Simple Harmonic Motion

Distinguish simple harmonic motion from other forms of periodic motion. Periodic motion is motion in which a body moves repeatedly over the same path in equal time intervals. Examples: uniform circular motion and simple harmonic motion.

Cont’d Simple Harmonic Motion (SHM) is a special type of periodic motion in which an object moves back and forth, along a straight line or arc. Examples: pendulum, swings, vibrating spring, piston in an engine. In SHM, we ignore the effects of friction. Friction damps or slows down the motion of the particles. If we included the affect of friction then it’s called damped harmonic motion.

Cont’d For instance a person oscillating on a bungee cord would experience damped harmonic motion. Over time the amplitude of the oscillation changes due to the energy lost to friction. http://departments.weber.edu/physics/amiri/di rector/DCRfiles/Energy/bungee4s.dcr

State the conditions necessary for simple harmonic motion. A spring wants to stay at its equilibrium or resting position. However, if a distorting force pulls down on the spring (when hanging an object from the spring, the distorting force is the weight of the object), the spring stretches to a point below the equilibrium position. The spring then creates a restoring force, which tries to bring the spring back to the equilibrium position.

Cont’d The distorting force and the restoring force are equal in magnitude and opposite in direction. FNET and the acceleration are always directed toward the equilibrium position.

Cont’d Applet showing the forces, displacement, and velocity of an object oscillating on a spring. http://www.ngsir.netfirms.com/englishhtm/Spr ingSHM.htm

Displacement Velocity Acceleration

Cont’d at equilibrium: speed or velocity is at a maximum displacement (x) is zero acceleration is zero FNET is zero (Restoring Force = Distorting Force) object continues to move due to inertia

Cont’d at endpoints: speed or velocity is zero displacement (x) is at a maximum equal to the amplitude acceleration is at a maximum restoring force is at a maximum FNET is at a maximum

State Hooke’s law and apply it to the solution of problems. Hooke’s Law relates the distorting force and the restoring force of a spring to the displacement from equilibrium.

Cont’d F – magnitude of the distorting or restoring force in Newtons k – spring constant or force constant (stiffness of a spring) in Newtons per meter (N/m) x – displacement from equilibrium in meters

Calculate the frequency and period of any simple harmonic motion. T – period (time required for a complete vibration) in seconds f – frequency in vibrations / second or Hertz

Relate uniform circular motion to simple harmonic motion. The reference circle relates uniform circular motion to SHM. The shadow of an object moving in uniform circular motion acts like SHM. The speed of an object moving in uniform circular motion may be constant but the shadow won’t move at a constant speed. The speed at the endpoints is zero and a maximum in the middle. The shadow only shows one component of the motion.

Cont’d Applet showing the forces, displacement, and velocity of an object oscillating on a spring and an object in uniform circular motion. http://www.ngsir.netfirms.com/englishhtm/Spr ingSHM.htm

Identify the positions of and calculate the maximum velocity and maximum accelerations of a particle in simple harmonic motion. The acceleration is a maximum at the endpoints and zero at the midpoint. The acceleration is directly proportional to the displacement, x. The radius of the reference circle is equal to the amplitude. The force and acceleration are always directed toward the midpoint.

Fmax – Force (N) m – mass (kg) A - Amplitude (m) T – period (seconds)

Cont’d This equation shows that the spring force according to Hooke’s Law is equal to the maximum force the spring experiences. K cancels on both sides.

Cont’d k – spring constant (N/m) m – mass (kg) T – period (s)

Cont’d Vmax = maximum velocity (m/s) A = amplitude or maximum displacement (m) T = period (s)

Cont’d Vmax = maximum velocity (m/s) A = amplitude or maximum displacement (m) T = period (s)

Cont’d a = acceleration (m/s2) v = velocity (m/s) A = amplitude (m)

Cont’d a = acceleration (m/s2) A = Amplitude(m) T = period (s)

Cont’d T = period (s) m = mass (kg) k = spring constant (N/m)

Relate the motion of a simple pendulum to simple harmonic motion. A pendulum is a type of SHM. A simple pendulum is a small, dense mass suspended by a cord of negligible mass. The period of the pendulum is directly proportional to the square root of the length and inversely proportional to the square root of the acceleration due to gravity.

Cont’d Applet of displacement, velocity, acceleration, force, etc acting on a pendulum. http://www.walter-fendt.de/ph11e/pendulum.htm

Cont’d T = period (s) l = length (m) g = acceleration due to gravity (m/s2)