7.3 Representing and Analyzing SHM

Describe the displacement, velocity, and acceleration of an object exhibiting SHM.

• For an object exhibiting SHM, the displacement of that object measured from its equilibrium position can be represented by the equations
  • i. Minima, maxima, and zeros of displacement, velocity, and acceleration are features of harmonic motion.
  • ii. Recognizing the positions or times at which the displacement, velocity, and acceleration for SHM have extrema or zeros can help in qualitatively describing the behavior of the motion.
• The position as a function of time for an object exhibiting SHM is a solution of the secondorder differential equation derived from the application of Newton’s second law. Derived equation:
• Characteristics of SHM, such as velocity and acceleration, can be determined by or derived from the equation TOPIC 7.3 Representing and Analyzing SHM
  • i. The acceleration of an object exhibiting SHM is related to the object’s angular frequency and position. Derived equation:
  • ii. It can be shown that the maximum velocity and acceleration of an object exhibiting SHM are related to the angular frequency of the object’s motion. Derived equations:
• In the presence of a sinusoidal external force, a system may exhibit resonance.
  • i. Resonance occurs when an external force is exerted at the natural frequency of an oscillating system.
  • ii. Resonance increases the amplitude of oscillating motion.
  • iii. The natural frequency of a system is the frequency at which the system will oscillate when it is displaced from its equilibrium position.
• Changing the amplitude of a system exhibiting SHM will not change its period.
• Properties of SHM can be determined and analyzed using graphical representations. BOUNDARY STATEMENT AP Physics C: Mechanics only expects students to know the solution to the second-order differential equation that describes SHM, as well as be able to identify SHM. AP Physics C: Mechanics does not expect students to mathematically prove that the solution is correct.