Further Mechanics

• What type of motion is described in section 3.6.1?


  Periodic motion
• What happens to the velocity of an object moving in a circular path at constant speed?


  The velocity is constantly changing due to a change in direction.
• What is centripetal acceleration?


  It is the acceleration experienced by an object moving in a circular path.
• According to Newton's first law, what must an object experience to accelerate?


  A resultant force
• What direction does the centripetal force act in circular motion?


  Towards the center of the circle
• How is angular speed (ω) defined?


  It is the angle an object moves through per unit time.
• How can angular speed (ω) be calculated using linear speed (v) and radius (r)?


  ω = v / r
• What is the relationship between angular speed (ω) and time period (T)?


  ω = 2π / T
• What unit is used to measure angles in circular motion?

  Radians
• How is one radian defined in relation to a circle?


  It is the angle when the arc length equals the radius of the circle.
• What is the angle in radians of a full circle?


  2π radians
• How can angles be converted from degrees to radians?


  By multiplying by $\frac{\pi}{180}$.
• How can angles be converted from radians to degrees?


  By multiplying by $\frac{180}{\pi}$.
• What is the relationship between displacement, velocity, and acceleration in periodic motion?


  • Velocity is the derivative of displacement.
  • Acceleration is the derivative of velocity.
  • Maximum speed occurs at equilibrium, while maximum acceleration occurs at maximum displacement.
• What is the formula for centripetal acceleration (a)?


  $a =$$\frac{v^2}{r}$
• How can centripetal force (F) be derived using Newton's second law?


  F = ma, where a is centripetal acceleration.
• What is the formula for centripetal force (F)?


  $F =$$m \cdot a =$$m \cdot \frac{v^2}{r}$
• What characterizes simple harmonic motion (SHM)?


  Acceleration is directly proportional to displacement and in the opposite direction.
• What is the equation that represents simple harmonic motion?


  $a =$$-\omega^2 x$
• What is an example of a simple harmonic oscillator?


  The simple pendulum
• How is the time period (T) of a pendulum measured?


  By measuring the time taken to complete one full oscillation.
• What is the formula for the time period (T) of a simple pendulum?


  $T =$$2\pi \sqrt{\frac{l}{g}}$
• Why must the angle of displacement for a pendulum be less than 10°?


  Because the small angle approximation is used in the derivation of the formula.
• What happens to gravitational potential energy during the oscillations of a simple pendulum?


  It is converted to kinetic energy and back to gravitational potential energy.
• What is the difference between vertical and horizontal mass-spring systems?


  Vertical systems convert kinetic energy to both elastic and gravitational potential energy, while horizontal systems convert it only to elastic potential energy.
• What is the formula for the time period (T) of a mass-spring system?


  $T =$$2\pi \sqrt{\frac{m}{k}}$
• What happens to energy in a simple harmonic motion system during oscillations?


  • Kinetic energy is converted to potential energy and back.
  • Maximum potential energy occurs at amplitude.
  • Maximum kinetic energy occurs at the equilibrium position.
  • Total energy remains constant when air resistance is negligible.
• What are forced vibrations?


  They occur when an external driving force causes a system to oscillate.
• What is resonance in the context of forced vibrations?


  It occurs when the driving frequency equals the natural frequency of a system.
• What are some applications of resonance?


  Instruments, radio tuning, and swings.
• What negative consequences can arise from resonance?


  It can cause damage to structures, such as bridges.
• What is damping in oscillating systems?


  It is the loss of energy to the environment, leading to reduced amplitude of oscillations.
• What are the types of damping in oscillating systems?


  • Light damping (under-damping): Amplitude decreases gradually.
  • Critical damping: Amplitude reduces to zero in the shortest time without oscillating.
  • Heavy damping (over-damping): Amplitude reduces slower than critical damping without oscillating.
• What are free vibrations?


  They occur when no external force is acting on the system, oscillating at its natural frequency.
• What is the natural frequency of a simple pendulum?


  It is the frequency at which the pendulum oscillates when no external force is acting.
• Why is the small angle approximation used for pendulums?


  It simplifies calculations for small angles, ensuring accuracy in the formula.
• How is energy transferred in a mass-spring system?


  Kinetic energy is converted to elastic potential energy and vice versa.
• What happens to the total energy of a simple harmonic motion system when air resistance is negligible?


  The total energy remains constant.
• What is the effect of damping on the resonant frequency?


  As damping increases, the resonant frequency decreases.
• What happens to the maximum amplitude as damping increases?


  The maximum amplitude decreases.