🏎️Engineering Mechanics – Dynamics Unit 2 – Kinetics of Particles in Engineering Mechanics

Key Concepts and Definitions

• Particle a body with a mass but negligible size and shape, treated as a point
• Kinematics the study of motion without considering the forces causing it, focusing on position, velocity, and acceleration
• Dynamics the study of forces and their effects on motion, building upon kinematics
• Inertia the resistance of an object to changes in its motion, related to its mass
• Force a push or pull acting on an object, causing it to accelerate if unbalanced
  • Contact forces (friction, normal force) result from direct contact between objects
  • Action-at-a-distance forces (gravity, electromagnetism) act without direct contact
• Momentum the product of an object's mass and velocity, a vector quantity
• Work the product of force and displacement, a scalar quantity measuring energy transfer
• Energy the capacity to do work, existing in various forms (kinetic, potential, thermal)
  • Kinetic energy $\frac{1}{2}mv^2$, energy of motion
  • Potential energy energy due to position or configuration (gravitational, elastic)

Kinematics of Particles

• Position the location of a particle in space, typically described using Cartesian coordinates $(x, y, z)$
• Displacement the change in position of a particle, a vector quantity
• Velocity the rate of change of position with respect to time, the first derivative of position $\vec{v} = \frac{d\vec{r}}{dt}$
  • Average velocity $\vec{v}_{avg} = \frac{\Delta\vec{r}}{\Delta t}$, the displacement divided by the time interval
  • Instantaneous velocity $\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta\vec{r}}{\Delta t}$, the velocity at a specific instant in time
• Acceleration the rate of change of velocity with respect to time, the second derivative of position $\vec{a} = \frac{d\vec{v}}{dt}$
  • Average acceleration $\vec{a}_{avg} = \frac{\Delta\vec{v}}{\Delta t}$, the change in velocity divided by the time interval
  • Instantaneous acceleration $\vec{a} = \lim_{\Delta t \to 0} \frac{\Delta\vec{v}}{\Delta t}$, the acceleration at a specific instant in time
• Kinematic equations a set of equations describing the motion of a particle under constant acceleration
  • $\vec{v} = \vec{v}_0 + \vec{a}t$
  • $\vec{r} = \vec{r}_0 + \vec{v}_0t + \frac{1}{2}\vec{a}t^2$
  • $\vec{v}^2 = \vec{v}_0^2 + 2\vec{a} \cdot (\vec{r} - \vec{r}_0)$

Newton's Laws and Particle Dynamics

• Newton's first law an object at rest stays at rest, and an object in motion stays in motion with constant velocity, unless acted upon by an unbalanced force
• Newton's second law the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass $\vec{F} = m\vec{a}$
• Newton's third law for every action (force), there is an equal and opposite reaction (force)
• Free body diagram a graphical representation of all the forces acting on an object, used to analyze the net force
• Friction a force that opposes the relative motion between two surfaces in contact
  • Static friction $f_s \leq \mu_s N$, the force preventing an object from starting to move
  • Kinetic friction $f_k = \mu_k N$, the force opposing the motion of an object sliding on a surface
• Tension the force transmitted through a rope, string, or cable pulling on an object
• Normal force the force exerted by a surface on an object in contact with it, perpendicular to the surface

Work, Energy, and Power

• Work the product of force and displacement in the direction of the force $W = \vec{F} \cdot \vec{d}$
  • Positive work done by a force in the same direction as the displacement
  • Negative work done by a force opposite to the direction of displacement
• Work-energy theorem the net work done on an object equals the change in its kinetic energy $W_{net} = \Delta KE$
• Conservative force a force for which the work done is independent of the path taken (gravity, spring force)
  • Potential energy associated with conservative forces, $\Delta PE = -W_{cons}$
• Non-conservative force a force for which the work done depends on the path taken (friction, air resistance)
• Conservation of energy the total energy of a closed system remains constant, energy cannot be created or destroyed, only converted between forms
• Power the rate at which work is done or energy is transferred $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$
  • Average power $P_{avg} = \frac{W}{\Delta t}$, the total work divided by the time interval
  • Instantaneous power $P = \lim_{\Delta t \to 0} \frac{\Delta W}{\Delta t}$, the power at a specific instant in time

Impulse and Momentum

• Impulse the product of force and time, equal to the change in momentum $\vec{J} = \vec{F}\Delta t = \Delta\vec{p}$
• Conservation of momentum the total momentum of a closed system remains constant, momentum is neither created nor destroyed, only transferred between objects
  • Elastic collision a collision in which both momentum and kinetic energy are conserved
  • Inelastic collision a collision in which momentum is conserved but kinetic energy is not, some energy is converted to other forms (heat, deformation)
• Coefficient of restitution a measure of the elasticity of a collision, the ratio of the final to initial relative velocities $e = -\frac{v_{2f} - v_{1f}}{v_{2i} - v_{1i}}$
  • Perfectly elastic collision $e = 1$, kinetic energy is conserved
  • Perfectly inelastic collision $e = 0$, objects stick together after the collision
• Center of mass the point at which the entire mass of a system can be considered to be concentrated $\vec{r}_{cm} = \frac{\sum m_i\vec{r}_i}{\sum m_i}$
• Rocket equation describes the motion of a rocket, based on the conservation of momentum $\Delta v = v_e \ln \frac{m_0}{m_f}$

Relative Motion and Moving Reference Frames

• Relative position the position of one object with respect to another, $\vec{r}_{A/B} = \vec{r}_A - \vec{r}_B$
• Relative velocity the velocity of one object with respect to another, $\vec{v}_{A/B} = \vec{v}_A - \vec{v}_B$
• Relative acceleration the acceleration of one object with respect to another, $\vec{a}_{A/B} = \vec{a}_A - \vec{a}_B$
• Galilean transformation equations relating position, velocity, and acceleration between two reference frames moving with constant relative velocity
  • $\vec{r}' = \vec{r} - \vec{v}_{frame}t$
  • $\vec{v}' = \vec{v} - \vec{v}_{frame}$
  • $\vec{a}' = \vec{a}$
• Coriolis acceleration an apparent acceleration experienced by an object moving in a rotating reference frame $\vec{a}_{cor} = -2\vec{\omega} \times \vec{v}'$
• Centrifugal acceleration an apparent acceleration directed away from the axis of rotation in a rotating reference frame $\vec{a}_{cf} = -\vec{\omega} \times (\vec{\omega} \times \vec{r}')$

Applications and Problem-Solving Strategies

• Identify the system and draw a free body diagram showing all the forces acting on the object(s) of interest
• Determine the known and unknown quantities, and choose appropriate coordinate axes
• Apply Newton's second law $\vec{F} = m\vec{a}$ to relate the forces to the acceleration
• Use kinematic equations to relate position, velocity, and acceleration
• Apply the work-energy theorem $W_{net} = \Delta KE$ or conservation of energy to solve for unknown quantities
• Apply the impulse-momentum theorem $\vec{J} = \Delta\vec{p}$ or conservation of momentum to solve for unknown quantities
• For relative motion problems, choose appropriate reference frames and apply Galilean transformations or consider Coriolis and centrifugal accelerations if necessary
• Check the units and reasonableness of the answer, and consider any limiting cases or special conditions

Advanced Topics and Extensions

• Angular kinematics and dynamics extending the concepts of linear motion to rotational motion, considering angular position, velocity, acceleration, and torque
• Rigid body dynamics analyzing the motion of extended objects, considering both translation and rotation
• Lagrangian mechanics an alternative formulation of mechanics based on the principle of least action, using generalized coordinates and energies
• Hamiltonian mechanics a reformulation of Lagrangian mechanics using generalized momenta and Hamilton's equations
• Variational principles the principle of least action and Hamilton's principle, which state that the path followed by a system minimizes a certain functional (action)
• Noninertial reference frames reference frames that are accelerating or rotating, requiring the inclusion of fictitious forces (Coriolis, centrifugal)
• Relativistic mechanics the extension of Newtonian mechanics to high-speed motion, considering the effects of special relativity (time dilation, length contraction, mass-energy equivalence)
• Continuum mechanics the study of the mechanics of continuous media, such as fluids and deformable solids, using concepts like stress, strain, and constitutive relations