Chapter 15 Oscillatory Motion PHYS 2211

Recall the Spring Since F=ma, this can be rewritten as: The direction of the force is negative because it is a restoring force. In other words, if x is positive, the force is negative and vice versa. This makes the object oscillate and as we will see, it undergoes simple harmonic motion.

Using the kinematic definitions… Remember that: can be rewritten as:So

To make this differential equation easier to solve, we write then A valid solution of this equation is: Note: Solving differential equations is beyond the scope of this class. It is more important for us to know what the solution is.

Simple Harmonic Motion An equation for distance (x) as time (t) changes; sin and cos are the basic components of any formula describing simple harmonic motion. Here A, ω, ϕ are constants A is Amplitude - For springs: max value of distance (x) (positive or negative) - Maximum value the wave alternates back and forth between ω is Angular frequency → - How rapidly oscillations occur - Units are rad/s Remember:

is the phase constant is the phaseEssentially is the shifts of the wave The phase constant determines the starting position of the wave at the time t = 0.

General Concepts Anything with behaviors which have formulas that look like these (not including the constants) are undergoing simple harmonic motion and can be described using the same method as we used for the spring.

Period and frequency Period: time for 1 full oscillation frequency: number of oscillations per second Measured in cycles per second - Hertz (Hz) For springs Note: frequency (f) and angular frequency (ω) measure the same thing but with different units. They differ by a factor of 2 pi.

Velocity and Acceleration Velocity of oscillation Acceleration of oscillation Note: magnitudes of maximum values are when the sine/cosine arguments equal 1

Energy of Simple Harmonic Oscillators Remember that: After substituting the equations of velocity(v) and distance(x) for simple harmonic oscillations, we get:

Applications: Simple Pendulum mg T The restoring force for a pendulum is where which is the arclength or the path the ball travels along thus

Simple Pendulums continued Notice thatalmost looks like According to the small angle approximation, which states that sinθ ≈ θ if θ is small (about less than 10°) We can rewrite the equation to be which is exactly in the form for simple harmonic motions whereso then we can now use all the other formulas for simple harmonic motions for the case of a pendulum for small angles

Applications: Torsional Pendulum When a torsion pendulum is twisted, there exists a restoring torque which is equal to: This looks just likebut in rotational form Thus, we can apply what we know about angular motion to get information about this object’s simple harmonic oscillations

Torsion Pendulum continued Remember: After substitutingwe get whereand