Transcript Examples

Chapter 15
Oscillatory Motion
EXAMPLES
Chapter 15
Oscillatory Motion: EXAMPLES
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
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
Example 15-2: Conceptual Example
(SHM Forces)
Which of the following represent an object undergoing
SHM
2
 (a). F = – 0.5x
 (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).
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
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
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

Material for the Midterm
Material from the book to Study!!!
Objective Questions: 12-14
Conceptual Questions: 3-5
Problems: 5-8-14-18-21-25