C.1- Simple Harmonic Motion

• What is the symbol for angular frequency?


  ω
• What is the formula for angular frequency in terms of period?


  ω = \frac{2\pi}{T}
• What is the formula for angular frequency in terms of frequency?


  ω = 2\pi f
• What does it mean for a motion to be isochronous?


  It means the time period remains constant.
• What is the base equation for acceleration in simple harmonic motion (SHM)?


  a = -ω²x
• How is simple harmonic motion (SHM) defined?


  SHM is an oscillation where acceleration is proportional and opposite to displacement.
• What is the maximum displacement in SHM called?

  Amplitude (X₀)
• What is the total mechanical energy in SHM represented by?


  Eₜ
• What is the relationship between kinetic energy (Eₖ) and potential energy (Eₚ) in SHM?


  Eₖ + Eₚ = Eₜ
• What is the formula for kinetic energy in SHM?


  Eₖ = \frac{1}{2} mw²(x₀² - x²)
• What is the formula for potential energy in SHM?


  Eₚ = \frac{1}{2} mw²x²
• What is the formula for total energy in SHM?


  Eₜ = \frac{1}{2} mw²x₀²
• What is the formula for velocity in SHM?


  v = ±ω\sqrt{x₀² - x²}
• What is the formula for the period of a simple pendulum?

  T = 2\pi\sqrt{\frac{l}{g}}
• What is the formula for the period of a mass-spring system?


  T = 2\pi\sqrt{\frac{m}{k}}
• What does the equation x = x₀ \sin(ωt + φ) represent in SHM?

  It represents the displacement as a function of time, starting at the origin.
• What is the relationship between angular frequency and spring constant?


  ω² = \frac{k}{m}
• What is the relationship between acceleration and displacement in SHM?


  a α -x
• What are the key characteristics of simple harmonic motion (SHM)?


  • Isochronous: time period remains constant
  • Acceleration is proportional and opposite to displacement
  • Examples include pendulums, mass-spring systems, and vibrating strings
• What are the equations for displacement, velocity, and acceleration in SHM?

  1. Displacement: \( x = x₀ \cos(ωt) \)
  2. Velocity: \( v = -ωx₀ \sin(ωt) \)
  3. Acceleration: \( a = -ω²x₀ \cos(ωt) \)
• How does the time period of a simple pendulum change with length?


  A shorter length results in a smaller time period.
• What is the hypothesis for the mass-spring experiment?


  The spring constant will remain the same for all masses used.
• What is the relationship between T² and mass in the mass-spring system?


  T² ∝ m
• What are the energy equations in SHM?


  1. Total Energy: \( E_T = \frac{1}{2} mω²x₀² \)
  2. Potential Energy: \( E_p = \frac{1}{2} mω²x² \)
  3. Kinetic Energy: \( E_k = \frac{1}{2} mω²(x₀² - x²) \)
• What does the graph of energy in SHM show over time?


  The total energy remains constant while kinetic and potential energy vary.
• If the maximum displacement is 4 m and the mass is 3 kg, what is the total energy in SHM if ω = 2 rad/s?


  Using \( E_T = \frac{1}{2} mω²x₀² = \frac{1}{2} \cdot 3 \cdot (2)^2 \cdot (4)^2 = 48 \text{ J} \)
• How do the equations for displacement, velocity, and acceleration change with position in SHM?


  The equations can change based on the phase of the motion.
• What is the significance of the phase difference in SHM?

  It indicates the starting point of the oscillation.
• What is the formula for the angular frequency in terms of spring constant and mass?


  ω = \sqrt{\frac{k}{m}}
• What does the equation \( a = -ω²\sqrt{\frac{k}{m}} \) represent in SHM?

  It represents the acceleration in terms of angular frequency and mass-spring constant.
• What is the relationship between the time period and mass in a mass-spring system?

  The time period increases with increasing mass.
• What is the effect of increasing the length of a pendulum on its time period?

  Increasing the length increases the time period.
• How does the energy in SHM relate to the displacement from equilibrium?


  Energy is maximum at maximum displacement and zero at equilibrium.
• What is the natural frequency of an oscillator?


  It is the frequency at which every oscillator vibrates after an initial disturbance.
• What is the forced or driving frequency?


  It is the frequency of an applied external force.
• What is free oscillation?


  Free oscillation occurs when a system oscillates without an external force being applied.
• What is forced or driving oscillation?

  Forced oscillation occurs when a system oscillates due to an external force.
• What is damping in oscillating systems?


  Damping is the loss of energy to surroundings, which decreases the amplitude of oscillation.
• What is resonance?


  Resonance is the increase in amplitude when the forced frequency is in phase with the natural frequency.
• What are the differences between free and forced oscillation?


  • Free oscillation: No external force applied, natural frequency.
  • Forced oscillation: External force applied, oscillates at forced frequency.