Kinetic Books – Principles of Physics Summary of Chapter 15 Chapter 15 Simple harmonic motion (SHM) is a kind of repeated, consistent back and forth motion, like the swinging of a pendulum. It is caused by a restoring force that varies linearly with displacement. The displacement associated with such motion can be described with a sinusoidal function, typically a cosine. The displacement is zero at equilibrium and maximum at the extreme positions. Just as with other types of repetitive motion, the period of SHM is the amount of time required to complete one cycle of motion. The frequency is the number of cycles completed per second. It is the reciprocal of the period. The unit of frequency is the hertz (Hz), equal to one inverse second. Angular frequency is the frequency measured in radians per second. It is represented by the Greek letter ω and is seen in the function for harmonic motion. If the object in simple harmonic motion is a mass on a spring, the spring constant and the mass determine the angular frequency. The amplitude of harmonic motion is the maximum displacement from equilibrium. It is represented by A and appears as the coefficient of the cosine in the displacement function for SHM. The phase constant, φ, specifies the position at the zero time, effectively shifting the graph to the left or right. The velocity and acceleration functions for SHM are also sinusoidal. The maximum velocity occurs at equilibrium, and it is zero at the extremes. Acceleration is the opposite: zero at equilibrium and maximum at the extremes. These relationships follow from the general nature of velocity as the instantaneous slope of the displacement graph, and acceleration as the slope of velocity. As an object like a mass on a spring moves in SHM, its total energy remains constant, although it transforms from potential to kinetic energy and back continuously. If you stretch the spring and release it to begin the motion, the amount of work you do on the spring is the amount of potential energy you have stored in it. All the energy is potential energy at the extremes, and kinetic energy at the equilibrium position. A simple pendulum displays simple harmonic motion in its angular displacement, provided that the amplitude of the motion is small. Instead of a restoring force, there is a restoring torque due to gravity. The period of a pendulum depends upon the length of the pendulum and the acceleration of gravity. The simple pendulum is a special case of the more complicated physical pendulum. In general, the period of a physical pendulum depends upon its moment of inertia, mass, and the distance from the pivot point to its center of mass, as well as the acceleration of gravity. Sometimes a damping force opposes oscillatory motion. A typical damping force is proportional to the velocity of the object, which changes with time. A force that acts with the restoring force can maintain or increase the amplitude of oscillations. Forced oscillations occur when such a driving force is present. The natural frequency of a system is the frequency at which it will oscillate in the absence of external force. As the frequency of the driving force approaches the natural frequency, energy is transferred more efficiently and the system’s oscillation amplitude increases. When these frequencies are approximately equal, resonance occurs. x(t) = A cos (ωt + φ) f = 1/T ω = 2πf v(t) = –Aω sin (ωt + φ) a(t) = –Aω2 cos (ωt + φ) PE = ½ kx2 PE = ½ kA2 cos2 ωt KE = ½ mA2ω2 sin2 ωt TE = ½ kA