Oscillations · ⁨التذبذبات⁩

Learning Objective 7.1.A: Describe simple harmonic motion.

• 7.1.A.1 Simple harmonic motion is a special case of periodic motion.
• 7.1.A.2 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: $ma_x = -k\Delta x$
  • 7.1.A.2.i A restoring force is a force that is exerted in a direction opposite to the object's displacement from an equilibrium position.
  • 7.1.A.2.ii An equilibrium position is a location at which the net force exerted on an object or system is zero.

Learning Objective 7.2.A: Describe the frequency and period of an object exhibiting SHM.

• 7.2.A.1 The period of SHM is related to the angular frequency, $\omega$, of the object's motion by the following equation:
  • Equation: $T = \dfrac{2\pi}{\omega} = \dfrac{1}{f}$
  • 7.2.A.1.i The period of an object–ideal-spring oscillator is given by the equation
    • Equation: $T_s = 2\pi\sqrt{\dfrac{m}{k}}.$
  • 7.2.A.1.ii The period of a simple pendulum displaced by a small angle is given by the equation
    • Equation: $T_p = 2\pi\sqrt{\dfrac{l}{g}}.$

Learning Objective 7.3.A: Describe the displacement, velocity, and acceleration of an object exhibiting SHM.

• 7.3.A.1 For an object exhibiting SHM, the displacement of that object measured from its equilibrium position can be represented by the equations $x = A\cos(2\pi ft)$ or $x = A\sin(2\pi ft)$.
  • 7.3.A.1.i Minima, maxima, and zeros of displacement, velocity, and acceleration are features of harmonic motion.
  • 7.3.A.1.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.
• 7.3.A.2 The position as a function of time for an object exhibiting SHM is a solution of the second-order differential equation derived from the application of Newton's second law.
  • Derived equation: $\dfrac{d^2 x}{dt^2} = -\omega^2 x$
• 7.3.A.3 Characteristics of SHM, such as velocity and acceleration, can be determined by or derived from the equation $x = A\cos(\omega t + \phi).$
  • 7.3.A.3.i The acceleration of an object exhibiting SHM is related to the object's angular frequency and position.
    • Derived equation: $a = -\omega^2 x$
  • 7.3.A.3.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: $v_{\max} = A\omega$
    • $a_{\max} = A\omega^2$
• 7.3.A.4 In the presence of a sinusoidal external force, a system may exhibit resonance.
  • 7.3.A.4.i Resonance occurs when an external force is exerted at the natural frequency of an oscillating system.
  • 7.3.A.4.ii Resonance increases the amplitude of oscillating motion.
  • 7.3.A.4.iii The natural frequency of a system is the frequency at which the system will oscillate when it is displaced from its equilibrium position.
• 7.3.A.5 Changing the amplitude of a system exhibiting SHM will not change its period.
• 7.3.A.6 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.

Learning Objective 7.4.A: Describe the mechanical energy of a system exhibiting SHM.

• 7.4.A.1 The total energy of a system exhibiting SHM is the sum of the system's kinetic and potential energies.
  • Relevant equation: $E_{\text{total}} = U + K$
• 7.4.A.2 Conservation of energy indicates that the total energy of a system exhibiting SHM is constant.
• 7.4.A.3 The kinetic energy of a system exhibiting SHM is at a maximum when the system's potential energy is at a minimum.
• 7.4.A.4 The potential energy of a system exhibiting SHM is at a maximum when the system's kinetic energy is at a minimum.
  • 7.4.A.4.i The minimum kinetic energy of a system exhibiting SHM is zero.
  • 7.4.A.4.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: $E_{\text{total}} = \dfrac{1}{2}kA^2$

Learning Objective 7.5.A: Describe the properties of a physical pendulum.

• 7.5.A.1 A physical pendulum is a rigid body that undergoes oscillation about a fixed axis.
• 7.5.A.2 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: $T_{\text{phys}} = 2\pi\sqrt{\dfrac{I}{mgd}}$
  • 7.5.A.2.i When displaced from equilibrium, the gravitational force exerted on a physical pendulum's center of mass provides a restoring torque.
    • Derived equation: $\tau = -mgd\sin\theta$
  • 7.5.A.2.ii For small amplitudes of motion, the small-angle approximation can be applied to the restoring torque.
    • Derived equation: $\sin\theta \approx \theta$
    • $\tau = -mgd\theta = I\alpha$
  • 7.5.A.2.iii The small-angle approximation and Newton's second law in rotational form yield a second-order differential equation that describes SHM:
    • Equation: $\dfrac{d^2\theta}{dt^2} = -\omega^2\theta$
• 7.5.A.3 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: $T_p = 2\pi\sqrt{\dfrac{\ell}{g}}$
• 7.5.A.4 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: $I\alpha = -k\Delta\theta$