Vibrations

• What is simple harmonic motion (SHM)?

  SHM occurs when an object moves such that its acceleration is always directed toward a fixed point and is proportional to its distance from that point.
• How can the acceleration in SHM be expressed mathematically?

  The acceleration can be expressed as $a =$$-w^2x$, where $w^2$ is a constant.
• What does the variable A represent in SHM?

  A represents the amplitude, which is the maximum value of the displacement.
• What is the period (T) in the context of SHM?

  The period (T) is the time taken for one complete cycle of motion.
• How is frequency (f) defined in SHM?

  Frequency (f) is the number of oscillations per second.
• What is the relationship between period (T) and frequency (f)?

  The relationship is given by $T =$$\frac{1}{f}$.
• What is the equation that defines SHM in terms of displacement and time?

  The equation is $x =$$A \cos(\omega t + \epsilon)$, where A, $\omega$, and $\epsilon$ are constants.
• What does the phase constant ($\epsilon$) indicate in SHM?


  The phase constant indicates the initial position of the object at time $t =$$0$.
• What is the maximum velocity in SHM?

  The maximum velocity is given by $v_{\text{max}} =$$A\omega$.
• How do displacement and velocity vary during SHM?

  Both displacement and velocity vary sinusoidally with time during SHM.
• What happens to kinetic and potential energy during SHM?

  Energy transfers between potential energy and kinetic energy, but the total energy remains constant.
• How can kinetic energy (Ek) during SHM be calculated?

  Kinetic energy can be calculated using $E_k =$$\frac{1}{2}mv^2$.
• What is the total energy of a body in SHM?

  The total energy is given by $E =$$\frac{1}{2}mA^2\omega^2$.
• What are two common examples of SHM?

  Two common examples are masses on a spring and a simple pendulum.
• What is the equation for the period of a mass on a spring?

  The period is given by $T =$$2\pi\sqrt{\frac{m}{k}}$.
• What is the equation for the period of a simple pendulum?

  The period is given by $T =$$2\pi\sqrt{\frac{l}{g}}$.
• What are free oscillations?

  Free oscillations occur when an oscillatory system is displaced and released.
• What is damping in the context of oscillations?

  Damping is the decrease in amplitude of oscillations over time due to resistive forces.
• What is critical damping?

  Critical damping occurs when resistive forces are just large enough to prevent oscillations when the system is displaced and released.
• What is resonance?

  Resonance occurs when a driving force's frequency matches the natural frequency of an oscillating system, resulting in large amplitude oscillations.
• How can resonance be useful in practical applications?

  Resonance can be useful in applications like microwave ovens, where the frequency matches the natural frequency of water molecules.
• What is a negative aspect of resonance?

  A negative aspect is that it can cause dangerous oscillations, as seen with the Millennium Bridge when people walked at a frequency close to its natural frequency.
• What does the variable k represent in the context of springs?

  The variable k represents the spring constant, which is the stiffness of the spring.
• What does the variable g represent in the context of pendulums?

  The variable g represents the acceleration due to gravity.
• What are the key definitions related to simple harmonic motion?

  • Amplitude (A): Maximum displacement from equilibrium.
  • Period (T): Time taken for one complete cycle.
  • Frequency (f): Number of oscillations per second.
• What are the equations related to the period of oscillation for a mass on a spring and a simple pendulum?

  • For a mass on a spring: $T =$$2\pi\sqrt{\frac{m}{k}}$
  • For a simple pendulum: $T =$$2\pi\sqrt{\frac{l}{g}}$
• What are the energy interchanges during simple harmonic motion?

  • Kinetic energy (Ek) and potential energy (Ep) interchange.
  • Total energy remains constant: $E =$$E_k +$$E_p$.
• What are the effects of damping in oscillatory systems?

  • Damping reduces amplitude over time.
  • Critical damping prevents oscillations entirely.
• What are the practical implications of resonance?

  • Useful in applications like microwave cooking.
  • Can be dangerous in structures like bridges.
• What is the equation for velocity in simple harmonic motion (SHM)?

  v = -A \sin( \omega t + \epsilon)
• What does the variable ε represent when the body starts at equilibrium in SHM?

  ε = \frac{\pi}{2} \text{ (as } \cos(\omega t) = 0\text{)}.
• How does the velocity in SHM relate to the position of the object?

  The velocity and displacement are always a quarter of a cycle out of phase.
• What is the formula for the force exerted by a spring when extended a distance x?

  F = kx
• What does Newton's second law state in the context of a mass on a spring?

  ma = -kx
• What is the relationship between angular frequency (ω), spring constant (k), and mass (m)?

  \omega^2 = \frac{k}{m}
• What is the formula for the time period (T) of a mass on a spring?

  T = 2\pi \sqrt{\frac{m}{k}}
• What is the approximate behavior of a pendulum for small angles?

  For small angles, the path is approximately a straight line.
• What is the total energy of a system in SHM composed of?

  Total energy is the sum of kinetic energy (KE) and potential energy (PE).
• What is the formula for kinetic energy in a mass-spring system?

  KE = \frac{1}{2} mv^2
• What is the formula for elastic potential energy stored in a spring when extended by x?

  PE = \frac{1}{2} kx^2