Engineering Mechanics – Dynamics Unit 7 Review: Energy and Momentum in Rigid Body Dynamics

Key Concepts and Definitions

• Rigid body a solid object where the distance between any two points remains constant, regardless of external forces
• Center of mass (COM) the point where the total mass of the body can be considered to be concentrated
• Moment of inertia (MOI) a measure of an object's resistance to rotational acceleration, depends on the object's mass distribution
  • Calculated using the integral $I = \int r^2 dm$, where $r$ is the distance from the axis of rotation
• Angular velocity $\omega$ the rate of change of angular displacement, measured in radians per second
• Angular momentum $L$ the product of an object's moment of inertia and its angular velocity, $L = I\omega$
• Kinetic energy (KE) the energy an object possesses due to its motion, consists of translational and rotational components
  • Translational KE $= \frac{1}{2}mv^2$, where $m$ is mass and $v$ is velocity
  • Rotational KE $= \frac{1}{2}I\omega^2$, where $I$ is moment of inertia and $\omega$ is angular velocity
• Potential energy (PE) the energy an object possesses due to its position in a force field (gravitational, elastic, etc.)

Rigid Body Motion Basics

• Rigid body motion involves both translation and rotation
• Translation occurs when all points in the body move along parallel paths, described by the motion of the COM
• Rotation occurs when the body spins about an axis, described by angular velocity and acceleration
• Instantaneous center of rotation (ICR) the point about which a body appears to be rotating at a given instant
• Pure translation no rotation, all points move with the same velocity and acceleration
• Pure rotation no translation, the body rotates about a fixed axis
• General plane motion a combination of translation and rotation in a single plane
• Degrees of freedom (DOF) the number of independent parameters needed to describe a body's motion (maximum of 6 for a 3D rigid body)

Energy in Rigid Body Systems

• Total energy the sum of kinetic and potential energy in a system
• Work-energy principle states that the net work done on a system equals the change in its kinetic energy
  • $W_{net} = \Delta KE = KE_f - KE_i$, where $W_{net}$ is net work, and $KE_f$ and $KE_i$ are final and initial kinetic energy
• Power the rate at which work is done or energy is transferred, measured in watts (W)
  • $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$, where $P$ is power, $W$ is work, $t$ is time, $\vec{F}$ is force, and $\vec{v}$ is velocity
• Conservative forces (gravity, springs) work done is independent of the path taken and can be stored as potential energy
• Non-conservative forces (friction, air resistance) work done depends on the path and dissipates energy as heat
• Mechanical energy the sum of kinetic and potential energy in a system, conserved in the absence of non-conservative forces

Momentum in Rigid Body Dynamics

• Linear momentum $p$ the product of an object's mass and velocity, $\vec{p} = m\vec{v}$
• Angular momentum $L$ the product of an object's moment of inertia and angular velocity, $\vec{L} = I\vec{\omega}$
  • Calculated about a specific point or axis
• Impulse the product of a force and the time over which it acts, equals the change in linear momentum
  • $\vec{J} = \int \vec{F} dt = \Delta \vec{p} = m(\vec{v}_f - \vec{v}_i)$, where $\vec{J}$ is impulse, $\vec{F}$ is force, and $\vec{v}_f$ and $\vec{v}_i$ are final and initial velocity
• Angular impulse the product of a torque and the time over which it acts, equals the change in angular momentum
  • $\vec{H} = \int \vec{\tau} dt = \Delta \vec{L} = I(\vec{\omega}_f - \vec{\omega}_i)$, where $\vec{H}$ is angular impulse, $\vec{\tau}$ is torque, and $\vec{\omega}_f$ and $\vec{\omega}_i$ are final and initial angular velocity
• Collisions interactions between bodies where forces are large and act over a short time
  • Elastic collisions conserve both kinetic energy and momentum
  • Inelastic collisions conserve momentum but not kinetic energy

Conservation Laws and Principles

• Conservation of energy energy cannot be created or destroyed, only converted between forms
  • In an isolated system, total energy remains constant
• Conservation of linear momentum in the absence of external forces, the total linear momentum of a system remains constant
  • $\sum \vec{p}_{initial} = \sum \vec{p}_{final}$
• Conservation of angular momentum in the absence of external torques, the total angular momentum of a system remains constant
  • $\sum \vec{L}_{initial} = \sum \vec{L}_{final}$
• Principle of work and energy the change in kinetic energy of a system equals the net work done on it
  • Applies to both translational and rotational motion
• Parallel axis theorem allows the calculation of moment of inertia about any parallel axis, given the MOI about the COM
  • $I_{parallel} = I_{COM} + md^2$, where $d$ is the distance between the parallel axis and the COM
• Perpendicular axis theorem relates the moments of inertia about three mutually perpendicular axes
  • For a plane body, $I_z = I_x + I_y$, where $z$ is perpendicular to the $x$-$y$ plane

Equations and Formulas

• Rotational kinetic energy $KE_{rot} = \frac{1}{2}I\omega^2$
• Translational kinetic energy $KE_{trans} = \frac{1}{2}mv^2$
• Gravitational potential energy $PE = mgh$, where $h$ is the height above a reference level
• Elastic potential energy $PE = \frac{1}{2}kx^2$, where $k$ is the spring constant and $x$ is the displacement from equilibrium
• Power $P = \vec{F} \cdot \vec{v}$ (translational) or $P = \vec{\tau} \cdot \vec{\omega}$ (rotational)
• Linear momentum $\vec{p} = m\vec{v}$
• Angular momentum $\vec{L} = I\vec{\omega}$
• Impulse-momentum theorem $\vec{J} = \Delta \vec{p}$
• Angular impulse-momentum theorem $\vec{H} = \Delta \vec{L}$
• Parallel axis theorem $I_{parallel} = I_{COM} + md^2$
• Perpendicular axis theorem (for plane bodies) $I_z = I_x + I_y$

Real-World Applications

• Vehicle dynamics analyzing the motion, stability, and control of cars, trucks, and motorcycles
  • Involves considering tire forces, suspension, and weight distribution
• Robotics designing and controlling the motion of robotic arms, legs, and grippers
  • Requires understanding of joint torques, inertia, and control algorithms
• Biomechanics studying the forces and motions in living organisms (human gait analysis, prosthetic design)
  • Applies rigid body dynamics principles to biological systems
• Spacecraft dynamics modeling the attitude control and orbital mechanics of satellites and space vehicles
  • Involves analyzing torques from thrusters, reaction wheels, and environmental disturbances
• Sports equipment design optimizing the performance of bats, clubs, rackets, and balls
  • Considers the effect of mass distribution and impact dynamics on player performance and equipment durability
• Wind turbines predicting the loads and power output of wind turbine blades under various operating conditions
  • Requires modeling the coupled aerodynamic and structural dynamics of the blades and tower

Problem-Solving Strategies

• Identify the system and draw a clear diagram showing all relevant forces, moments, and dimensions
• Determine the type of motion (pure translation, pure rotation, or general plane motion) and choose appropriate coordinates
• Isolate the body or system of interest and apply Newton's laws (force/moment balance) in each direction/axis
  • For translation $\sum \vec{F} = m\vec{a}$, where $\vec{a}$ is the acceleration of the COM
  • For rotation $\sum \vec{M} = I\vec{\alpha}$, where $\vec{\alpha}$ is the angular acceleration about the axis of rotation
• Use the work-energy principle to relate changes in kinetic and potential energy to the work done by forces
  • Identify conservative and non-conservative forces and calculate the corresponding work
• Apply conservation of momentum (linear and angular) for systems with no external forces/torques
  • Determine initial and final momenta and solve for unknown velocities or angular velocities
• Check units, signs, and orders of magnitude to ensure the solution is physically reasonable
• Verify that the solution satisfies the original equations and constraints