Chapter VIIPeriodic MotionsA. Periodic MotionB. Simple Harmonic MotionC. Energy of The Simple Harmonic OscillatorD. Simple Pendulum

A. Periodic Motion

The amplitude of the motion, denoted by A, is the maximum magnitude of displacement from equilibrium-that is, the maximum value of Ixl. It is always positive.

• A complete vibration, or cycle, is one complete round trip say, from A to -A and back to A, or from 0 to A, back through 0 to -A, and back to O.

• The period, T, is the time for one cycle. It is always positive. The SI unit is the second, but it is sometimes expressed as "seconds per cycle."

• The frequency, f, is the number of cycles in a unit of time. It is always positive. The SI unit of frequency is the hertz.

1 hertz = I Hz = 1 cycle/s = 1 S-l

The angular frequency (rad/s), , is 2 times the frequency (cycle/s): = 2 f The number 2 as having units rad/cycle.

B. Simple Harmonic Motion

The minus sign means the acceleration and displacemem always have opposite signs.

= phase angle(t + ) = Phase

The frequency and period depend only on the mass of the block and on the force constant of the spring

C. Energy of The Simple Harmonic Oscillator

• The total mechanical energy of a simple harmonic oscillator is a constant of the motion and is proportional to the square of the amplitude.

• Note, that U is small when K is large, and vice versa, because the sum must be constant.

D. Simple Pendulum

x

• The period and frequency of a simple pendulum depend only on the length of the string and the acceleration due to gravity.

• Because the period is independent of the mass, we conclude that all simple pendulums that are of equal length and are at the same location (so that g is constant) oscillate with the same period.