Vibrations and Waves
Simple Harmonic Motion Vibrations about an equilibrium position which a restoring force is proportional to the displacement from equilibrium What does this mean? Periodic motion is back and forth over the same path.
Simple Harmonic Motion Equilibrium position, velocity reaches a maximum. Let’s look at an example. Spring
Simple Harmonic Motion Equilibrium position is when the spring is unstretched When a spring has a mass attached at the end
Simple Harmonic Motion If the spring is stretched and released, the spring exerts a force on the mass towards the equilibrium position.
Simple Harmonic Motion The spring force decrease, as the spring reaches the equilibrium position. Eventually reaching zero.
Simple Harmonic Motion The mass’s acceleration also reaches zero at equilibrium.
Simple Harmonic Motion However, the velocity of the mass increases as it approaches equilibrium. Reaching a maximum velocity at equilibrium.
Simple Harmonic Motion At maximum displacement, spring force and acceleration reach a maximum.
Simple Harmonic Motion This can occur in the opposite direction (compression).
Simple Harmonic Motion Spring’s compression is equal to the distance the spring was stretched.
Simple Harmonic Motion Ideally, the mass-spring system would oscillate indefinitely.
Simple Harmonic Motion But friction slows, the motion and eventually the mass-spring will come to a rest. (Damping)
Simple Harmonic Motion Restoring force- spring pushes and pulls a mass back to equilibrium.
Simple Harmonic Motion Restoring force- directly proportional to the displacement of the mass.
Hooke’s Law F elastic = -kx Spring force = - (spring constant)(displacement) Negative sign signifies direction of the spring force opposite the direction of the mass displacement.
Hooke’s Law K = spring constant (always positive) Measure of the stiffness of the spring Greater value = stiffer the spring SI units of K is N/m
Simple Harmonic Motion All systems we will consider mechanical energy is conserved. PE initial = KE final
Simple Pendulum Pendulum consist of a mass (bob) attached to a fixed string Disregard friction and the mass of the string
Simple Pendulum Is a simple pendulum, simple harmonic motion? What is the restoring force? Is the restoring force proportional to the displacement? If so, then YES!
Simple Pendulum Is a simple pendulum, simple harmonic motion? What is the restoring force? Is the restoring force proportional to the displacement? If so, then YES!
Simple Pendulum Does the force acting on the bob act as the restoring force? Force exerted by the string acts along the y-axis Any point other than equilibrium, bob’s weight can be resolved into two components
Simple Pendulum Does the force acting on the bob act as the restoring force? Force exerted by the string and the y component of the bob’s weight are perpendicular to the motion X component of the bob’s weight is the net force acting on the bob in the direction of motion.
Simple Pendulum Does the force acting on the bob act as the restoring force? Therefore, the X component pushes or pulls the bob towards equilibrium It is a restoring force.
Simple Pendulum Restoring force of a SP is SHM Small angles, the pendulum is SHM Less than 15°