FE Dynamics: Kinematics Review

Mastering kinematics—the geometry of motion—is non-negotiable for the FE exam. It’s the fundamental language of dynamics, forming the bedrock for solving complex problems in mechanics of materials, vibrations, and machine design. A rapid, accurate command of kinematic principles allows you to dissect any motion scenario efficiently, saving precious minutes and boosting your confidence across multiple exam sections. This review is designed to sharpen that skill, focusing on the systematic approaches you need to select and execute under time pressure.

Particle Kinematics: Describing a Point's Motion

Particle kinematics deals with the motion of a point, ignoring its size or rotation. The first critical decision is choosing your coordinate system. For rectilinear motion (motion along a straight line), you describe position with a single coordinate, often $s(t)$. Velocity is the time derivative of position, $v=ds/dt$, and acceleration is the derivative of velocity, $a=dv/dt=d^{2}s/d{t}^2$. Crucially, you can often relate acceleration directly to velocity or position using the chain rule: $a=vdv/ds$. A common exam question presents an acceleration expressed as a function of velocity (e.g., $a=-kv$) and asks for the distance to stop. The solution path is to use $a=vdv/ds$, separate variables, and integrate.

Curvilinear motion describes a point moving along a curved path. Here, your coordinate system choice dictates your analysis method. The Cartesian (x-y) approach is best when the motion's components are independent, described by $x(t)$ and $y(t)$. You find velocity components $v_x=dx/dt$, $v_y=dy/dt$ and acceleration components $a_x=d^{2}x/d{t}^2$, $a_y=d^{2}y/d{t}^2$. Use this for projectile motion without air resistance.

When a particle's path is known, normal and tangential (n-t) coordinates are powerful. Here, velocity is always tangent to the path: $v=\dot{s}$ (where $s$ is the path coordinate). Acceleration has two components: tangential ($a_t=dv/dt$) changes the speed, and normal ($a_n=v^2/\rho$) changes the direction. The term $\rho$ is the radius of curvature of the path at that point. For a known circular path of radius $r$, $\rho =r$. This system is ideal for problems involving cars on curved roads or particles on tracks.

Finally, polar coordinates (r-θ) excel for motion involving radial lines, like a sliding collar on a rotating rod. Position is defined by a radial distance $r$ and an angle $\theta$. The velocity components are radial ($v_r=\dot{r}$) and transverse ($v_{\theta }=r\dot{\theta }$). The acceleration components are more complex: radial acceleration $a_r=\ddot{r}-r{\dot{\theta }}^2$ and transverse acceleration $a_{\theta }=r\ddot{\theta }+2\dot{r}\dot{\theta }$. The $-r{\dot{\theta }}^2$ term is the centripetal acceleration, and $2\dot{r}\dot{\theta }$ is the Coriolis acceleration, a critical concept for rotating frame analysis.

Relative Motion and Dependent Motion Constraints

Engineering systems often involve multiple moving parts. Relative motion analysis breaks down complex motion by relating the motions of two particles. The fundamental equation is a vector relationship: ${\vec{v}}_B={\vec{v}}_A+{\vec{v}}_{B/A}$. Here, ${\vec{v}}_{B/A}$ is the velocity of $B$ as seen by an observer moving with $A$. The same applies to acceleration: ${\vec{a}}_B={\vec{a}}_A+{\vec{a}}_{B/A}$. This is the goto method for analyzing interconnected rigid bodies or particles on moving platforms. For example, to find the velocity of a passenger walking on a moving ship, you'd add the ship's velocity (motion of the frame) to the passenger's velocity relative to the ship.

Dependent motion uses physical constraints—typically ropes, cables, or linkages—to relate the positions of particles. The key is to define position coordinates from a fixed datum to each object, then write an equation for the constant length of the connecting element(s). Differentiating this constraint equation with respect to time yields relationships between velocities and accelerations. A classic FE setup is a system of pulleys and weights. The trick is to recognize that the total length of the inextensible rope is constant. For a simple two-pulley system with one weight hanging and one on an incline, you might find that if weight $A$ moves down a distance $s_A$, weight $B$ moves up the incline a distance $s_B/2$. Therefore, $v_A=2v_B$ and $a_A=2a_B$. Always check your sign convention: positive direction for each coordinate.

Rigid Body Kinematics: Rotation and General Plane Motion

A rigid body is an object where the distance between any two points remains constant. Its planar motion falls into three categories: translation, rotation about a fixed axis, and general plane motion (a combination of translation and rotation). For fixed-axis rotation, every point on the body moves in a circular path about the axis. The body's angular position is $\theta$, angular velocity is $\omega =d\theta /dt$, and angular acceleration is $\alpha =d\omega /dt=d^2\theta /d{t}^2$. A point at a distance $r$ from the axis has tangential velocity $v=r\omega$ and tangential acceleration $a_t=r\alpha$. Its normal acceleration, directed toward the axis, is $a_n=r{\omega }^2=v^2/r$.

General plane motion is analyzed using relative motion, but applied to two points on the same rigid body. The governing equation is: ${\vec{v}}_B={\vec{v}}_A+{\vec{v}}_{B/A}$. Here, ${\vec{v}}_{B/A}$ is not arbitrary; it is the velocity of $B$ relative to A due only to the body's rotation. Its magnitude is $v_{B/A}=\omega \cdot r_{B/A}$ and its direction is perpendicular to the line $AB$. In practice, you often know the direction of ${\vec{v}}_A$ and ${\vec{v}}_B$ (e.g., a wheel rolling without slipping: the contact point with the ground has instantaneously zero velocity). Writing the vector equation in two perpendicular component directions (usually x and y) allows you to solve for two unknowns, typically a speed and an angular velocity.

The Instantaneous Center of Rotation

For a body in general plane motion at a given instant, there exists a point where the velocity is zero. This point is the instantaneous center of zero velocity (IC). It is a powerful shortcut for finding velocities. If you know the direction of the velocities of two points on a body, you can locate the IC: it lies at the intersection of lines drawn perpendicular to each velocity vector. Once located, the body is treated as purely rotating about the IC at that instant. The velocity of any point $P$ is perpendicular to the line from the IC to $P$, with magnitude $v_P=\omega \cdot (r_{P/IC})$. For a wheel rolling without slipping, the IC is the contact point with the ground. The center's velocity is then $v_O=\omega \cdot r$, a fundamental result. Remember: The IC is a geometric point that can be on or off the physical body, and it changes location from instant to instant. Its acceleration is not zero, so you cannot use it for acceleration analysis.

Common Pitfalls

1. Mixing Coordinate Systems Incoherently: The most frequent error is using components from different systems in a single vector equation. You cannot directly add a normal-tangential component to a Cartesian x-component. Decide on one analysis frame (Cartesian, n-t, or relative) for the entire equation and resolve all vectors into that frame's components before solving.
2. Misapplying the Relative Motion Equation: Confusing ${\vec{v}}_{B/A}$ (velocity of B relative to A) with ${\vec{v}}_B-{\vec{v}}_A$ is a semantic trap, but the bigger error is misidentifying the reference frame. In rigid body kinematics (${\vec{v}}_B={\vec{v}}_A+\vec{\omega }\times {\vec{r}}_{B/A}$), points A and B must be on the same rigid body. Do not use this form for particles on different, independently moving objects.
3. Forgetting the Chain Rule in Rectilinear Motion: When acceleration is given as a function of velocity or position, blindly trying to integrate $a=dv/dt$ will fail if you don't have $a$ as a function of time. Immediately recognize the need for the kinematic relationship $a=vdv/ds$ or $a=dv/dt$.
4. Using the Instantaneous Center for Acceleration: The IC is only for velocity analysis. Because the IC's location moves, it generally has a non-zero acceleration. Attempting to use $a_P=\alpha \times r_{P/IC}-{\omega }^2{r}_{P/IC}$ is incorrect. For acceleration, you must return to the full relative acceleration equation: ${\vec{a}}_B={\vec{a}}_A+\vec{\alpha }\times {\vec{r}}_{B/A}-{\omega }^2{\vec{r}}_{B/A}$.

Summary

• Systematic Setup is Key: Your first step for any kinematics problem is to choose the most efficient coordinate system or method—Cartesian for independent components, n-t for known paths, relative motion for interconnected parts, or the IC for quick velocities.
• Distinguish Particle vs. Rigid Body: Particle kinematics uses derivatives of path coordinates. Rigid body kinematics requires the relative motion equation ${\vec{v}}_B={\vec{v}}_A+\vec{\omega }\times {\vec{r}}_{B/A}$, which treats motion as translation plus rotation.
• Leverage Physical Constraints: For dependent motion, write a geometric constraint equation based on constant rope length or no-slip conditions, then differentiate to relate velocities and accelerations.
• The IC is a Velocity-Only Tool: The instantaneous center provides a fast way to find velocities by treating motion as pure rotation at that instant, but it cannot be used for acceleration calculations.
• Avoid Component Confusion: Never combine vector components from different coordinate systems. Resolve all vectors into a single, consistent set of directions before solving your equations.