1. Periodic Motion

2. Definition of Terms Periodic Motion: Motion that repeats itself in a regular pattern. Periodic Motion: Motion that repeats itself in a regular pattern. Cycle: One complete vibration or oscillation. Cycle: One complete vibration or oscillation. Equilibrium position: Position of the object when it is at rest. No energy is stored, and no net force acts on the object. Equilibrium position: Position of the object when it is at rest. No energy is stored, and no net force acts on the object. Restoring Force: Force that brings an oscillating object back to its equilibrium position. Restoring Force: Force that brings an oscillating object back to its equilibrium position.

3. Robert Hooke (1635-1703) In 1678, he determined that the deformation of an elastic object is directly proportional to the force causing the deformation. (HOOKE’S LAW) In 1678, he determined that the deformation of an elastic object is directly proportional to the force causing the deformation. (HOOKE’S LAW)

4. Hooke’s Law F  x or F = -kx F = applied force (N) F = applied force (N) x = amount of deformation (m) x = amount of deformation (m) k = spring (force) constant k = spring (force) constant Units = N/m Units = N/m

5. Ex: A 0.25 kg mass is suspended from a spring with a force constant of 48 N/m. How far does the spring stretch? Given: m = 0.25 kg F = mg = -2.4 N k = 48 N/m Find: x = ? F = -kx F/k = x (2.4 N) / (48 N/m) = x 0.050 m = x

6. Elastic Potential Energy (PE e ) Energy stored in an elastic object (usually a spring) by deforming it (doing work on it). Energy stored in an elastic object (usually a spring) by deforming it (doing work on it). PE e = ½ kx 2 x = distance spring is deformed (stretched or compressed) x = distance spring is deformed (stretched or compressed) k = spring constant: How resistant an elastic object is to being stretched or compressed (stiffness). Units = N/m k = spring constant: How resistant an elastic object is to being stretched or compressed (stiffness). Units = N/m Units = N/m (m 2 ) = N*m = Joules Units = N/m (m 2 ) = N*m = Joules

7. Ex: A spring with a spring constant of 160 N/m is compressed by 8.0 cm. How much energy is stored in the spring? Given: k = 160 N/m x = 0.080 m Find: PE e = ? PE e = ½ kx 2 = ½ (160 N/m)(.080 m) 2 PE e = 0.51 J

8. Simple Harmonic Motion (SHM) Any periodic motion in which the restoring force is proportional to the displacement from equilibrium. Any periodic motion in which the restoring force is proportional to the displacement from equilibrium. Mass on a Spring Mass on a Spring Mass on a Spring Mass on a Spring Simple Pendulum (for small angles) Simple Pendulum (for small angles) Simple Pendulum (for small angles) Simple Pendulum (for small angles)

10. Measures of Simple Harmonic Motion Amplitude (A): Maximum displacement from equilibrium position (meters) Amplitude (A): Maximum displacement from equilibrium position (meters) Period (T): Time it takes to execute one complete cycle of motion (seconds) Period (T): Time it takes to execute one complete cycle of motion (seconds) Frequency (f): Number of cycles or vibrations per unit of time Frequency (f): Number of cycles or vibrations per unit of time Units = Hertz (Hz) or s -1 or 1/s Units = Hertz (Hz) or s -1 or 1/s

11. Calculating Period Simple Pendulum: Simple Pendulum: T = period (s) L = length of pendulum (m) g = free fall acceleration (m/s 2 )

12. Ex: What is the period on Earth of a simple pendulum that has a length of 1.20 m? Given: L = 1.20 m g = 9.81 m/s 2 Find: T =? T = 2π√(L/g) = 2π√(1.20 m / 9.81 m/s 2 ) = 2.20 s

13. Resonance Forced Vibrations: One vibrating object causes another object to vibrate at the same frequency. Forced Vibrations: One vibrating object causes another object to vibrate at the same frequency. Natural Frequency: Frequency at which minimum energy is required to produce forced vibrations. An object “prefers” to vibrate at this frequency. Natural Frequency: Frequency at which minimum energy is required to produce forced vibrations. An object “prefers” to vibrate at this frequency. Resonance occurs when the frequency of a forced vibration matches the natural frequency of a system. Resonance occurs when the frequency of a forced vibration matches the natural frequency of a system.