,\phi,\psi \)

location description

point mass

position vector $ \vec{r}_P $

rigid body in 2D

position vector $ \vec{r}_P $

angle $ \theta $

rigid body in 3D

position vector $ \vec{r}_P $

angles $ \theta,\phi,\psi $

Neither point masses nor rigid bodies can physically exist, as no body can really be a single point with no extent, and no extended body can be exactly rigid. Despite this, these are very useful models for mechanics and statics.

Solving Equilibrium Problems

This section is adapted from Engineering Statics by Daniel Baker and William Haynes.

The vast majority of problems in this class are equilibrium problems. Here is the general procedure you should follow when going about solving these problems.

Read the problem and make sure you understand the situation.
Identify what is given from the problem and what the problem is asking you to find.
Stop, think, and come up with an approach to solve the problem.
Draw a free-body diagram with all of the following required elements:
1. An isolated body or group of bodies
2. All applied loads
3. All support reactions
4. A coordinate system
5. Labeled vectors
Write the equilibrium equations based on the free-body diagram.
Check if the number of equations is equal to the number of unknowns. If they are equal, the system is considered statically determinate and you can proceed in solving the problem. If they are not equal, the system is considered statically indeterminate, so something is missing. You may need additional free-body diagrams or other relationships.
Solve for unknowns, then check if your answer(s) are reasonable in terms of magnitude, units, and sign/direction.

Using these steps does not guarantee that you will get the right solution, and they might not be enough alone for every problem, but they will help you be critical and conscious when choosing your approach. This reflection will help you learn more quickly and increase the odds that you choose a winning strategy.