7 Oscillations

Describe simple harmonic motion.

• Simple harmonic motion is a special case of periodic motion.
• SHM results when the magnitude of the restoring force exerted on an object is proportional to that object’s displacement from its equilibrium position. Derived equation:
  • i. A restoring force is a force that is exerted in a direction opposite to the object’s displacement from an equilibrium position.
  • ii. An equilibrium position is a location at which the net force exerted on an object or system is zero. TOPIC 7.1 Defining Simple Harmonic Motion (SHM) TOPIC 7.2 Frequency and Period of SHM

Describe the frequency and period of an object exhibiting SHM.

• The period of SHM is related to the angular frequency, , of the object’s motion by the following equation:
  • i. The period of an object–ideal-spring oscillator is given by the equation
  • ii. The period of a simple pendulum displaced by a small angle is given by the equation

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.

Describe the mechanical energy of a system exhibiting SHM.

• The total energy of a system exhibiting SHM is the sum of the system’s kinetic and potential energies. Relevant equation: EU Ktotal =+
• Conservation of energy indicates that the total energy of a system exhibiting SHM is constant.
• The kinetic energy of a system exhibiting SHM is at a maximum when the system’s potential energy is at a minimum.
• The potential energy of a system exhibiting SHM is at a maximum when the system’s kinetic energy is at a minimum.
  • i. The minimum kinetic energy of a system exhibiting SHM is zero.
  • ii. Changing the amplitude of a system exhibiting SHM will change the maximum potential energy of the system and, therefore, the total energy of the system. Relevant equation for a spring–object system: Ek A 1 2 total 2= TOPIC 7.4 Energy of Simple Harmonic Oscillators TOPIC 7.5 Simple and Physical Pendulums

Describe the properties of a physical pendulum.

• A physical pendulum is a rigid body that undergoes oscillation about a fixed axis.
• For small amplitudes of motion, the period of a physical pendulum is derived from the application of Newton’s second law in rotational form. Relevant equation:
  • i. When displaced from equilibrium, the gravitational force exerted on a physical pendulum’s center of mass provides a restoring torque. Derived equation:
  • ii. For small amplitudes of motion, the smallangle approximation can be applied to the restoring torque. Derived equation: sin
  • iii. The small-angle approximation and Newton’s second law in rotational form yield a second-order differential equation that describes SHM:
• A simple pendulum is a special case of physical pendulums in which the hanging object can be modeled as a point mass at a distance, l, from the pivot point. Relevant equation:
• A torsion pendulum is a case of SHM where the restoring torque is proportional to the angular displacement of a rotating system. For example, a horizontal disk that is suspended from a wire attached to its center of mass may undergo rotational oscillations about the wire in the horizontal plane. Derived equation: