Oscillatory Motion

Definitions Amplitude, A - Maximum displacement from equilibrium. Period, T - Time for one full oscillation. Frequency, f - Number of oscillations per unit of time. T 2T 3T 4T 5T -A A Units (s-1 or Hz)

Harmonic Oscillator Restoring force is proportional to displacement. Equilibrium (1) (2) (3) t = 0s (4) Try (5) X -A A

Angular Frequency In , w is angular frequency. In one time period, 2p = wT. Units for w is rad/s.

Spring Constant A 500 g mass is attached to a spring and it oscillates 10 times in 15 seconds. What is the spring constant of the mass? How would this change on the moon?

Kinematics Displacement Velocity Acceleration T 2T -A A T 2T -Aw Aw T T 2T -A A T 2T -Aw Aw T 2T -Aw2 Aw2

Jack-in-a-Box While initially at rest, a Jack-in-a-Box head is struck by a child's hand. The impulse of the strike gives the head an initial velocity of 0.3 m/s. What is the maximum displacement of the head from equilibrium? (The mass of the head is 0.10 kg and the spring constant has a value of 1.5 N/m.)

Mass on a Spring At equilibrium, Otherwise, D.E. where x = Dx Solution

Physical Pendulum Restoring force Derive D.E. Small angle: Solution d q d mg Derive D.E. Small angle: Solution

Simple Pendulum All mass is concentrated at one point l q Solution mg

Energy in a Spring Potential Energy Kinetic Energy Total Energy then, Therefore,

Damped Oscillators Damping Force Newton’s 2nd Law Solution to D.E. A -A 2T T 3T

Forced Oscillations Oscillating system has a natural frequency of w0 Driving force is w The closer w0 and w are, the larger the amplitude Amplitude, A Driving Frequency, w w0 2w0 3w0

Resonance Maximum energy transfer between two objects that have similar oscillatory frequencies. Tacoma Narrows Bridge

Phase Angle Phase, f , shifts the location of the peak displacement. f >0 (shift to the left) f <0 (shift to the right) f A Amplitude -A T 2T time

Torsion Pendulum Restoring force q Derive D.E. Solution

Meter Stick Oscillations A meter stick has a length of 1 meter, a thickness of 8 mm and a width of 2.5 cm. The mass of the meter stick is 165 g. How many different ways can the meter stick be set up to oscillate and what is the angular frequency in each of these situations?