Harmonic Motion

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

Velocity Components  The velocity vector can also be described by components. v x = -v sin  v y = v cos   This is the derivative of the position.  v -v sin  v cos  

Acceleration Components  For uniform circular motion the acceleration vector points inward. a x = -a cos  a y = -a sin   This is the derivative of the velocity.  a -a sin  -a cos  

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. x = A cos  t 1 period

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 law of action the force is proportional to the acceleration.  Harmonic motion has a position-dependent force. Force is negativeForce is negative Restoring forceRestoring force

Springboard  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?  Find the spring constant from the mass and frequency.  With values: k = 4  2 (5.0 /s) 2 (70. kg)k = 4  2 (5.0 /s) 2 (70. kg) K = 6.9 x 10 4 N/mK = 6.9 x 10 4 N/m next