Mass on a Spring Dynamics Quiz
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Questions and Answers

Match the following terms with their definitions in oscillatory motion:

Kinetic Energy = Energy associated with motion Potential Energy = Energy stored due to position Acceleration = Rate of change of velocity Mass = Amount of matter in an object

Match the following equations with their corresponding physical concepts:

$F_{spring,x}= -k * x$ = Hooke's Law for springs $E_{total} = K + U_{elastic}$ = Total mechanical energy of oscillating system $v_{max} = A / ext{m}$ = Maximum speed at midpoint of oscillation $x(t) = A \cos(t)$ = Displacement in simple harmonic motion

Match the following types of oscillations to their characteristics:

Under-damping = Completes a single oscillation with decreasing energy Overdamping = Never completes a single oscillation Driven oscillations = Oscillates at the driving frequency Damping = Gradually reduces total mechanical energy

Match the following symbols with their physical quantities:

<p>$K$ = Kinetic energy $U_{grav}$ = Gravitational potential energy $T_0$ = Period of natural frequency $f_0$ = Natural frequency of the system</p> Signup and view all the answers

Match the following conditions with their outcomes in oscillations:

<p>Maximum force and acceleration = Occurs at endpoints of oscillation Zero force and acceleration = Occurs at midpoint (x = 0) Decreasing amplitude = Results from under-damping Unchanged frequency = Occurs in damped oscillations</p> Signup and view all the answers

Match the following spring dynamics concepts with their expressions:

<p>Total mechanical energy at endpoints = $E_{total} = m * g * y_{max}$ Elastic potential energy = $U_{elastic} = \frac{1}{2} k x^2$ Maximum kinetic energy = $K = \frac{1}{2} m v^2$ Velocity at position x = $v_x = km(A^2 - x^2)$</p> Signup and view all the answers

Match the following terms with their respective forces in driven oscillations:

<p>Driving force = Periodic force exerted on system Natural frequency = Frequency at which a system naturally oscillates Resonance = Condition when driving frequency matches natural frequency Negative work = Function performed by damping on the system</p> Signup and view all the answers

Match the following pendulum concepts with their equations:

<p>Total mechanical energy at midpoint = $E_{total} = \frac{1}{2} mv_{max}^2$ Velocity at maximum height = $v_{max} = \sqrt{2g y_{max}}$ Gravitational potential energy = $U_{grav} = m g y$ Kinetic energy = $K = \frac{1}{2} mv^2$</p> Signup and view all the answers

Match the following thermal phenomena with their descriptions:

<p>Conduction = Transfer of heat via direct contact Convection = Bulk motion and mixing of fluids Radiation = Transport of energy via light waves Thermal Equilibrium = Heat flow from hot to cold until equal</p> Signup and view all the answers

Match the following temperature scales with their important characteristics:

<p>Celsius = Centigrade scale with 100 degrees between freezing and boiling Kelvin = Absolute temperature scale starting from absolute zero Absolute Zero = Cessation of all thermal motion Thermal Energy = Related to microscopic kinetic energy of particles</p> Signup and view all the answers

Match the following heat units with their values:

<p>Calorie = 4184 Joules Dietary Calorie = 1000 calories British Thermal Unit = 252 calories Kilocalorie = 1 dietary calorie</p> Signup and view all the answers

Match the following thermal expansion formulas with their concepts:

<p>Linear Expansion = $L = L_0 eta riangle T$ Volume Expansion = $V = V_0 eta riangle T$ Coefficient of Thermal Expansion = Measures size increase due to temperature Change in Temperature = Can be in Celsius or Kelvin</p> Signup and view all the answers

Match the following laws with their descriptions:

<p>Zeroth Law of Thermodynamics = If A and B are equal, and B and C are equal, then A and C are equal Ideal Gas Law = $P_1 V_1 T_1 = P_2 V_2 T_2$ Equivalency of Heat and Mechanical Energy = Heat equals thermal energy change in a substance Calorimetry = The sum of heat changes in a system equals zero</p> Signup and view all the answers

Match the following specific heat capacity components with their symbols:

<p>Q = Heat added or removed m = Mass of the substance c = Specific heat in J/kg·C T = Change in temperature</p> Signup and view all the answers

Match the following types of conduction with their characteristics:

<p>Natural Convection = Hotter parcels rise due to lower density Forced Convection = Bulk motion caused by non-buoyant forces Thermal Conductivity = Depends on material's ability to conduct heat Insulators = Materials that do not conduct heat well</p> Signup and view all the answers

Match the following energy transport mechanisms with their speed:

<p>Conduction = Usually slow Natural Convection = Faster than conduction Forced Convection = Much faster than natural convection Radiation = Very fast, does not require matter</p> Signup and view all the answers

Match the following definitions with their terms:

<p>Thermal Equilibrium = Heat transfer until temperature equalization Co-efficient of Linear Expansion = Describes length change with temperature Heat Q = Amount of thermal energy change Absolute Temperature = Must use Kelvin for thermodynamic calculations</p> Signup and view all the answers

Match the wave types with their characteristics:

<p>Transverse waves = Displacement is perpendicular to propagation Longitudinal waves = Displacement is parallel to propagation Mechanical waves = Require a medium to propagate Electromagnetic waves = Do not require a medium to propagate</p> Signup and view all the answers

Match the terms with their definitions:

<p>Amplitude = Maximum displacement from equilibrium Wavelength = Distance between successive crests Frequency = Number of oscillations per unit time Period = Time taken for one complete cycle</p> Signup and view all the answers

Match the type of wave interference with its description:

<p>Constructive interference = Displacements add to become larger Partially destructive interference = Displacements add to become smaller Completely destructive interference = Displacements add to become zero Superposition = Net displacement is the sum of individual displacements</p> Signup and view all the answers

Match the wave concepts with their formulas:

<p>Wave speed = v = f × λ Frequency of harmonics = f_n = n × f_1 Beat frequency = f_{beat} = |f_1 - f_2| Continuous wave equation = v = λ / T</p> Signup and view all the answers

Match the types of standing wave nodes and antinodes with their characteristics:

<p>Nodes = Locations of total destructive interference Antinodes = Locations of constructive interference Fundamental mode = Fewest nodes and antinodes Overtones = More nodes and antinodes than the fundamental</p> Signup and view all the answers

Match the properties of sound waves with their effects:

<p>Amplitude = Perceived loudness Frequency = Perceived pitch Wave speed in dense medium = Slower wave speed Wave speed in tensioned mediums = Faster wave speed</p> Signup and view all the answers

Match the Doppler effect scenarios with their outcomes:

<p>Source and listener approaching = Observed frequency increases Source and listener receding = Observed frequency decreases Moving observer towards stationary source = Pitch rises Moving observer away from stationary source = Pitch falls</p> Signup and view all the answers

Match the wave phenomena with their descriptions:

<p>Beats = Amplitude oscillation from two waves with close frequencies Shock fronts = Formed when source speed exceeds wave speed Standing waves = Patterns formed from reflecting waves Superposition principle = Multiple waves combine in a medium</p> Signup and view all the answers

Match each wave category with an example:

<p>Transverse wave = Vibrating strings Longitudinal wave = Sound waves Electromagnetic wave = Light waves Surface wave = Waves at water-air interface</p> Signup and view all the answers

Match the descriptors to the type of wave:

<p>Pressure wave = Longitudinal wave Shear wave = Transverse wave Gravitational wave = Does not require a medium Sound wave = Requires a medium to travel</p> Signup and view all the answers

Match the wave characteristics with their effects during a transition between mediums:

<p>Wave speed = May increase or decrease Frequency = Remains unchanged Amplitude = Can vary significantly Wavelength = Changes based on speed and frequency</p> Signup and view all the answers

Match the terms related to wave mechanics with their functions:

<p>Node = Point of no displacement Antinode = Point of maximum displacement Fundamental frequency = Lowest frequency mode Harmonic = Multiple of the fundamental frequency</p> Signup and view all the answers

Match the acoustic variables with their definitions:

<p>Loudness = Perception of amplitude Pitch = Perception of frequency Tension in medium = Impacts wave speed Density of medium = Affects wave propagation speed</p> Signup and view all the answers

Flashcards

Hooke's Law

The force exerted by a spring is proportional to the displacement from its equilibrium position and acts in the opposite direction.

Total Mechanical Energy (Mass on Spring)

The sum of kinetic energy and elastic potential energy in a mass-spring system is constant and depends on the spring constant and amplitude.

Maximum Speed of Mass on Spring

The mass on a spring attains maximum speed at the equilibrium position (x=0) and is determined by the amplitude and spring constant.

Period of Simple Harmonic Motion

The time it takes for a mass on a spring to complete one full cycle of oscillation is determined by the mass and spring constant.

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Natural Frequency

The frequency at which an oscillating system vibrates freely without any external driving force.

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Damping

A force that gradually reduces the total mechanical energy of an oscillating system, leading to a decrease in amplitude over time.

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Overdamping

A type of damping where the system never completes a full oscillation and reaches equilibrium gradually due to strong damping forces.

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Driving Force

An external periodic force applied to an oscillating system, causing it to oscillate at the driving frequency rather than its natural frequency.

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Mach Number

The ratio of the speed of an object to the speed of sound in the surrounding medium.

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Temperature

A measure of the average kinetic energy of the particles in a substance. It reflects how fast the particles are moving.

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Celsius and Kelvin

Two temperature scales. Celsius is a centigrade scale with 100 degrees between the freezing and boiling points of water. Kelvin is a centigrade scale with the same size degree as Celsius, but with absolute zero as its starting point.

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Linear Thermal Expansion

The change in length of a solid due to a change in temperature.

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Volume Thermal Expansion

The change in volume of a solid or liquid due to a change in temperature.

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Heat (Q)

The amount of thermal energy transferred to or from a substance, resulting in a temperature change.

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Specific Heat Capacity (c)

The amount of heat energy required to raise the temperature of 1 kg of a substance by 1 degree Celsius.

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Thermal Equilibrium

A state where two objects in contact have reached the same temperature, resulting in no net heat transfer.

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Conduction

Heat transfer through direct contact, where energy is passed by collisions between particles.

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Convection

Heat transfer by the movement of a fluid (liquid or gas). Hotter, less dense fluid rises, while cooler, denser fluid sinks.

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Resonance

When the driving frequency of a system matches its natural frequency, leading to a dramatic increase in amplitude of oscillations. Can be desirable (amplification in musical instruments) or undesirable (buildings in earthquakes).

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Mechanical Waves

Waves that require a medium to propagate. The medium's particles oscillate around their equilibrium positions as the energy passes through.

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Amplitude

The maximum displacement of a particle from its equilibrium position in a wave.

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Wavelength

The distance between two consecutive crests or troughs in a wave.

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Wave Speed

The speed at which a wave propagates through a medium.

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Period (T)

The time it takes for one complete cycle of a wave.

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Frequency (f)

The number of wave cycles per second.

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Transverse Wave

A wave where the displacement of particles is perpendicular to the direction of wave propagation.

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Longitudinal Wave

A wave where the displacement of particles is parallel to the direction of wave propagation.

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Wave Speed on a String

Depends on the tension in the string and the density of the string. Higher tension means faster speed. Higher density means slower speed.

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Principle of Superposition

When multiple waves pass through the same medium, the net displacement at any point is the sum of the individual displacements caused by each wave.

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Constructive Interference

Occurs when waves combine, resulting in a larger amplitude than either individual wave.

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Destructive Interference

Occurs when waves combine, resulting in a smaller amplitude than either individual wave. Total destructive interference results in zero amplitude.

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Beats

A phenomenon produced when two waves with slightly different frequencies interfere, creating a periodic variation in amplitude.

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Standing Waves

A wave pattern created by the superposition of two identical waves traveling in opposite directions. Characterized by nodes (zero displacement) and antinodes (maximum displacement) at fixed locations.

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Study Notes

Mass on a Spring Dynamics

  • Hooke's Law: Force exerted by a spring is proportional to its displacement from equilibrium: Fspring, x= -k * x (where k is the spring constant).

  • Acceleration: If only the spring force acts, acceleration is proportional to displacement: ax= -km*x (m is mass).

  • Forces & Acceleration: Maximum force and acceleration occur at the endpoints of oscillation; zero force and acceleration at the midpoint (x=0).

  • Kinetic Energy: K= 12mv^2

  • Potential Energy (Elastic/Spring): Uelastic or Uspring= 12kx^2

  • Total Mechanical Energy: Etotal = K + Uelastic = 12kA^2 or 12mvmax^2 (where A is the amplitude of oscillation, vmax is the maximum speed).

  • Conservation of Energy: In a frictionless system, the total mechanical energy is conserved.

  • Maximum Speed: vmax = A√(k/m) at the midpoint of oscillation.

  • Velocity at any position: vx = √(k/m)(A^2-x^2)

  • Natural Frequency & Period: f0= 12√(k/m) ; T0= 2π√(m/k)

Mass on a Spring Kinematics

  • Period & Frequency: T = 1/f; T=2Ï€ √(m/k)
  • Oscillation Equations:
    • Displacement/position: x(t) = Acos(ωt) (where ω= 2Ï€f = 2π√(k/m))
    • Velocity: v(t) = -ωAsin(ωt) = -vmaxsin(ωt)
    • Acceleration: a(t) = -ω^2Acos(ωt) = -amaxcos(ωt)
  • Phase Relationship: The displacement, velocity, and acceleration are each 1/4 cycle out of phase with the next.

Pendulum Energy

  • Kinetic Energy: K=12mv^2
  • Potential Energy (Gravity): Ugrav=mgy (y = 0 at the bottom of the swing).
  • Total Mechanical Energy: Etotal = K + Ugrav = mgymax = 12mvmax^2 , evaluated at endpoints and midpoint.
  • Maximum Speed: vmax =√(2g*ymax)
  • Endpoints and Midpoint Energy: Total energy is conserved everywhere in the oscillation, including at the endpoints (maximum potential energy) and the lowest point (maximum kinetic energy)

Dampened Oscillations

  • Damping: A function that performs negative work, reducing mechanical energy (E) gradually.
  • Overdamping: System never completes a single oscillation. Example: shock absorbers.
  • Underdamping: System oscillates with decreasing amplitude and maximum speed per oscillation, maintaining a constant frequency (and unchanged period).

Driven Oscillations & Resonance

  • Driving Force: A periodic force causing oscillation at the driving frequency (d), not the natural frequency (f0).
  • Resonance: When the driving frequency approaches the natural frequency, energy builds up and oscillation amplitude increases dramatically. This can be desirable or undesirable (e.g., musical instruments vs. building in earthquakes).

Waves

  • Mechanical Waves: Energy carried through a medium, requiring a medium to propagate.
  • Continuous Waves: Anatomy: amplitude (A), wavelength (λ), wave speed (v), period (T), frequency (f), displacement (y(x,t))
  • Simple Wave Equation: v = λ/T = fλ.
  • Transverse Waves: Displacement is perpendicular to propagation (e.g., vibrating strings, electromagnetic waves).
  • Longitudinal Waves: Displacement is parallel to propagation (e.g., sound waves).
  • Wave Speed: Wave speed depends on the characteristics of the medium, e.g., tension for strings and density of the material. Transverse waves on a string have a non-dispersive speed.
  • Surface Waves: Gravity-driven waves on waterbodies.
    • In deep water: v = √(gλ/2Ï€). Dispersive.
    • In shallow water: v = √(gh). Dispersive; speed depends on depth and wavelength.
  • Frequency and Amplitude: frequency (pitch) and amplitude (loudness) are related to the wave properties.

Principle of Superposition

  • Waves passing each other: net displacement is the sum of individual displacements.
  • Constructive interference: larger displacement.
  • Partially destructive interference: smaller displacement.
  • Completely destructive interference: zero displacement.
  • Beats: Amplitude oscillation (beat) from overlapping waves of close frequencies (f1-f2).

Standing Waves (Normal Modes)

  • Superimposition of reflected waves in a bound medium.
  • Certain wavelengths produce constructive interference, resulting in fixed positions of maximum and zero displacement (standing waves).
  • Nodes: Zero displacement.
  • Antinodes: maximum displacement.
  • Rules for strings/air columns: Fixed ends have nodes, open ends have antinodes; nodes and antinodes alternate evenly spaced.
  • Fundamental Mode: Fewest nodes and antinodes. Successive overtones have more than the previous.
  • Numbering Harmonics: Frequencies of normal modes: fn = n * f1 (n = harmonic number), f1=fundamental mode.

Open-Open Pipes (Standing Waves)

  • Rules similar to strings/air columns

Doppler Effect

  • Observed frequency (fL) can differ from emitted frequency (fS) when source (S) and/or observer (L) are moving.
  • Formula: fL = (vw + vL) / (vw + vS) fS (vw is speed of wave, vL is velocity of observer, vS is velocity of source).
  • Radial Velocity: Only radial velocity (along line of sight) affects the Doppler shift.

Shock Fronts

  • If source speed exceeds wave speed, a cone-shaped shock front forms.
  • Mach Number (M): Ratio of source speed to wave speed.

Temperature and Heat

  • Temperature: Measure of microscopic kinetic energy of particles; related to average speed of particles.
  • Temperature Scales: Celsius, Kelvin (Celsius + 273.15), Absolute zero.
  • Thermal Expansion: Materials expand with increasing temperature (nearly all).
    • Linear Thermal Expansion: ΔL = αL0ΔT
    • Volume Thermal Expansion: ΔV = βV0ΔT (β = coefficient of volume expansion)
    • Gases: Ideal Gas Law (P1V1/T1 = P2V2/T2)
  • Heat (Q): Amount of thermal energy transferred.
  • Units: 1 calorie (cal) = 4184 J; 1 dietary calorie = 1000 cal; 1 Btu = 1055 J.
  • Specific Heat Capacity: Q = mcΔT (heat = mass × specific heat × temperature change)

Thermal Equilibrium & Heat Transport

  • Thermal Equilibrium: When two objects at different temperatures are in contact, heat flows from hotter to colder until they reach the same temperature.
  • Zeroth Law of Thermodynamics: If A = B, and B = C, then A = C.
  • Heat Transport:
    • Conduction: Heat transfer via particle collisions (solids, liquids, gases). Speed depends on thermal conductivity.
    • Convection: Heat transfer via fluid motion (liquids, gases). Speed depends on viscosity and density.
    • Radiation: Heat transfer via electromagnetic waves (all substances). Speed is very fast.

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Test your understanding of mass on a spring dynamics, including Hooke's Law and the principles of potential and kinetic energy. Dive into concepts like total mechanical energy and natural frequency, and see how they relate to oscillatory motion. Perfect for students studying physics topics on springs and oscillations.

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