FE Dynamics: Kinetics Review

Mastering kinetics is essential for success on the FE Exam, as it forms the backbone of many mechanical and civil engineering problems. This review focuses on the force-acceleration methods derived from Newton's Second Law, moving systematically from particle dynamics to the more complex motion of rigid bodies. You will learn to efficiently set up equations of motion across different coordinate systems and calculate key inertial properties, transforming intricate word problems into solvable equations under time-pressured exam conditions.

Newton's Second Law: The Foundation for Particles and Rigid Bodies

Newton's Second Law of Motion provides the fundamental link between the forces acting on a body and its resulting acceleration. For a particle (a point mass with no rotation), the law is straightforward: the vector sum of all forces equals the mass times the acceleration vector, expressed as $\sum \vec{F}=m\vec{a}$. This is a vector equation, meaning you must apply it independently in each relevant direction (e.g., x, y, or n, t).

For a rigid body (an object with size and shape that does not deform), the law extends to two key equations. The first governs translational motion: the sum of all external forces equals the mass of the body times the acceleration of its center of mass (G): $\sum \vec{F}=m{\vec{a}}_G$. The second governs rotational motion: the sum of the moments about the center of mass equals the mass moment of inertia about G times the angular acceleration: $\sum M_G=I_G\alpha$. It is critical to remember that for a rigid body, the translational equation uses the acceleration of the center of mass, not an arbitrary point on the body.

Choosing Your Coordinate System: Normal-Tangential and Polar

The chosen coordinate system dramatically simplifies the equations of motion for curved paths. The normal-tangential (n-t) coordinate system is ideal for known paths, such as a car on a road or a block on a curved track. Here, acceleration is resolved into two perpendicular components: tangential ($a_t$) and normal ($a_n$). The tangential component is the rate of change of speed ($a_t=dv/dt$). The normal, or centripetal, component is directed toward the path's center of curvature and is calculated as $a_n=v^2/\rho$, where $\rho$ is the radius of curvature. Applying Newton's Second Law in these directions gives $\sum F_t=m{a}_t$ and $\sum F_n=m{a}_n$.

For problems involving central force motion or rotation about a fixed point, the polar coordinate system (r-θ) is more effective. In this system, a point's position is defined by a radial distance ($r$) from a fixed origin and an angular coordinate ($\theta$). The acceleration components are more complex: radial acceleration $a_r=\ddot{r}-r{\dot{\theta }}^2$ and transverse acceleration $a_{\theta }=r\ddot{\theta }+2\dot{r}\dot{\theta }$. On the FE exam, you'll often encounter special cases like circular motion where $\dot{r}=0$ and $\ddot{r}=0$, simplifying these expressions significantly to $a_r=-r{\dot{\theta }}^2$ and $a_{\theta }=r\ddot{\theta }$.

Moments: About Fixed Points, Moving Points, and the Center of Mass

Correctly summing moments is crucial for rigid body rotation. The fundamental equation is $\sum M_G=I_G\alpha$. However, you can sum moments about other points if you account for the "inertial moment" caused by the translation of the center of mass. For a point A, the moment equation becomes: $$\sum M_A=I_G\alpha +m\vec{d}\times {\vec{a}}_G$$ where $\vec{d}$ is the position vector from A to G. The cross-product term $m\vec{d}\times {\vec{a}}_G$ is often called the "inertia" of the linear acceleration.

A major shortcut exists for two special cases. You can use $\sum M_P=I_P\alpha$ directly (without the cross-product term) if point P is either: 1) the center of mass G, or 2) a fixed point (or a point with zero acceleration, like the instantaneous center of zero velocity for general plane motion). Identifying these cases on the exam can save valuable time and reduce algebraic errors.

Calculating Mass Moment of Inertia

The mass moment of inertia, $I$, is a measure of a body's resistance to angular acceleration. It depends on the object's mass distribution relative to the axis of rotation. The fundamental definition is the integral $I=\int r^{2}dm$, where $r$ is the perpendicular distance from the axis to the mass element $dm$. For the FE exam, you must know the formulas for common shapes. These are typically provided in the NCEES FE Reference Handbook, but recognizing and applying them quickly is key.

The most essential formula is for a slender rod. For an axis through its center, $I_G=\frac{1}{12}m{L}^2$. For an axis at its end, $I_{end}=\frac{1}{3}m{L}^2$. For other common shapes:

• Solid cylinder/disk about its central axis: $I=\frac{1}{2}m{r}^2$
• Thick-walled circular tube about its central axis: $I=\frac{1}{2}m(r_{outer}^2+r_{inner}^2)$
• Solid sphere about its centroid: $I=\frac{2}{5}m{r}^2$

To find the moment of inertia about any parallel axis, use the parallel axis theorem: $I_A=I_G+m{d}^2$, where $d$ is the perpendicular distance between the axis through the center of mass (G) and the parallel axis through A. This theorem is indispensable for composite bodies.

Common Pitfalls

1. Using $\sum F=ma$ for a Point Other Than the Center of Mass: For a rigid body, the translational equation $\sum \vec{F}=m\vec{a}$ is only valid if $\vec{a}$ is the acceleration of the center of mass. A common trap is to incorrectly apply it using the acceleration of another point, like the point of contact on a rolling object.

• Correction: Always identify the center of mass (G) first. For translation, use $\sum \vec{F}=m{\vec{a}}_G$.

1. Misapplying the Simple Moment Equation $\sum M_P=I_P\alpha$: It is tempting to sum moments about any convenient point P and use this simple form. This is only valid if P is the center of mass or a fixed point (or point with zero acceleration).

• Correction: If you choose a point that is accelerating, you must use the full equation: $\sum M_P=I_G\alpha +m\vec{d}\times {\vec{a}}_G$.

1. Confusing Radial Acceleration in Polar Coordinates: In circular motion, the radial acceleration $a_r$ is not zero; it is the centripetal acceleration. Students often see $\ddot{r}=0$ for constant radius and incorrectly conclude $a_r=0$.

• Correction: Remember the full expression: $a_r=\ddot{r}-r{\dot{\theta }}^2$. For constant radius circular motion, $\ddot{r}=0$, so $a_r=-r{\omega }^2$ (directed inward).

1. Forgetting the Direction of Normal Force: In n-t coordinates, the normal force $F_n$ is not always "up"; it is always perpendicular to the path and directed toward the center of curvature. On the crest of a hill, the normal force from the road points downward.

• Correction: Draw your n-axis consistently toward the local center of curvature. The sign of $F_n$ in your equation $\sum F_n=m(v^2/\rho )$ will then follow from your chosen positive direction.

Summary

• The core kinetics approach uses Newton's Second Law: for translation, $\sum \vec{F}=m{\vec{a}}_G$; for rotation about G, $\sum M_G=I_G\alpha$.
• Select your coordinate system based on the motion: use normal-tangential (n-t) for known curved paths, and polar (r-θ) for motion about a fixed center or rotational problems.
• When summing moments for a rigid body, you may use $\sum M_P=I_P\alpha$ directly only if point P is the center of mass (G) or a fixed point. Otherwise, you must include the $m\vec{d}\times {\vec{a}}_G$ term.
• Mass moment of inertia resists angular acceleration. Know the formulas for common shapes (rod, disk, sphere) and use the parallel axis theorem ($I_A=I_G+m{d}^2$) to find I about any parallel axis.
• On the FE exam, carefully define your system (particle vs. rigid body), identify the correct acceleration, and double-check the applicability of simplified moment equations to avoid classic traps.