Engineering Dynamics - Kinematics and Kinetics of Particles and Rigid Bodies

Comprehensive practice flashcards defining foundational vocabulary terms, core principles, kinematic relationships, and kinetic laws from Engineering Mechanics: Dynamics.

Mechanics

A branch of the physical sciences that deals with bodies at rest or in motion subjected to the action of forces.

Statics

The branch of mechanics concerned with the equilibrium of a body either at rest or moving with constant velocity.

Dynamics

The branch of mechanics that deals with the accelerated motion of a body, subdivided into kinematics and kinetics.

Kinematics

The branch of dynamics that treats only the geometric aspects of motion without considering the forces causing the motion.

Kinetics

The branch of dynamics that analyzes the forces causing the motion of a body.

Rectilinear Kinematics

The study of the kinematics of a particle moving along a straight-line path, characterized by its position, displacement, velocity, and acceleration at any instant.

Displacement (Rectilinear)

The vector change in position of a particle defined as $\Delta s = s' - s$, where a positive sign indicates a position to the right of the origin and a negative sign indicates a position to the left.

Average Speed

The total distance traveled by a particle, $s_T$, divided by the elapsed time interval $\Delta t$, expressed as $(v_{\text{sp}})_{\text{avg}} = \frac{s_T}{\Delta t}$.

Deceleration

The condition occurring when a particle's speed is decreasing, resulting in a negative change in velocity $\Delta v = v' - v$ such that the acceleration vector $\mathbf{a}$ acts in the opposite direction to velocity $\mathbf{v}$.

Curvilinear Motion

The motion that occurs when a particle travels along a curved trajectory in two or three dimensions.

Hodograph

A locus curve constructed by plotting the endpoints of the particle's velocity vectors $\mathbf{v}$ radiating from a fixed origin $O'$, such that the acceleration vector $\mathbf{a}$ is always tangent to this curve.

Osculating Plane

The plane containing the normal ($n$) and tangential ($t$) axes that is fixed in the plane of motion and locally tangent to the path curve at any instant.

Tangential Acceleration ($a_t$)

The acceleration component directed tangent to the path representing the time rate of change in the velocity's magnitude, formulated as $a_t = \frac{dv}{dt}$ or $a_t\,ds = v\,dv$.

Normal Acceleration ($a_n$)

The acceleration component directed perpendicular to the path toward the center of curvature, representing the time rate of change in the velocity's direction, defined as $a_n = \frac{v^2}{\rho}$.

Radius of Curvature ($\rho$)

The radius of the osculating circle for a path described by $y = f(x)$, evaluated as $\rho = \frac{[1 + (\frac{dy}{dx})^2]^{3/2}}{|\frac{d^2y}{dx^2}|}$.

Projectile Motion

Curvilinear motion of a launched body acted upon solely by downward gravitational acceleration ($a_c = g = 9.81\,\text{m/s}^2$) with no horizontal acceleration ($a_x = 0$), neglecting aerodynamic drag.

Absolute Dependent Motion

Motion where the position, velocity, and acceleration of one particle depend directly on the motion of another interconnected particle, related through a fixed cord length measured from a datum.

Relative Motion (Translating Axes)

The motion analysis of a particle $B$ observed from a moving frame attached to particle $A$ that translates without rotation, defined by $\mathbf{r}_B = \mathbf{r}_A + \mathbf{r}_{B/A}$.

Newton's Second Law of Motion

The law stating that a particle acted upon by an unbalanced force experiences an acceleration in the same direction as the force with a magnitude directly proportional to it, expressed as $\mathbf{F} = m\mathbf{a}$.

Kinetic Diagram

A diagram that graphically displays the resultant inertial vector $m\mathbf{a}$ produced by the unbalanced forces represented on the free-body diagram.

Kinetic Friction Force ($f_k$)

The contact force opposing the relative sliding of a particle on a rough surface, defined by $f_k = \mu_k N$, where $\mu_k$ is the coefficient of kinetic friction and $N$ is the normal reaction.

Spring Force ($F_s$)

The restoring force exerted by a linearly elastic spring toward its unstretched equilibrium position, governed by $F_s = ks$, where $k$ is stiffness in $\text{N/m}$ and $s = |l - l_0|$ is stretch or compression.

Work of a Force

The scalar quantity of energy transferred when a force $\mathbf{F}$ moves through a displacement $d\mathbf{r}$, defined as $dU = \mathbf{F}\cdot d\mathbf{r} = F\,ds\cos(\theta)$.

Principle of Work and Energy

The theorem stating that the initial kinetic energy of a particle plus the net work done by all forces equals its final kinetic energy: $T_1 + \sum U_{1-2} = T_2$.

Power

The amount of work performed per unit time, evaluated as $P = \frac{dU}{dt} = \mathbf{F}\cdot\mathbf{v}$, measured in watts ($1\,\text{W} = 1\,\text{J/s} = 1\,\text{N}\cdot\text{m/s}$).

Mechanical Efficiency ($\eta$)

The dimensionless ratio of the useful power output to the power input supplied to a machine: $\eta = \frac{\text{power output}}{\text{power input}}$.

Conservative Force

A force whose work done in moving a particle between two points depends only on the initial and final positions and is independent of the path followed (e.g., weight and linear spring forces).

Gravitational Potential Energy ($V_g$)

The potential energy of a particle of weight $W$ at vertical distance $y$ relative to a chosen datum, given by $V_g = \pm W y$, where elevation above the datum is positive and below is negative.

Elastic Potential Energy ($V_e$)

The energy stored in an elastic spring deformed by a distance $s$ from its unstretched position, expressed as $V_e = +\frac{1}{2}ks^2$, which is always positive.

Linear Momentum ($\mathbf{L}$)

A vector quantity defined by the product of a particle's mass and its velocity, $\mathbf{L} = m\mathbf{v}$, directed along the velocity vector with units of $\text{kg}\cdot\text{m/s}$.

Linear Impulse ($\mathbf{I}$)

The integral of a force acting over a time interval, defined as $\mathbf{I} = \int \mathbf{F}\,dt$, representing the force-time effect on a body with units of $\text{N}\cdot\text{s}$.

Principle of Linear Impulse and Momentum

The vector relation stating that the initial linear momentum of a particle plus the sum of all external impulses equals its final linear momentum: $m\mathbf{v}_1 + \sum \int_{t_1}^{t_2} \mathbf{F}\,dt = m\mathbf{v}_2$.

Impulsive Force

A force of very large magnitude that acts over a minute time interval, producing a finite and significant change in momentum.

Line of Impact

The common normal line passing through the centers of mass and the contact point of colliding particles along which internal impulsive collision forces act.

Central Impact

A collision where the velocities of both particles are directed along the line of impact immediately before contact.

Oblique Impact

A collision in which the velocity vector of one or both particles makes an angle with the line of impact immediately prior to contact.

Coefficient of Restitution ($e$)

The ratio of the restitution impulse to the deformation impulse during collision, formulated as $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$, ranging from $e = 1$ (elastic) to $e = 0$ (plastic).

Angular Momentum ($\mathbf{H}_O$)

The moment of a particle's linear momentum vector about a fixed point $O$, defined as $\mathbf{H}_O = \mathbf{r} \times m\mathbf{v}$, measured in $\text{kg}\cdot\text{m}^2/\text{s}$.

General Plane Motion

The planar motion of a rigid body that represents a simultaneous combination of translation and rotation about a reference axis.

Instantaneous Center of Zero Velocity (IC)

A point on or off a rigid body undergoing planar motion that momentarily has zero velocity, about which every point on the body rotates with angular velocity $\omega$, such that $v = \omega r$.

Coriolis Acceleration

The additional acceleration vector $2\boldsymbol{\Omega} \times (\mathbf{v}_{B/A})_{xyz}$ produced when a particle moves with relative velocity inside a rotating reference frame of angular velocity $\boldsymbol{\Omega}$.