Simple Harmonic Motion | Jamb(UTME)

Jamb(utme) key points on definition and explanation of simple harmonic motion SHM; examples of system that execute SHM; period, frequency, and amplitude of SHM

Definition and Explanation of Simple Harmonic Motion

1. Simple harmonic motion (SHM) is a type of oscillatory motion where an object moves back and forth about a mean position.
2. The motion is periodic, meaning it repeats itself at regular intervals.
3. In SHM, the restoring force acting on the object is directly proportional to its displacement from the equilibrium position.
4. The restoring force is always directed toward the equilibrium position.
5. Mathematically, SHM is described by:
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   $F = -kx$ where $F$ is the restoring force, $k$ is the force constant, and $x$ is the displacement.
6. The negative sign indicates that the force opposes the displacement.
7. SHM can be visualized as the projection of uniform circular motion onto a straight line.
8. The displacement $(x)$ of an object in SHM varies sinusoidally with time:
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   $x(t) = A \cos(\omega t + \phi)$ where $A$ is the amplitude, $\omega$ is the angular frequency, and $\phi$ is the phase constant.
9. The velocity $(v)$ in SHM is maximum at the equilibrium position and zero at the extreme positions.
10. The acceleration $(a)$ in SHM is maximum at the extreme positions and zero at the equilibrium position.
11. SHM is characterized by a constant period $(T)$ and frequency $(f)$.
12. The energy in SHM oscillates between kinetic energy and potential energy, but the total energy remains constant.
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Examples of Systems That Execute Simple Harmonic Motion

13. Mass-Spring System: A block attached to a spring oscillates when displaced from its equilibrium position.
14. Pendulum: A simple pendulum exhibits SHM for small angular displacements.
15. Oscillating Ruler: A ruler clamped at one end and displaced exhibits SHM when released.
16. Tuning Fork: The vibrating prongs of a tuning fork execute SHM.
17. Molecular Vibration: Atoms in a diatomic molecule oscillate about their equilibrium positions.
18. Bungee Jumping: The motion of a jumper on an elastic cord resembles SHM.
19. Guitar Strings: Plucked strings vibrate in SHM, producing sound waves.
20. Water Waves: Small particles in water waves exhibit SHM as they move up and down.
21. Electrical Oscillators: An LC circuit (inductor and capacitor) shows SHM in the flow of electric charge.
22. Swing: A swing behaves like a pendulum, undergoing SHM for small displacements.
23. Seismic Waves: During earthquakes, the oscillatory motion of the ground can resemble SHM.
24. Car Suspension: The shock absorber and spring system in vehicles execute SHM.
25. Bridge Oscillations: Under specific conditions, bridges can oscillate in SHM-like motion (e.g., Tacoma Narrows Bridge collapse).
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Period, Frequency, and Amplitude in SHM

26. The period $(T)$ is the time taken for one complete oscillation.
27. The formula for the period of SHM depends on the system:
    • For a mass-spring system:
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      $T = 2\pi \sqrt{\frac{m}{k}}$
    • For a simple pendulum:
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      $T = 2\pi \sqrt{\frac{L}{g}}$
28. The frequency $(f)$ is the number of oscillations per second:
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    $f = \frac{1}{T}$
29. The SI unit of period is seconds $(s)$.
30. The SI unit of frequency is hertz $(Hz)$.
31. The amplitude $(A)$ is the maximum displacement of the object from its equilibrium position.
32. Amplitude determines the energy of the oscillating system; higher amplitudes mean more energy.
33. The period and frequency of SHM are independent of amplitude (as long as the motion remains harmonic).
34. Angular frequency $(\omega)$ is related to the frequency:
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    $\omega = 2\pi f$
35. For a spring-mass system, the period increases if the mass increases or the spring constant decreases.
36. For a pendulum, the period increases with the length of the pendulum or decreases with stronger gravitational acceleration.
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Key Characteristics of SHM

37. At the equilibrium position, the speed is maximum, and the restoring force is zero.
38. At the extreme positions, the speed is zero, and the restoring force is maximum.
39. The phase constant $(\phi)$ determines the initial position and direction of motion of the oscillating object.
40. The total energy $(E)$ in SHM is the sum of kinetic energy $(KE)$ and potential energy $(PE)$:
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    $E = \frac{1}{2}kA^2$
41. Kinetic energy is maximum at the equilibrium position, while potential energy is maximum at the extreme positions.
42. The displacement, velocity, and acceleration are sinusoidal functions of time in SHM.
43. The phase difference between displacement and velocity is $\pi/2$ radians.
44. The phase difference between displacement and acceleration is $\pi$ radians.
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Applications of SHM

45. SHM is used to model oscillations in clocks, such as pendulum clocks and quartz watches.
46. Vibrating molecules in physics and chemistry are analyzed using SHM principles.
47. Sound waves are modeled as longitudinal SHM in air molecules.
48. Electrical signals in LC circuits oscillate in SHM, crucial for radio and communication systems.
49. Understanding SHM helps in designing stable structures that can withstand vibrations.
50. Engineers use SHM concepts to study natural frequencies and avoid resonance in buildings and bridges.
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Jamb(utme) key points on velocity and acceleration of SHM; energy change in SHM; force vibration and resonance

Velocity in Simple Harmonic Motion

1. Velocity $(v)$ in SHM is the rate of change of displacement with respect to time.
2. The velocity is maximum at the equilibrium position, where the object moves fastest.
3. At the extreme positions (maximum displacement), the velocity is zero.
4. The velocity in SHM is given by: $v = \pm \omega \sqrt{A^2 - x^2}$ where $(A)$ is the amplitude, $x$ is the displacement, and $(\omega)$ is the angular frequency.
5. The positive sign indicates motion toward the equilibrium position, and the negative sign indicates motion away from it.
6. Velocity is sinusoidal, reaching its peak when $\sin(\omega t + \phi) = \pm 1$.
7. Maximum velocity occurs at:
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   $v_{max} = \omega A$
8. The direction of velocity changes as the object passes through the equilibrium position.
9. Velocity is out of phase with displacement by $\pi/2$ radians.
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Acceleration in Simple Harmonic Motion

10. Acceleration $(a)$ in SHM is the rate of change of velocity with respect to time.
11. Acceleration is maximum at the extreme positions, where the restoring force is greatest.
12. At the equilibrium position, the acceleration is zero because the restoring force is zero.
13. The acceleration in SHM is given by:
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    $a = -\omega^2 x$
14. The negative sign indicates that the acceleration is directed toward the equilibrium position (restoring).
15. Maximum acceleration occurs at:
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    $a_{max} = \omega^2 A$
16. Acceleration is directly proportional to displacement but opposite in direction.
17. Acceleration is out of phase with displacement by $\pi$ radians (they are in opposite directions).
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Energy Changes in Simple Harmonic Motion

18. In SHM, the total mechanical energy $(E)$ remains constant.
19. Total energy is the sum of kinetic energy $(KE)$ and potential energy $(PE)$.
20. At the equilibrium position, $KE$ is maximum, and $PE$ is zero.
21. At the extreme positions, $PE$ is maximum, and $KE$ is zero.
22. The total energy in SHM is given by:
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    $E = \frac{1}{2} k A^2$ where $k$ is the spring constant and $A$ is the amplitude.
23. Kinetic energy is given by:
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    $KE = \frac{1}{2} mv^2$
24. Potential energy is given by:
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    $PE = \frac{1}{2} k x^2$
25. The energy oscillates between $KE$ and $PE$ during the motion.
26. When the object is halfway to the extreme position, $KE = PE$.
27. Energy conservation in SHM ensures that the system never gains or loses energy, assuming no damping.
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Forced Vibration

28. Forced vibration occurs when an external periodic force is applied to a system.
29. In forced vibration, the system oscillates at the frequency of the external force, not its natural frequency.
30. A swing pushed periodically by an external force exhibits forced vibration.
31. Forced vibration can occur in mechanical systems, electrical circuits, and even buildings during earthquakes.
32. The amplitude of forced vibration depends on the frequency of the external force and the system's natural frequency.
33. If the external force is close to the natural frequency, resonance occurs, amplifying the oscillations.
34. Forced vibration requires continuous input of energy to sustain oscillation.
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Resonance

35. Resonance is a phenomenon that occurs when the frequency of the external force matches the natural frequency of the system.
36. At resonance, the amplitude of oscillation becomes significantly large.
37. Resonance occurs because energy transfer is most efficient when the frequencies match.
38. In mechanical systems, resonance can cause structures to vibrate dangerously, as in the Tacoma Narrows Bridge collapse.
39. Resonance in musical instruments enhances sound quality (e.g., vibrating strings of a guitar).
40. In electrical circuits, resonance allows selective frequencies to pass in filters and radios.
41. Resonance can be both beneficial and destructive, depending on the application.
42. Engineers design systems to avoid unwanted resonance (e.g., damping in car suspensions).
43. In resonance, the energy input from the external force matches the energy required to overcome restoring forces.
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Practical Applications

Applications of SHM and Energy Changes

44. The balance wheel in mechanical clocks uses SHM to regulate time.
45. Quartz crystals in watches vibrate in SHM, providing precise timing.
46. Oscillating pendulums are used in metronomes to measure time intervals.
47. Understanding energy changes in SHM helps design shock absorbers.
48. Energy conservation in SHM aids in designing oscillatory systems, like springs in vehicles.
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Applications of Forced Vibration

49. Seismic isolation systems reduce forced vibrations in buildings during earthquakes.
50. Vibrations in machines are controlled by adjusting the frequencies of applied forces.
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