Further Mechanics

• What is the unit of angle used in circular motion?

  Radian
• How do you convert degrees to radians?

  Use the conversion 2π radians = 360°
• How do you find the angle in radians?

  Use the formula $\theta =$$\frac{s}{r}$
• What is the formula for angular speed?

  $\omega =$$2\pi f$
• What is required for an object to change velocity in circular motion?

  A resultant force must act
• Why is a centripetal force necessary in circular motion?

  Because the direction of velocity is changing
• What direction does the centripetal force point?

  Towards the center of the orbit
• What is the equation for centripetal force using mass and velocity?

  $F =$$\frac{mv^2}{r}$
• What is the alternative equation for centripetal force using angular speed?

  $F =$$m\omega^2 r$
• What does the variable $\omega$ represent?


  Angular speed
• What contributes to centripetal force at position A of a ball on a string?

  The weight of the ball
• At position B, how does the weight of the ball affect centripetal force?

  It makes no contribution
• At position C, what must the string's force do?

  Overcome the weight of the ball
• What are the characteristics of simple harmonic motion (SHM)?

  • Oscillates around an equilibrium position
  • Restoring force acts towards equilibrium
  • Force is proportional to displacement
  • Acceleration is proportional to displacement
• What is the defining equation for SHM?

  $F =$$-kx$
• What does frequency represent in SHM?

  Number of cycles per second
• What is the time period in SHM?

  Time to complete one full cycle
• What is the equation for acceleration in SHM?

  $a =$$-\omega^2 x$
• What is the equation for displacement in SHM?

  $x =$$A \cos(\omega t)$
• What is the equation for velocity in SHM?

  $v =$$-A\omega \sin(\omega t)$
• What is the equation for acceleration in terms of displacement in SHM?

  $a =$$-A\omega^2 \cos(\omega t)$
• What is the formula for the period of a spring oscillator?

  $T =$$2\pi \sqrt{\frac{m}{k}}$
• How does gravitational field strength affect the period of a spring oscillator?

  It does not affect the period
• What is the formula for the period of a pendulum oscillator?

  $T =$$2\pi \sqrt{\frac{l}{g}}$
• How does mass affect the period of a pendulum oscillator?

  It does not affect the period
• How does displacement affect the period of a pendulum oscillator?

  It does not affect the period
• What are the energy transfers in SHM?

  • Maximum potential energy at maximum displacement
  • Potential energy transfers to kinetic energy towards equilibrium
  • Maximum kinetic energy at equilibrium
  • Maximum potential energy at maximum negative displacement
• What is the relationship between kinetic and potential energy in SHM?

  They transfer energy back and forth
• What is the total energy in SHM assuming no damping?

  Constant
• What causes damping in SHM systems?

  Energy transfer out of the system
• What are the types of damping?

  1. Light damping: Slow oscillation reduction
  2. Heavy damping: Quick stop of oscillations
  3. Critical damping: Quickest stop of oscillations
  4. Overdamping: Slow return to equilibrium
• What is the natural frequency in free vibrations?

  Frequency at which a system vibrates freely
• What causes forced vibrations?

  A driving force acting on the system
• What happens to phase difference at higher driving frequencies?

  It rises to π radians
• What occurs when resonance happens?

  Driving frequency equals natural frequency
• What are the effects of resonance?

  • Maximum rate of energy transfer
  • Driving force is π/2 out of phase
  • Maximum amplitude of oscillation
• Why is resonance sometimes unwanted?

  It can cause damage to structures
• How do structures like bridges deal with resonance?

  They are damped to prevent oscillations
• What is the defining equation of SHM?

  SHM equation: $a =$$-\omega^2 x$
• What is simple harmonic motion?

  Periodic motion with acceleration opposite to displacement