6.1 Periodic motion

• The maximum displacement from the mean position in periodic motion is called the amplitude
• Periodic motion is found in systems such as a pendulum, a vibrating guitar string, and the Earth's rotation around the Sun.


  True
• Understanding oscillations is crucial for analyzing systems like pendulums, springs, and vibrating molecules.


  True
• The period of a spring-mass system is given by $T =$$2\pi\sqrt{\frac{m}{k}}$.


  True
• What does the amplitude of SHM represent?

  Maximum displacement
• Match the SHM equation with its description:

  Displacement ↔️ Distance from equilibrium at time \(t\)

  Velocity ↔️ Rate of change of displacement

  Acceleration ↔️ Rate of change of velocity
• Match the equation with its description:

  Displacement \(x(t)\) ↔️ \( x(t) = A \cos(\omega t + \phi) \)

  Velocity \(v(t)\) ↔️ \( v(t) = -A\omega \sin(\omega t + \phi) \)

  Acceleration \(a(t)\) ↔️ \( a(t) = -A\omega^2 \cos(\omega t + \phi) \)
• What is the defining characteristic of periodic motion?

  Repeats at regular intervals
• What drives the repeated back-and-forth motion in oscillations?

  Restoring force
• How does SHM differ from general oscillations?

  Linear restoring force
• The period of a simple pendulum depends on its length and the acceleration due to gravity.

  True
• What is the unit of angular frequency in SHM?

  Radians per second
• The equations of SHM can be used to analyze the motion of a pendulum.


  True
• The acceleration of an object in SHM is proportional to its displacement but in the opposite direction.

  True
• The kinetic energy in SHM is maximum at the equilibrium position.


  True
• The total energy in SHM is the sum of kinetic and potential energy.
• In underdamped oscillations, the amplitude decreases until the object eventually comes to rest.
• In resonance, the phase of the driving force and the system's motion are in phase with each other.


  True
• What does \(\omega_0\) represent in the amplitude equation for forced oscillations?

  Natural frequency
• What is the period in periodic motion?

  Time for one complete cycle
• What drives the repeated back-and-forth motion in oscillations?

  Restoring force
• What type of motion is characterized by a restoring force proportional to displacement?

  Simple harmonic motion
• Match the SHM equation with its variable:

  Displacement ↔️ \( x(t) = A \cos(ωt + φ) \)

  Velocity ↔️ \( v(t) = -Aω \sin(ωt + φ) \)

  Acceleration ↔️ \( a(t) = -Aω^2 \cos(ωt + φ) \)
• The angular frequency in SHM is given by \( ω = \sqrt{\frac{k}{m}} \), where \( k \) is the spring constant and \( m \) is the mass of the object.
• The angular frequency of SHM is measured in radians per second.
• What does \(A\) represent in the equations of SHM?

  Amplitude
• Match the property with its definition:

  Amplitude ↔️ Maximum displacement from mean position

  Period ↔️ Time for one complete cycle

  Frequency ↔️ Number of cycles per time unit
• In simple harmonic motion, the restoring force is proportional to the displacement.
• Hooke's Law states that the restoring force in SHM is equal to -kx.
• What is the formula for the angular frequency of SHM in terms of the spring constant and mass?

  \omega = \sqrt{\frac{k}{m}}</latex>
• In SHM, the amplitude represents the maximum displacement from equilibrium.
• What does the displacement equation in SHM describe?

  The distance from equilibrium
• Match the variables in SHM equations with their descriptions:

  A ↔️ Amplitude of motion

  \(\omega\) ↔️ Angular frequency

  t ↔️ Time

  \(\phi\) ↔️ Phase constant
• The potential energy in SHM is given by U = (1/2)kx².
• At the maximum displacement in SHM, all energy is kinetic.

  False
• Match the types of damping with their descriptions:

  Underdamped ↔️ Oscillations continue with decreasing amplitude

  Critically Damped ↔️ Returns to equilibrium without oscillation

  Overdamped ↔️ Returns to equilibrium slowly
• What happens to the amplitude of forced oscillations when the driving frequency matches the natural frequency?

  Maximum amplitude occurs
• What are examples of systems where resonance is observed?

  Mechanical systems like pendulums
• What type of driving force causes forced oscillations?

  External periodic force
• The amplitude of forced oscillations is given by A = \frac{F_{0} / k}{\sqrt{(1 - (\omega / \omega_{0})^{2})^{2} + (2\zeta\omega / \omega_{0})^{2}}}</latex>, where $F_{0}$ is the amplitude of the driving force