1. Oscillatory Motion EXAMPLES
Chapter 15
Oscillatory Motion EXAMPLES

2. Chapter 15 Oscillatory Motion: EXAMPLES

3. Example 15-1: Motion of the Block (Horizontal)
The block continues to oscillate between –A and +A
These are turning points of the motion
The force F is conservative
In the absence of friction, the motion will continue forever
Real systems are generally subject to friction, so they do not actually oscillate forever

4. Example 15-1: Motion of the Block (Vertical)
When the block is hung from a vertical spring, its weight will cause the spring to stretch
Figure (a): Free spring, hang vertically
Figure (b): Mass m attached to spring in new equilibrium position, which occurs when:
F = mg – kxo  xo = mg/k
If the resting position of the spring is defined as x = 0, the same analysis as was done with the horizontal spring will apply to the vertical spring-mass system

5. Example 15-2: Conceptual Example (SHM Forces)
Which of the following represent an object undergoing
SHM
(a). F = – 0.5x2
(b). F = –2.3y
(c). F = 8.6x
(d). F = – 4θ
Answer:
(b). F = – 2.3y & (d). F = – 4θ
Force has minus sign required to restore the system to equilibrium.
Force is proportional to a displacement
(which need not to be x).

6. 15-3: SHM Example 1 Initial conditions at t = 0 : x (0)= A v (0) = 0
This means f = 0
The velocity reaches extremes of : ± ωA
The acceleration reaches extremes of: ± ω2A

7. 15-3: SHM Example 2 Initial conditions at t = 0 : x (0)=0 v (0) = vi
This means f = - π/2
The graph is shifted one-quarter cycle to the right compared to the graph of x (0) = A

8. Example 15-4: Conceptual Example (Doubling Amplitude)
Suppose the spring from figure is stretched twice as far (to x = 2A). What happened to: E, vmax, & amax?
(a). E = ½kA2 (15.21)
(b). vmax = ωA = (k/m)½A (15.17)
(c). amax = ω2A = (k/m)A (15.18)
Answers:
(a). E = ½k(2A)2 = 4( kA2)
So stretching A twice as far, quadruples the energy
(b). vmax = ω(2A) = 2ωA
So stretching A twice as far, doubles the maximum velocity
(c). amax = ω2(2A) = 2ω2A
So stretching A twice as far, doubles the maximum acceleration

9. Material from the book to Study!!!
Material for the Midterm
Material from the book to Study!!!
Objective Questions: 12-14
Conceptual Questions: 3-5
Problems: