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Dynamics Particle and Rigid Body Motion cheat sheet - grade college

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Physics Grade college

Dynamics Particle and Rigid Body Motion Cheat Sheet

A printable reference covering Newton’s laws, momentum, work-energy, rigid-body kinematics, moments of inertia, and Lagrange equations for college.

TopicsPhysicsNewton's LawsForcesMotionMomentumEnergyTorqueRotation
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This cheat sheet summarizes the core tools used to analyze particle motion and rigid-body plane motion in college dynamics. It connects forces, motion, energy, momentum, and rotation in a compact reference format. Students need it because dynamics problems often require choosing the right model before doing any calculation.

The goal is to make common equations and decision points easy to scan during review.

Key Facts

  • Newton’s second law for a particle is ∑F⃗=ma⃗\sum \vec{F} = m\vec{a}, where the net external force determines the particle acceleration.
  • Linear impulse and momentum are related by ∫t1t2∑F⃗ dt=mv⃗2−mv⃗1\int_{t_1}^{t_2} \sum \vec{F}\,dt = m\vec{v}_2 - m\vec{v}_1.
  • The work-energy equation for a particle is T1+U1→2=T2T_1 + U_{1 \to 2} = T_2, with translational kinetic energy T=12mv2T = \frac{1}{2}mv^2.
  • For a rigid body in plane motion, the velocity of point BB relative to point AA is v⃗B=v⃗A+ω⃗×r⃗B/A\vec{v}_B = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}.
  • For a rigid body in plane motion, the acceleration relation is a⃗B=a⃗A+α⃗×r⃗B/A+ω⃗×(ω⃗×r⃗B/A)\vec{a}_B = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} + \vec{\omega} \times (\vec{\omega} \times \vec{r}_{B/A}).
  • The rotational equation of motion about a fixed axis through the mass center is ∑MG=IGα\sum M_G = I_G\alpha.
  • Rigid-body kinetic energy in plane motion is T=12mvG2+12IGω2T = \frac{1}{2}mv_G^2 + \frac{1}{2}I_G\omega^2.
  • Lagrange’s equation for generalized coordinate qiq_i is ddt(∂L∂q˙i)−∂L∂qi=Qi\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = Q_i, where L=T−VL = T - V.

Vocabulary

Particle
A body whose size and rotation are neglected so its motion is described only by the position, velocity, and acceleration of its mass center.
Rigid body
An idealized body whose particles keep fixed distances from one another during motion.
Mass moment of inertia
The quantity I=∫r2 dmI = \int r^2\,dm that measures how strongly a body resists angular acceleration about an axis.
Instantaneous center of zero velocity
The point in a plane-moving rigid body or its extension that has zero velocity at one instant.
Generalized coordinate
An independent variable qiq_i chosen to describe the configuration of a system using the fewest convenient coordinates.
Lagrangian
The scalar function L=T−VL = T - V used in analytical dynamics to derive equations of motion.

Common Mistakes to Avoid

  • Using ∑F⃗=mv⃗\sum \vec{F} = m\vec{v} instead of ∑F⃗=ma⃗\sum \vec{F} = m\vec{a} is wrong because force is related to acceleration, not velocity.
  • Mixing inertial and non-inertial reference frames is wrong because ∑F⃗=ma⃗\sum \vec{F} = m\vec{a} applies directly only in an inertial frame unless fictitious forces are included.
  • Using ∑MO=IOα\sum M_O = I_O\alpha about any moving point OO is wrong because this simple form is valid only for a fixed point or the mass center in standard plane-motion analysis.
  • Forgetting the centripetal acceleration term ω⃗×(ω⃗×r⃗)\vec{\omega} \times (\vec{\omega} \times \vec{r}) is wrong because rotating bodies can have acceleration even when α=0\alpha = 0.
  • Treating energy as conserved when nonconservative work is present is wrong because friction, motors, and applied forces can change the mechanical energy according to T1+V1+Wnc=T2+V2T_1 + V_1 + W_{nc} = T_2 + V_2.

Practice Questions

  1. 1 A 4.0 kg4.0\,\mathrm{kg} particle is acted on by a constant horizontal net force of 18 N18\,\mathrm{N}. Find its acceleration using ∑Fx=max\sum F_x = ma_x.
  2. 2 A solid disk with m=6.0 kgm = 6.0\,\mathrm{kg} and R=0.20 mR = 0.20\,\mathrm{m} rotates about its central axis with α=12 rad/s2\alpha = 12\,\mathrm{rad/s^2}. Using IG=12mR2I_G = \frac{1}{2}mR^2, find the required moment MGM_G.
  3. 3 A rigid body has vG=3.0 m/sv_G = 3.0\,\mathrm{m/s}, ω=8.0 rad/s\omega = 8.0\,\mathrm{rad/s}, m=10 kgm = 10\,\mathrm{kg}, and IG=0.75 kg m2I_G = 0.75\,\mathrm{kg\,m^2}. Compute T=12mvG2+12IGω2T = \frac{1}{2}mv_G^2 + \frac{1}{2}I_G\omega^2.
  4. 4 Explain when it is better to use Newton’s laws, work-energy, impulse-momentum, or Lagrange’s equation for a dynamics problem.

Understanding Dynamics Particle and Rigid Body Motion

The hardest part of dynamics is not usually the algebra. It is deciding what object is being studied and what counts as an external influence on it. A particle model treats size and shape as unimportant.

It works well for a car moving along a road, a ball in flight, or a satellite orbiting Earth. A rigid-body model is needed when turning matters. A door, ladder, gear, wheel, and connecting rod can translate while rotating.

Start every problem by drawing the body separately. Show weight, contact forces, tension, applied forces, and any known direction of motion. This free-body diagram prevents many sign errors before they begin.

Different methods answer different kinds of motion questions. Force and acceleration methods are most useful when the problem asks for acceleration, contact force, or tension at a particular instant. Momentum methods are useful during short events such as a hammer strike, a ball bounce, or a vehicle collision.

In these events, forces can be very large but act for little time. Energy methods are often simpler when motion occurs between two positions and time is not important. Friction, springs, gravity, and applied forces may transfer energy into or out of motion.

Students should state clearly whether a force does work. A force perpendicular to an object’s displacement changes its direction but does no work at that instant.

Rotation introduces two ideas that are easy to mix up. Angular velocity describes how fast orientation changes. Angular acceleration describes how fast the spinning rate changes.

A point farther from the axis has a greater linear speed when the body spins at the same angular velocity. Its inward acceleration is greater too. This is why the rim of a bicycle wheel moves faster than a point near the hub.

In acceleration problems, the tangential part comes from changing spin rate. The normal part points toward the center of rotation and comes from the current spin rate.

Draw these parts separately. Their directions are often more important than their sizes at first.

Mass distribution controls how readily an object changes its rotation. Two objects can have equal mass and equal radius but behave differently if one has more mass near its edge. A hoop is harder to speed up in rotation than a solid disk of the same mass and radius.

This matters in flywheels, figure skating, tools, and rolling objects on ramps. For rolling without slipping, translation and rotation are linked by the wheel radius. Do not assume every rolling object has the same acceleration down a slope.

Its moment of inertia matters. Lagrange equations offer a more organized route when a system has constraints, such as a pendulum attached to a moving support or several linked bars.

Choose coordinates that describe the allowed motion with as few variables as possible. That choice can turn a crowded force diagram into a manageable energy calculation.