Harmonic Motion (II) Mass and Spring Energy in SHM Physics 1D03 - Lecture 33

SHM: x(t) A = amplitude t f = phase constant w = angular frequency A is the maximum value of x (x ranges from +A to -A) f gives the initial position at t=0: x(0) = A cosf w is related to the period T and the frequency f = 1/T T (period) is the time for one complete cycle (seconds). Frequency f (cycles per second or hertz, Hz) is the number of complete cycles per unit time. Physics 1D03 - Lecture 33

What is the oscillation period of a FM radio station with a signal at Example - frequency What is the oscillation period of a FM radio station with a signal at 100MHz ? Example - frequency A mass oscillating in SHM starts at x=A and has a period of T. At what time, as a fraction of T, does it first pass through x=A/2? Physics 1D03 - Lecture 33

Example The block is at its equilibrium position and is set in motion by hitting it (and giving it an initial velocity) at time t = 0. Its motion is SHM with amplitude 5 cm and period 2 seconds. Write the function x(t). M v0 x Result: x(t) = (5 cm) cos[π t – π/2] Physics 1D03 - Lecture 33

Example The block is at x0 = +5 cm, with positive velocity v0, at time t = 0. Its motion is SHM with amplitude 10 cm and period 2 s. If x(t) = A cos (wt + f), the phase constant f should be: M v0 x0 0o 30o 60o -30o -60o Physics 1D03 - Lecture 33

Velocity and Acceleration a(t) =- w 2 x(t) Physics 1D03 - Lecture 33

Position, Velocity and Acceleration x(t) t v(t) t a(t) t Physics 1D03 - Lecture 33

Question: Where in the motion is the velocity largest? Where in the motion is acceleration largest? When do these happen ? Physics 1D03 - Lecture 33

When do we have Simple Harmonic Motion ? Position: Differentiate: and we find that acceleration is proportional to displacement: a(t) = - w 2 x(t) SHM is also called ‘oscillatory’ motion, and is ‘periodic’. Physics 1D03 - Lecture 33