Dynamics: Kinetics of Rigid Bodies

Rigid body kinetics explains how solid objects move when forces and moments act on them. Unlike particle dynamics, where all mass is treated as concentrated at a point, rigid body dynamics must account for rotation, distributed mass, and the way a body’s geometry resists changes in angular motion. The central tools are Newton’s laws expressed for translation and rotation, mass moment of inertia, and energy methods that often simplify real engineering problems.

What “rigid body” means in kinetics

A rigid body is an idealization: distances between any two points on the body are constant. Real materials deform, but for many machines, mechanisms, vehicles, and structures, those deformations are small enough that treating the body as rigid gives accurate results.

Kinetics is the part of dynamics that connects motion to its causes. You may know the kinematics (how the body moves), but kinetics answers why it moves that way given applied forces, torques, and constraints.

Translation and rotation happen together

Any planar rigid body motion can be viewed as:

• Translation of a reference point, often the center of mass $G$
• Rotation about that point with angular velocity $\omega$ and angular acceleration $\alpha$

This separation matters because Newton’s laws have clean forms when written about the center of mass.

Newton’s second law for translation

For the body treated as a whole, the sum of external forces equals mass times acceleration of the center of mass:

$$\sum {\mathbf{F}}_{ext}=m{\mathbf{a}}_G$$

This is the same idea as for a particle, but the key is that ${\mathbf{a}}_G$ is the acceleration of the mass center, not of an arbitrary point.

Newton’s second law for rotation

For planar motion (rotation about an axis perpendicular to the plane), the rotational equation of motion is:

$$\sum M_G=I_G\alpha$$

Here, $\sum M_G$ is the sum of external moments about the center of mass, $I_G$ is the mass moment of inertia about the same axis, and $\alpha$ is angular acceleration.

A practical advantage of summing moments about $G$ is that internal forces do not appear, and the inertial “response” is simply $I_G\alpha$.

Mass moment of inertia: what it is and why it matters

Mass moment of inertia is the rotational analogue of mass. It quantifies how strongly a body resists angular acceleration about a given axis.

For a rigid body with mass distribution, the definition is:

$$I=\int r^{2}dm$$

where $r$ is the perpendicular distance from the axis of rotation to each mass element $dm$. Two important implications follow:

1. The farther mass is from the axis, the larger the inertia, because of the $r^2$ weighting.
2. Inertia depends on the chosen axis. The same object can be “easy” to spin about one axis and “hard” about another.

Common examples (qualitative)

• A solid disk has a smaller moment of inertia than a thin ring of the same mass and radius because more mass in the disk is closer to the axis.
• A slender rod has very different inertia about its center compared with about an end, which is why a door swings more “heavily” if you try to rotate it about a point far from its hinges.

The parallel-axis theorem

Engineers rarely integrate directly for every new axis. The parallel-axis theorem relates inertia about a centroidal axis to inertia about a parallel axis offset by distance $d$:

$$I_O=I_G+m{d}^2$$

This is indispensable in mechanisms where the rotation axis is at a pin or bearing, not through the center of mass.

Composite bodies

Real components often consist of multiple simple shapes. In that case:

• Compute each part’s centroidal inertia
• Shift to the desired axis with the parallel-axis theorem
• Sum them: $I_{total}=\sum I_i$

The same strategy applies to mass and center of mass calculations, which are often needed before doing kinetics.

Rotational equations of motion in practice

In planar rigid body kinetics, you typically solve three unknowns using:

• $\sum F_x=m{a}_{Gx}$
• $\sum F_y=m{a}_{Gy}$
• $\sum M_G=I_G\alpha$

If you choose a different point $O$ for moments, the rotational equation can be written as:

$$\sum M_O=I_O\alpha +{\mathbf{r}}_{G/O}\times m{\mathbf{a}}_G$$

This is still correct, but it is easier to make mistakes because of the extra term. Many problems become cleaner when moments are taken about $G$, provided $I_G$ is known.

Example context: a braking wheel

Consider a wheel with a brake applying a tangential force at the rim. The brake produces a moment about the wheel’s center. If the wheel rolls without slipping, translation and rotation are coupled. The external forces determine $a_G$, while the braking torque and friction determine $\alpha$. The mass moment of inertia decides how quickly the wheel’s angular speed drops for a given braking moment.

This is why heavier rims (larger inertia) tend to “hold speed” longer, and why lightweight wheels can accelerate and decelerate faster.

Energy methods for rigid body motion

Force and moment balances are not always the shortest path, especially when forces vary with position or when you care about speeds rather than accelerations. Energy methods provide another powerful route.

Kinetic energy of a rigid body

For planar motion, total kinetic energy can be split into translation of the center of mass plus rotation about the center of mass:

$$T=\frac{1}{2}m{v}_G^2+\frac{1}{2}{I}_G{\omega }^2$$

This form is both intuitive and practical. It makes clear that a body can have significant kinetic energy even if the center of mass moves slowly, as long as it spins rapidly (or has a large $I_G$).

Work-energy principle

The work done by external forces and moments equals the change in kinetic energy:

$$W_{ext}=\Delta T$$

In rotational terms, a moment $M$ doing work through angular displacement $\theta$ contributes $W=\int Md\theta$.

This approach is especially useful when:

• You want angular speed after a given rotation
• Forces are conservative or can be expressed as functions of position
• Constraint forces do no work (common with ideal pins and rolling without slip)

Conservative forces and potential energy

When gravity or springs are involved, you can use potential energy $V$ and write:

$$T_1+V_1+W_{nc}=T_2+V_2$$

where $W_{nc}$ is work by non-conservative effects such as friction or applied motors. For many mechanical systems, this provides a clean, physically transparent solution without explicitly computing reaction forces.

Choosing a method: forces and moments vs energy

A good rigid body dynamics solution often starts with a deliberate choice of tools:

• Use Newton-Euler equations when you need forces at supports, contact forces, or accelerations at a specific instant.
• Use energy methods when you need speeds after moving through a distance or angle, and the forces are easier to integrate as work than to resolve in time.
• Combine them when necessary. For example, energy can give speed, then Newton’s laws can give forces at that speed.

Practical insight: why inertia dominates design decisions

Mass moment of inertia is not an abstract classroom quantity. It is a design driver.

• In robotics and automation, reducing inertia improves responsiveness and reduces actuator torque requirements.
• In rotating machinery, higher inertia can smooth speed fluctuations, which is useful in flywheels and some engine systems.
• In vehicles, wheel and driveline inertia affects acceleration, braking, and energy efficiency.
• In safety and comfort, inertia shapes how systems respond to disturbances, from spinning tools to rotating platforms.

Understanding rigid body kinetics means understanding how force, torque, and mass distribution collaborate to produce motion. Once you can compute or estimate moments of inertia, apply rotational equations of motion, and use energy methods confidently, you can analyze most planar rigid body systems encountered in mechanical engineering and applied physics.