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    Introduction to Statics

    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). There is no derivation for Newton's equations, because they are an assumed model for dynamics. We can only verify them by comparing with experimental evidence, which confirms that Newtonian dynamics are accurate for non-relativistic and non-quantum systems.

    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:
    1. The mass of the particle is not 0.
    2. 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).

    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 dynamics.

    location description velocity description
    point mass position vector \( \vec{r}_P \) velocity vector \( \vec{ v}_P \)
    rigid body in 2D

    position vector \( \vec{r}_P \)

    angle \( \theta \)

    velocity vector \( \vec{v}_P \)

    angular velocity \( \omega \)
    rigid body in 3D

    position vector \( \vec{r}_P \)

    angles \( \theta,\phi,\psi \)

    velocity vector \( \vec{v}_P \)

    angular velocity vector \( \vec{\omega} \)