Springs Hooke’s Law (Fs) Spring Constant (k) Spring Force – is a restoring force because it always pushes or pulls towards the equilibrium position.

Active Figure 15.1  A block attached to a spring moving on a frictionless surface. (a) When the block is displaced to the right of equilibrium (x > 0), the force exerted by the spring acts to the left. (b) When the block is at its equilibrium position (x = 0), the force exerted by the spring is zero. (c) When the block is displaced to the left of equilibrium (x < 0), the force exerted by the spring acts to the right. At the Active Figures link at http://www.pse6.com, you can choose the spring constant and the initial position and velocities of the block to see the resulting simple harmonic motion.

Simple Harmonic Motion Simple harmonic motion occurs when the net force along the direction of motion obeys Hooke’s Law In other words, when the net force is proportional to the displacement from the equilibrium point and is always directed towards the equilibrium point.

Figure 15.3  An experimental apparatus for demonstrating simple harmonic motion. A pen attached to the oscillating object traces out a sinusoidal pattern on the moving chart paper. Fig. 15.3, p.456

Active Figure 15.2  (a) An x–t curve for an object undergoing simple harmonic motion. The amplitude of the motion is A, the period is T, and the phase constant is . At the Active Figures link at http://www.pse6.com, you can adjust the graphical representation and see the resulting simple harmonic motion on the block in Figure 15.1.

Terminology Amplitude (A) – maximum displacement Period (T) – time it takes the object to move through one complete cycle Frequency (f) – the number of complete cycles per unit of time Acceleration (a)

The Equations

Active Figure 15.7  A block–spring system that begins its motion from rest with the block at x = A at t = 0. In this case,  = 0 and thus x = A cos t. At the Active Figures link at http://www.pse6.com, you can compare the oscillations of two blocks starting from different initial positions to see that the frequency is independent of the amplitude.

Equations of Motion for the object-spring system

Figure 15. 6 Graphical representation of simple harmonic motion Figure 15.6  Graphical representation of simple harmonic motion. (a) Position versus time. (b) Velocity versus time. (c) Acceleration versus time. Note that at any specified time the velocity is 90° out of phase with the position and the acceleration is 180° out of phase with the position.

Active Figure 15.17  When  is small, a simple pendulum oscillates in simple harmonic motion about the equilibrium position  = 0. The restoring force is –mg sin , the component of the gravitational force tangent to the arc. At the Active Figures link at http://www.pse6.com, you can adjust the mass of the bob, the length of the string, and the initial angle and see the resulting oscillation of the pendulum.

Pendulum Equations

Active Figure 15.11  Simple harmonic motion for a block-spring system and its analogy to the motion of a simple pendulum (Section 15.5). The parameters in the table at the right refer to the block-spring system, assuming that at t = 0, x = A so that x = A cos t.

Figure 15.21 One example of a damped oscillator is an object attached to a spring and submersed in a viscous liquid.

Figure 15. 22 Graph of position versus time for a damped oscillator Figure 15.22 Graph of position versus time for a damped oscillator. Note the decrease in amplitude with time. At the Active Figures link at http://www.pse6.com, you can adjust the spring constant, the mass of the object, and the damping constant and see the resulting damped oscillation of the object.

Figure 15.23  Graphs of position versus time for (a) an underdamped oscillator, (b) a critically damped oscillator, and (c) an overdamped oscillator.

Types of Traveling Waves Transverse wave – the displacement of the wave is perpendicular to the motion of the wave Sine and cosine graphs Light waves (electromagnetic waves) Longitudinal wave – the displacement of the wave is parallel to the motion of the wave Sound waves

Figure 16. 2 A transverse pulse traveling on a stretched rope Figure 16.2  A transverse pulse traveling on a stretched rope. The direction of motion of any element P of the rope (blue arrows) is perpendicular to the direction of wave motion (red arrows). Fig. 16.2, p.488

Figure 16.5  A one-dimensional pulse traveling to the right with a speed v. (a) At t = 0, the shape of the pulse is given by y = f(x). (b) At some later time t, the shape remains unchanged and the vertical position of an element of the medium any point P is given by y = f(x – vt). Fig. 16.5, p.489

Figure 16. 3 A longitudinal pulse along a stretched spring Figure 16.3  A longitudinal pulse along a stretched spring. The displacement of the coils is in the direction of the wave motion. Each compressed region is followed by a stretched region. Fig. 16.3, p.488

Active Figure 16.10  One method for producing a sinusoidal wave on a string. The left end of the string is connected to a blade that is set into oscillation. Every element of the string, such as that at point P, oscillates with simple harmonic motion in the vertical direction. At the Active Figures link at http://www.pse6.com, you can adjust the frequency of the blade. Fig. 16.10, p.495

Terminology and Equations Wavelength (l) Wave speed (v) Mass per unit length (m) If the wave is traveling on a string then the wave velocity is defined as:

Active Figure 16.8  (a) The wavelength  of a wave is the distance between adjacent crests or adjacent troughs. At the Active Figures link at http://www.pse6.com, you can change the parameters to see the effect on the wave function. Fig. 16.8a, p.492

IMPORTANT There are two different velocities for a traveling transverse wave. The wave speed, which is literally how fast the wave is moving to the left or to the right. The transverse velocity, which is how fast the wave (rope, string) is moving up and down.

Wave Interference Superposition Principle – when two or more waves encounter each other while traveling through a medium, the resultant wave is found by adding together the displacements of the individual waves point by point.

Active Figure 18.1 (a–d) Two pulses traveling on a stretched string in opposite directions pass through each other. When the pulses overlap, as shown in (b) and (c), the net displacement of the string equals the sum of the displacements produced by each pulse. Because each pulse produces positive displacements of the string, we refer to their superposition as constructive interference. Fig. 18.1, p.545

Active Figure 18.2 (a–e) Two pulses traveling in opposite directions and having displacements that are inverted relative to each other. When the two overlap in (c), their displacements partially cancel each other. At the Active Figures link at http://www.pse6.com, you can choose the amplitude and orientation of each of the pulses and watch the interference as they pass each other. Fig. 18.2, p.546