Harmonic Motion

Vector Components Circular motion can be described by components. x = r cos q y = r sin q For uniform circular motion the angle is related to the angular velocity. q = w t The motion can be described as a function of time. x = r cos wt y = r sin wt r r sin q q r cos q

Velocity Components The velocity vector can also be described by components. vx = -v sin q vy = v cos q This velocity is related to the angular frequency. v v cos q q -v sin q q

Acceleration Components For uniform circular motion the acceleration vector points inward. ax = -a cos q ay = -a sin q The acceleration is also related to the angular frequency. -a cos q q a -a sin q q

Changing Angle to Position If only one component is viewed the motion is sinusoidal in time. This is called harmonic motion. Springs and pendulums also have harmonic motion. 1 period x = A cos wt

Acceleration and Position In uniform circular motion acceleration is opposite to the position from the center . In harmonic motion the acceleration is also opposite to the position. This is true for all small oscillations

Spring Oscilations From the Newton’s law the force is proportional to the acceleration. Harmonic motion has a position-dependent force. Force is negative Restoring force

Oscillation Frequency Oscillations use the same terms as rotation. The period (T) is the time it takes to complete one oscillation. The frequency (f) is the number of oscillations completed in a given time. Cycles per second (cps or Hz) Inverse of period (f = 1/T) Angular frequency () is the same as angular velocity.

Spring Energy The spring force has a potential energy U = ½ kx2 . U U Near the minimum all curves are approximately a spring force.

Springboard Find the spring constant from the mass and frequency. With values: k = 42(5.0 /s)2(70. kg) K = 6.9 x 104 N/m A diving board oscillates with a frequency of 5.0 cycles per second with a person of mass 70. kg. What is the spring constant of the board?