Kinematics and dynamics of the rigid body
An extended body does not only translate: it also rotates. Every quantity of translation has its twin in rotation, and mass becomes moment of inertia.
01Every quantity has its twin
A rigid body is a model: a solid whose particles always keep the same distance from one another. It does not deform. With that idealization, any motion breaks down into translation of the center of mass plus rotation about it.
And the remarkable thing is that rotation repeats, term by term, the structure of translation:
| Translation | Rotation | Relation |
|---|---|---|
| Position x | Angle \( \theta \) | \( s = \theta \cdot R \) |
| Velocity v | Angular velocity \( \omega \) | \( v = \omega \cdot R \) |
| Acceleration a | Angular acceleration \( \alpha \) | \( a = \alpha \cdot R \) |
| Mass m | Moment of inertia I | \( I = \sum m_i \, r_i^2 \) |
| Force F | Torque \( M = F \cdot d \) | \( M = r \times F \) |
| \( \Sigma F = m\cdot a \) | \( \Sigma M = I\cdot \alpha \) | The second law, twice |
| \( p = m\cdot v \) | \( L = I\cdot \omega \) | Angular momentum |
| \( E_c = \tfrac{1}{2} m v^2 \) | \( E_c = \tfrac{1}{2} I \omega^2 \) | They add up if both are present |
| \( P = F\cdot v \) | \( P = M\cdot \omega \) | Power |
The last row is the one used every day: a 1.5 kW motor turning at 1500 rpm delivers a torque of M = P/ω = 1500/(1500·2π/60) = 9.55 N·m. That calculation decides which motor is needed to move a load, and it is the same for a 3D-printer servo as for a grinding mill.
02Moment of inertia
Mass measures the resistance to changes in velocity; the moment of inertia measures the resistance to changes in angular velocity. And unlike mass, it depends on the axis: the same part is harder or easier to spin depending on where you hold it.
| Body (axis through the center) | Moment of inertia | \( k = \dfrac{I}{m\cdot R^{2}} \) |
|---|---|---|
| Hoop or thin-walled hollow cylinder | \( m\cdot R^{2} \) | 1 |
| Disk or solid cylinder | \( \tfrac{1}{2}\cdot m\cdot R^{2} \) | 0.5 |
| Hollow sphere | \( \tfrac{2}{3}\cdot m\cdot R^{2} \) | 0.667 |
| Solid sphere | \( \tfrac{2}{5} m R^2 \) | 0.4 |
| Rod about its center | \( \dfrac{1}{12}\cdot m\cdot L^{2} \) | — |
| Rod about one end | \( \tfrac{1}{3}\cdot m\cdot L^{2} \) | — |
For an axis parallel to the one through the center of mass, at a distance d: I = Icm + m·d². That is why a rod held at one end has ⅓·m·L² and not 1/12: 1/12 + 1/4 = 1/3.
03Rotational dynamics and rolling
A body that rolls without slipping satisfies v = ω·R, and its kinetic energy is split between translation and rotation. Writing the conservation of energy for a drop through a height h:
This gives a result that always surprises: in a race down an inclined plane, the solid sphere beats the disk, and the disk beats the hoop, regardless of how much each weighs or how big it is. The winner is the one that puts less energy into spinning.
All five start together from the same height. Move with the time slider and watch the finishing order.
04Angular momentum and its conservation
If the net external torque is zero, L = I·ω is conserved. Since I can change —by moving mass closer to or farther from the axis—, the angular velocity changes in the opposite direction:
- The skater who pulls in her arms reduces I and spins faster.
- The cat that falls on its back rights itself in midair with no external torque: it changes its shape.
- A satellite is stabilized by spinning an internal wheel; the spacecraft turns the opposite way. This is the principle of the reaction wheel, which orients space telescopes without using fuel.
- A gyroscope resists changes in the orientation of its axis, and that property is what a MEMS gyroscope in a phone or a drone measures.
When a torque is applied to a spinning body, the axis does not move in the direction the force pushes but perpendicular to it: this is precession. It is what keeps a moving bicycle upright and what makes a tilted spinning top turn instead of falling over.
05Where it shows up in electronics
- Sizing a motor. The starting torque must overcome the moment of inertia of the load: M = I·α. A stepper motor that falls short skips steps, and the error accumulates.
- Encoders. They measure θ and, by differentiating, ω. The speed control of a motor is a loop around that measurement.
- MEMS gyroscopes. They detect rotation through the Coriolis effect on a vibrating mass: the image stabilization of a camera and the control of a drone depend on them.
- Disks and fans. The time a disk or a fan takes to reach operating speed comes from I·α = Mmotor − Mfriction.
- Flywheels. They store energy as ½·I·ω², and are used as a reserve in large uninterruptible power supply systems.
06In the lab
Let a hoop, a disk and a sphere of the same diameter roll down an inclined board. Verify the finishing order and check that it does not change when bodies of different mass are used.
Hang a weight from a string wound around the axle of a wheel and measure the acceleration with which it descends. Solve for I from a = g/(1 + I/(m·r²)). Compare with the theoretical value for a disk.
With an MPU-6050, record the angular velocity while rotating the board 90°. Integrate ω numerically to obtain the angle and check how much it drifts because of the sensor’s drift: that accumulated error is the reason it is combined with the accelerometer in a complementary filter.
07Common mistakes
- Using a moment of inertia without saying about which axis. It makes no sense.
- Forgetting the parallel-axis theorem when the axis does not pass through the center of mass.
- Adding only the translational energy for a rolling body: ½·I·ω² is missing.
- Believing the heavier body wins the race. The mass cancels.
- Confusing the torque of a force with angular momentum. One is the cause, the other is what changes: M = dL/dt.
- Applying v = ω·R when there is slipping. It only holds in pure rolling.
08Self-assessment
Which reaches the bottom of a plane first: a solid sphere or a hoop?
The sphere: k = 0.4 versus k = 1, so its acceleration is g·sinθ/1.4 versus g·sinθ/2. It arrives first: it takes 16% less time (and at the bottom it moves about 20% faster).
A 2 kW motor turns at 3000 rpm. What is its torque?
ω = 3000·2π/60 = 314 rad/s, so M = 2000/314 ≈ 6.4 N·m.
How much energy does a flywheel with I = 0.5 kg·m² store at 6000 rpm?
ω = 628 rad/s, E = ½·0.5·628² ≈ 98.6 kJ. Doubling the speed quadruples the energy.
A 1 m, 2 kg rod rotates about one end. What is its moment of inertia?
I = ⅓·m·L² = ⅓·2·1 = 0.667 kg·m².
Why does the skater speed up when she pulls in her arms?
Because L = I·ω is conserved: as I decreases, ω increases in the same proportion. The kinetic energy, on the other hand, increases, and comes from the muscular work of pulling the arms in.
In what proportion is the energy of a rolling disk divided?
Two thirds in translation and one third in rotation, because ½·I·ω² = ¼·m·v², half of the translational energy.
09Further reading
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics, Volume 1. 9th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage Learning, 2014. Chapters 10 and 11: rotation, rolling and angular momentum.
- Paul A. Tipler and Gene Mosca. Physics for Scientists and Engineers, Volume 1. 6th ed., W. H. Freeman, 2008 (Spanish edition, “Física para la ciencia y la tecnología”, Reverté, 2010). Good development of the parallel-axis theorem and of gyroscopic precession.
- Ferdinand P. Beer and E. Russell Johnston. Vector Mechanics for Engineers: Dynamics. 10th ed., McGraw-Hill, 2013 (Spanish edition, “Mecánica vectorial para ingenieros: dinámica”). The engineering approach, with many cases of real machines.
- Marcelo Alonso and Edward J. Finn. Physics. Addison-Wesley, 1992 (Spanish edition, “Física”, Addison-Wesley Iberoamericana, 1995). Treats rotation starting from angular momentum, not from the moment of inertia.