Simple Harmonic Motion (SHM) Chapter 11: Section 1 Simple Harmonic Motion (SHM)

Periodic Motion A repeated motion that moves back and forth over the same path. Examples include: pendulum, vibrating string, orbiting planet

Simple Harmonic Motion (SHM) A special kind of periodic motion Results from a restoring force that is proportional to displacement. Restoring Force: force that tries to reestablish equilibrium. During SHM, an object oscillates (vibrates back and forth) around a central equilibrium position.

Springs: stretching m m -Fs When you stretch a spring, you must apply an increasing amount of force as the displacement increases. That means the spring is pulling back with an equal and opposite force m m x -Fs

Springs: compressing m m When you compress a spring, you must apply an increasing amount of force as the displacement increases. That means the spring is pushing back with an equal and opposite force m m x Fs

Displaced Position -Force is at maximum -Since F=ma a = max -v = 0 Equilibrium Position -Net Force= 0 -Since F=ma a =0 -v = maximum

Felastic= - k Δ x Hook’s Law K = Spring Constant (measures stiffness of spring) Δ x = displacement from equilibirum) - sign indicates that the force is opposite displacement

A stretched or compressed spring has elastic potential energy. Work needed to stretch or compress a string: W=½ kΔx2 Since work = change in PE PE =½ kΔx2 Taking various forms of energy into account, the total energy is given by: E = ½ mv2 + mgh + ½ kΔx2

Sample Problem If a mass of 0.25 kg attached to a vertical spring stretches the spring 3.0 cm from its equilibrium position, what is the spring constant?

Sample problem A 1.80 kg object is connected to a spring of force constant 120 N/m. How far is the spring stretched if it is used to drag the object across a floor at constant velocity? Assume the coefficient of kinetic friction is 0.60.

A Simple Pendulum Consists of a mass, called a bob The restoring force of a pendulum is a component of the bob’s weight. When the angle is <15 degrees, the restoring force is proportional to the displacement. For small angles, the pendulum’s motion is simple harmonic.

A Simple Pendulum Cont. Please note: FT = tension force, F r = restoring force and U is another symbol for Potential Energy