Notes in Intro to Mechanical Systems

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Published	10/15/2024	A change in position is called {{c1::displacement}}, and it is a {{c2::vector quantity}}, meaning it has both magnitude and direction.
Published	10/15/2024	{{c1::Average velocity}} is the ratio of displacement to the time interval over which it occurs, calculated as \( v_{\text{avg}} = \frac{\Delta x}{\De…
Published	10/15/2024	{{c1::Average speed}} differs from average velocity in that it uses the total distance traveled, independent of direction.
Published	10/15/2024	{{c1::Instantaneous velocity}} is the velocity of a particle at a given instant, obtained as the limit of average velocity as the time interval approa…
Published	10/15/2024	{{c1::Acceleration}} is the rate at which velocity changes with time, and it can be calculated as \( a = \frac{dv}{dt} \).
Published	10/15/2024	When velocity and acceleration have the same sign, the object's speed {{c1::increases}}; when they have opposite signs, the speed {{c1::decreases}}.
Published	10/15/2024	In {{c1::constant acceleration}}, the velocity changes at a steady rate, and equations of motion can be derived using integration.
Published	10/15/2024	The position of a moving object can be calculated by integrating {{c1::velocity}}
Published	10/15/2024	In {{c1::polar coordinates}}, the position of an object is described by its distance \( r \) from the origin and the angle \( \theta \) between \( r \…
Published	10/15/2024	{{c1::Newton's first law}} states that if no net force acts on an object, the object will maintain its state of rest or constant velocity.
Published	10/15/2024	A {{c1::force}} is a push or pull on an object that can cause a change in its velocity, and it is represented as a vector.
Published	10/15/2024	{{c1::Newton's second law}} relates force, mass, and acceleration: \( F_{\text{net}} = m \cdot a \).
Published	10/15/2024	The unit of force is the {{c1::newton}}, where \( 1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2 \).
Published	10/15/2024	{{c1::Newton's third law}} states that for every action force, there is an equal and opposite reaction force.
Published	10/15/2024	Mass is an intrinsic characteristic of an object that determines its resistance to acceleration, while {{c1::weight}} is the force exerted by gravity …
Published	10/15/2024	The principle of {{c1::superposition}} states that the net force on an object is the vector sum of all individual forces acting on it.
Published	10/15/2024	An object is in {{c1::equilibrium}} when the net force acting on it is zero, meaning it is either at rest or moving with constant velocity.
Published	10/15/2024	Forces are represented as {{c1::vectors}}, and their components can be summed along each axis to calculate the net force and resulting motion.
Published	10/15/2024	The acceleration of an object is inversely proportional to its {{c1::mass}} and directly proportional to the {{c2::net force}} acting on it.
Published	10/15/2024	A {{c1::free-body diagram}} is a visual representation of the forces acting on a single object, often used to solve problems involving Newton's second…
Published	10/15/2024	In a free-body diagram, each force is represented as a {{c1::vector}}, with its tail anchored on the object of interest and direction indicating the f…
Published	10/15/2024	The acceleration of an object can be found by {{c1::summing}} the forces acting on it and using the equation {{c2::\( F_{\text{net} } = m \cdot a \).}…
Published	10/15/2024	A free-body diagram helps in resolving forces into components along the {{c1::x and y axes}}, allowing for easier application of Newton's laws.
Published	10/15/2024	The {{c1::gravitational force}} is the pull exerted by Earth that acts on an object toward the center of the planet.
Published	10/15/2024	In {{c1::free fall}}, the only force acting on a body is the gravitational force, and the acceleration is equal to \( g \), the acceleration due to gr…
Published	10/15/2024	{{c1::Weight}} is the force required to prevent an object from falling, equal to the gravitational force \( W = mg \).
Published	10/15/2024	The {{c1::normal force}} is the force exerted perpendicular to a surface to support the weight of an object resting on it.
Published	10/15/2024	{{c1::Friction}} is the force that opposes the motion of two surfaces sliding against each other, and it can be either static or kinetic.
Published	10/15/2024	Static friction has a maximum value given by {{c1::\( f_s = \mu_s F_N \),}} where \( \mu_s \) is the coefficient of static friction and \( F_N \)…
Published	10/15/2024	{{c1::Centripetal force}} is the force required to keep an object moving in a circular path, directed toward the center of the circle.
Published	10/15/2024	The equation for centripetal force is {{c1::\( F = \frac{mv^2}{r} \),}} where \( m \) is the mass, \( v \) is the speed, and \( r \) is the radiu…
Published	10/15/2024	{{c1::Uniform circular motion}} occurs when an object moves in a circular path at constant speed, but the velocity vector changes direction.
Published	10/15/2024	In a {{c1::vertical circular loop}}, the minimum speed required at the top to maintain contact with the loop is {{c2::\( v = \sqrt{gR} \),}} wher…
Published	10/15/2024	A {{c1::spring element}} is a mechanical component that deforms when subjected to force and stores energy as potential energy, described by \( F = -kx…
Published	10/15/2024	In a {{c1::mass-spring system}}, the equation of motion is \( m \ddot{x} = -kx \), which leads to simple harmonic motion.
Published	10/15/2024	{{c1::Simple harmonic motion (SHM)}} occurs when the {{c4::restoring force}} is proportional to {{c2::displacement}} and directed toward the {{c3::equ…
Published	10/15/2024	The angular frequency \( \omega \) of SHM is related to the spring constant and mass as {{c1::\( \omega = \sqrt{\frac{k}{m}} \).}}
Published	10/15/2024	In SHM, the displacement \( x(t) \) is given by {{c1::\( x(t) = x_m \cos(\omega t + \phi) \),}} where \( x_m \) is the amplitude and \( \phi \) is the…
Published	10/15/2024	{{c1::Damping}} occurs when an external force dissipates energy, reducing the amplitude of oscillations over time.
Published	10/15/2024	A {{c1::critically damped}} system returns to equilibrium without oscillating, while an {{c2::overdamped}} system takes longer to return.
Published	10/15/2024	In {{c1::forced oscillations}}, an external periodic force drives the system, and the oscillations occur at the driving frequency \( \omega_f \).
Published	10/15/2024	{{c1::Resonance}} occurs when the driving frequency matches the natural frequency of the system, leading to large amplitude oscillations.
Published	10/15/2024	In SHM, the velocity and acceleration are related to displacement by {{c1::\( v = -\omega x_m \sin(\omega t + \phi) \)}} and {{c2::\( a = -\omega…
Published	10/15/2024	Once an object starts moving, {{c1::kinetic friction}} takes over and is generally smaller than the maximum static friction.
Published	10/15/2024	{{c1::Static friction}} resists the initial motion of an object and increases with the applied force until it reaches its maximum value.
Published	10/15/2024	Friction is caused by the microscopic {{c1::interactions between the rough surfaces}} of two objects, including {{c2::cold-welding}} at contact points…
Published	10/15/2024	The coefficient of friction \( \mu \) depends on the {{c1::materials}} in contact and does not depend on the {{c2::size}} of the contact area.
Published	10/15/2024	The three basic types of mechanical elements in a system are {{c1::inertia elements}}, {{c2::spring elements}}, and {{c3::damper elements}}.
Published	10/15/2024	Inertia elements, such as masses, are associated with {{c1::linear motion}} and are described by the equation \( F = m \ddot{x} \).
Published	10/15/2024	A linear spring stores mechanical energy in the form of {{c1::potential energy}} and exerts a force given by \( F_s = -kx \).
Published	10/15/2024	{{c1::Dampers}} dissipate energy as heat and oppose motion with a force proportional to velocity, described by \( F_d = -b \dot{x} \).
Published	10/15/2024	In a {{c1::mass-spring-damper system}}, the equation of motion is \( m \ddot{x} + b \dot{x} + kx = 0 \).
Published	10/15/2024	The {{c1::natural frequency}} \( \omega_n \) of a system is the frequency at which it oscillates without external forces, given by {{c2::\( \omega_n =…
Published	10/15/2024	The {{c1::damping ratio}} \( \zeta \) determines whether a system is underdamped, critically damped, or overdamped.
Published	10/15/2024	An {{c1::underdamped}} system oscillates with gradually decreasing amplitude, while an overdamped system returns to equilibrium without oscillating.
Published	10/15/2024	In a {{c1::critically damped}} system, the system returns to equilibrium as quickly as possible without oscillating.
Published	10/15/2024	{{c1::Multi-degree-of-freedom (DOF) systems}} involve more than one mass and spring, leading to complex motion that can be described by a set of coupl…
Published	10/15/2024	A {{c1::vibration absorber}} is used to reduce vibrations caused by rotating machinery, typically by attaching an additional mass-spring system.
Published	10/15/2024	{{c1::Vibration isolation}} aims to minimize the transmission of vibratory motion from a machine to its surroundings, using components like springs an…
Published	10/15/2024	In a dynamic vibration absorber, the absorber's mass and spring constant must be tuned to match the {{c1::natural frequency}} of the external force.
Published	10/15/2024	A dynamic vibration absorber significantly reduces vibration amplitude when the absorber's frequency matches the {{c1::external forcing frequency}}.
Published	10/15/2024	The {{c2::steady-state response}} of a {{c3::system to a sinusoidal forcing function}} depends on the {{c4::damping ratio}} and the ratio of the forci…
Published	10/15/2024	{{c3::Vibration absorbers}} are particularly effective when the damping ratio \( \zeta \) is {{c1::low}} and the forcing frequency is close to the {{c…
Published	10/15/2024	To reduce vibrations at specific frequencies, the {{c1::absorber mass}} and spring constant must be carefully selected to match the frequency of the e…
Published	10/15/2024	Without a vibration absorber, the system exhibits {{c1::large displacements}} near resonance; adding an absorber {{c2::reduces}} the amplitude signifi…
Published	10/15/2024	{{c1::Kinetic energy}} is the energy associated with the motion of an object, given by \( K = \frac{1}{2} m v^2 \).
Published	10/15/2024	The {{c1::work-energy theorem}} states that the net work done on an object is equal to the change in its kinetic energy: \( W = \Delta K \).
Published	10/15/2024	{{c1::Work}} is the energy transferred to or from an object by applying a force over a distance, calculated as \( W = Fd \cos(\theta) \).
Published	10/15/2024	Work done by a {{c1::variable force}} can be calculated by integrating the force over the distance moved: \( W = \int_{x_i}^{x_f} F(x) dx \).
Published	10/15/2024	Energy is conserved, meaning it can be transformed from one form to another, but the total amount of energy in a closed system remains {{c1::constant}…
Published	10/15/2024	{{c1::Linear momentum}} is a vector quantity defined as \( \vec{p} = m \vec{v} \), where \( m \) is the mass and \( \vec{v} \) is the velocity.
Published	10/15/2024	Newton's second law can be expressed in terms of momentum as {{c1::\( \vec{F}_{\text{net} } = \frac{d\vec{p} }{dt} \),}} meaning the rate of chan…
Published	10/15/2024	{{c1::Impulse}} is the change in momentum, given by \( \vec{J} = \Delta \vec{p} \), and can be calculated as the integral of force over time.
Published	10/15/2024	In an {{c1::elastic collision}}, both momentum and kinetic energy are conserved, whereas in an inelastic collision, only momentum is conserved.
Published	10/15/2024	In a {{c1::completely inelastic collision}}, the colliding bodies stick together after the collision, resulting in the maximum loss of kinetic energy.
Published	10/15/2024	The {{c1::conservation of momentum}} states that the total momentum of a system remains constant if no external forces act on it.
Published	10/15/2024	The velocity of the {{c1::center of mass}} of a system remains constant in a closed and isolated system, even during collisions.
Published	10/15/2024	In one-dimensional collisions, the final velocities of the two objects can be determined using the equations for {{c1::conservation of momentum}} and …
Published	10/15/2024	For a {{c1::completely inelastic collision}}, the final velocity \( V \) is given by \( V = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \), where \( m_1 \) an…
Published	10/15/2024	{{c1::Kinetic energy}} is the energy associated with the motion of an object, defined as {{c2::\( K = \frac{1}{2} m v^2 \).}}
Published	10/15/2024	{{c1::Work}} is defined as the energy transferred by a force acting over a distance, calculated as {{c2::\( W = F d \cos \theta \).}}
Published	10/15/2024	{{c1::Potential energy}} is energy stored in an object due to its position or configuration, such as gravitational or elastic potential energy.
Published	10/15/2024	{{c1::Conservative forces}} allow the conversion of kinetic energy to potential energy and vice versa, without loss of mechanical energy.
Published	10/15/2024	The principle of {{c1::conservation of mechanical energy}} states that the total mechanical energy of a system (kinetic + potential) remains constant …
Published	10/15/2024	In a closed system with only conservative forces, the sum of kinetic energy and potential energy is {{c1::constant}}.
Published	10/15/2024	{{c1::Nonconservative forces}} like friction convert mechanical energy into other forms, such as heat, leading to a loss of mechanical energy in the s…
Published	10/15/2024	The equation for {{c1::gravitational potential energy}} is {{c2::\( U = mgh \)}}, where \( m \) is mass, \( g \) is the acceleration due to gravity, a…
Published	10/15/2024	The {{c1::elastic potential energy}} stored in a spring is given by {{c2::\( U = \frac{1}{2} k x^2 \),}} where \( k \) is the spring constant and…
Published	10/15/2024	The {{c1::law of conservation of energy}} states that the total energy of an isolated system remains constant, though it can change forms (e.g., from …
Published	10/15/2024	{{c1::Frictionless motion}} assumes no energy is lost to friction, meaning mechanical energy is fully conserved between kinetic and potential energy.
Published	10/15/2024	{{c1::Rotational motion}} occurs when an object turns about a fixed axis, with each point moving in a circle around the axis.
Published	10/15/2024	The {{c1::angular displacement}} \( \Delta \theta \) is the change in the angular position of a rotating object, measured in radians.
Published	10/15/2024	{{c1::Angular velocity}} \( \omega \) is the rate of change of angular displacement, defined as \( \omega = \frac{d\theta}{dt} \).
Published	10/15/2024	{{c1::Angular acceleration}} \( \alpha \) is the rate of change of angular velocity, given by \( \alpha = \frac{d\omega}{dt} \).
Published	10/15/2024	The {{c1::moment of inertia}} \( I \) is a measure of an object's resistance to changes in its rotational motion, dependent on mass distribution.
Published	10/15/2024	The kinetic energy of a rotating object is given by {{c1::\( K = \frac{1}{2} I \omega^2 \)}}, where \( I \) is the moment of inertia and \( \omega \) …
Published	10/15/2024	{{c1::Torque}} is the rotational equivalent of force and is calculated as {{c2::\( \tau = r \cdot F \sin \theta \),}} where \( r \) is the distance fr…
Published	10/15/2024	Newton's second law for rotational motion is {{c1::\( \tau = I \alpha \),}} where \( \tau \) is the net torque and \( \alpha \) is the angular ac…
Published	10/15/2024	The {{c1::parallel-axis theorem}} states that the moment of inertia about any axis parallel to the center of mass is {{c2::\( I = I_{\text{com} } + Mh…
Published	10/15/2024	When a force causes {{c2::rotational acceleration}}, the work done by the force can be expressed as {{c1::\( W = \tau \cdot \Delta \theta \),}} where …
Published	10/15/2024	The {{c1::Lagrangian}} \( L \) is defined as {{c2::\( L = K - U \),}} where \( K \) is the kinetic energy and \( U \) is the potential energy of …
Published	10/15/2024	The {{c1::Euler-Lagrange equation}} is derived from the Lagrangian and provides the equations of motion for a system: {{c2::\( \frac{d}{dt} \frac{\par…
Published	10/15/2024	In a {{c3::mass-spring system}}, the potential energy is given by {{c1::\( U = \frac{1}{2} k x^2 \),}} and the kinetic energy is {{c2::\( K = \fr…
Published	10/15/2024	The {{c1::Rayleigh dissipation function}} models energy losses due to friction in a system, where {{c2::\( G(\dot{x}) = \frac{1}{2} b \dot{x}^2 \)}} a…
Published	10/15/2024	For a system with multiple degrees of freedom, the Euler-Lagrange equations must be written for each {{c1::generalized coordinate}} \( q_k \).
Published	10/15/2024	{{c1::Rolling motion}} is a combination of translation and rotation, where the object moves linearly while rotating about its center.
Published	10/15/2024	The kinetic energy of a rolling object consists of two parts: {{c1::translational energy}} and {{c2::rotational energy}}, with the total energy formul…
Published	10/15/2024	For an object rolling without slipping, the relationship between linear speed and angular speed is {{c1::\( v_{\text{com} } = \omega R \),}} where \( …
Published	10/15/2024	When an object rolls down a ramp, the acceleration is given by {{c1::\( a_{\text{com}} = \frac{g \sin \theta}{1 + \frac{I_{\text{com}}}{m R^2}} \),}}&…
Published	10/15/2024	{{c1::Static friction}} at the point of contact prevents slipping, ensuring smooth rolling motion without energy loss due to sliding.
Published	10/15/2024	{{c1::Fictitious forces}} arise in non-inertial reference frames and include forces like the Coriolis force and centrifugal force.
Published	10/15/2024	The {{c1::Coriolis force}} is a fictitious force that acts on objects moving in a rotating frame of reference, given by {{c2::\( F_{\text{Coriolis} } …
Published	10/15/2024	The {{c1::Foucault pendulum}} is an experiment demonstrating Earth's rotation by observing the slow rotation of a pendulum's swing plane.
Status	Last Update	Fields