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

2. 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

3. 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

4. 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

5. 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

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

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

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

9. 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