Chapter 9: Simple Harmonic Motion

Exercise 1: Displacement of an Object

• Expression of displacement:
  [ x = 3.5 \sin(8\pi t + 0.25\pi) ]
  where x (cm) and t (seconds).
• Determine the amplitude:
  • Amplitude (a) = 3.5 cm.

Finding the Period

• Given: ( \omega = 8\pi )
• Formula:
  [ \omega = \frac{2\pi}{T} ]
• Calculation:
  [ T = \frac{2\pi}{8\pi} = 0.25 \text{ seconds} ]

Finding the Frequency

• Formula:
  [ f = \frac{1}{T} ]
• Calculation:
  [ f = \frac{1}{0.25} = 4 \text{ Hz} ]

Finding the Phase Angle

• Phase angle:
  [ \phi = 8\pi(0.02) + 0.25\pi = 0.41\pi \text{ radians} ]

Finding the Displacement

• Displacement equation:
  [ x = 3.5 \sin(8\pi(0.02) + 0.25\pi) ]
• Calculation:
  [ x \approx 3.4 \text{ cm} ]
  • Ensure calculator is in radians mode.

Exercise 2: Vertical Spring Motion

• Given:
  • Stretched distance: 8.8 cm (Amplitude)
  • Period: 0.66 seconds.
• Finding Displacement after 1.8 seconds:
  • Equation:
    [ x = -8.8 \cos(\omega t) ]
    where ( \omega = \frac{2\pi}{T} = 3.03\pi ).
  • Calculation:
    [ x = -8.8 \cos(3.03\pi \times 1.8) \approx 2.72 \text{ cm} ]

Exercise 3: Particle Motion

• Given:
  • Completes 20 cycles in 2 seconds.
  • Amplitude = 3 cm.
• Finding angular frequency:
  • ( \omega = \frac{20 \times 2\pi}{2} = 20\pi ext{ radians/second} )
• Displacement equation:
  [ x = 3 \sin(20\pi t) ]

Displacement at t = 1/16 seconds

• Calculation:
  [ x \approx -2.1 \text{ cm} ]

Finding time for x = 1.5 cm

• Substituting in equation:
  [ 1.5 = 3 \sin(20\pi t) ]
• Calculation:
  [ t \approx 8.33 \times 10^{-3} ext{ seconds} ]

Exercise 4: Force vs Displacement

• Given: Mass = 0.15 kg; Force vs Displacement graph.
• Relationship:
  [ F = m a ]
  [ a = -\omega^2 x ]
• Finding frequency:
  • Determine omega from ( F = 3 ext{ N} ) and ( x = -0.2 ext{ m} ).
  • Calculation:
    [ \omega = 10 \text{ radians/second} ]
• Frequency:
  [ f = \frac{10}{2\pi} \approx 1.6 \text{ Hz} ]
• Finding amplitude:
  • Maximum displacement = 0.2 m (greater than 0.02 m).

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• Conclusion: Review concepts of amplitude, frequency, period, displacement, and phase angle in simple harmonic motion.