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LNK2LRN™ 2008/09 AP Physics C: December 10th to 19th Ch.15 - Oscillatory Motion. Semester Review. |
Plans for the Week and Assignments:
1. WEDNESDAY(12/10): Motion of an Object Attached to a Spring and
Mathematical representation of Simple Harmonic Motion. Energy and
Simple Harmonic Motion. HW: Read pages 452-465 and solve prob. 3, 5,
9,11, 17, and 23 on pages 476-8.
2. THURSDAY(12/11): Harmonic vs. Circular Motion. Simple Pendulum,
Damped vs. Driven Oscillators. HW: Read pages 465-475 and solve prob.
25, 31, 35, 41, and 45 on pages 294-6.
3. FRIDAY(12/12): Bring all Homework Problems to Class on Monday,
NO Exceptions. They will be collected and graded as a 50 Point Quiz.
We will also begin our Semester Exam Review I. HW: Complete Review
Handout #1.
4. MONDAY(12/15): Review II for Semester Exam. All assigned
Homework Problems due, NO Exceptions. HW: Complete Review
Handout #2.
5. TUESDAY(12/16): Review III for Semester Exam. Period 1 Semester
Exam. Shortened classes for Periods 2-7. HW: Study for Semester Exams.
6. WEDNESDAY(12/17): Late Start (10:30). Period 2&3 Semester Exams.
Semester Folder check (25 pts.) due, NO Exceptions, for 2nd & 3rd period
classes. HW: Study for Semester Exams.
7. THURSDAY(12/18): Late Start (10:30). Period 4 & 5 Semester Exams.
HW: Study for Semester Exams.
8. FRIDAY(12/19): Late Start (10:30). Period 6 & 7 Semester Exams.
HW: Have a Safe and Happy Holiday Break!
Very Important: If you have any questions or miss a class, see
me before school (8:00 - 8:30 AM), during Lunch, or after school.
Best to send an email to rpersin@fau.edu.
WEBSITE NOTES: Ch.15 - Oscillatory Motion.
1. A very special kind of motion occurs when the force on a body is
proportional to the displacement of the body from equilibrium. If this
force always acts toward the equilibrium position of the body, there
is a repetitive back-and-forth motion about this position.
2. Such motion is an example of what is called periodic or oscillatory
motion. You are most likely familiar with several examples of periodic
motion, such as the oscillations of a mass on a spring, the motion of a
pendulum, and the vibrations of a stringed musical instrument.
3. Most of the material in this chapter deals with idealized periodic
motion, called Simple Harmonic Motion (SHM). In this type of motion,
an object oscillates between two spatial positions for an indefinite
period of time with no loss in mechanical energy.
4. In real mechanical systems, retarding (frictional) forces are always
present and these forces are considered in section 15.6 of the chapter.
We call these systems "Damped Oscillations." Consequently, in
order to maintain their motion they must be "driven", hence the term
"Forced Oscillations" in section 15.7.
5. The most common system which undergoes simple harmonic motion
is the mass-spring system. The mass is assumed to move on a horizontal,
frictionless surface while the spring is fastened to a wall. Or, the spring
can be hung from a beam and the mass is attached to the free end of
the spring.
6. The point x = 0 is the equilibrium position of the mass; that is, the
point where the mass would reside if left undisturbed. In this position,
there is no net force on the mass.
7. When the mass is displaced a distance x from its equilibrium position,
the spring produces a linear restoring force given by Hooke's Law, F = -kx,
where k is the force constant of the spring, and has SI units of N/m.
8. This law is named after Robert Hooke, a British scientist and
mathematician who lived from 1635 to 1703, and was a contemporary of
Isaac Newton. In simple terms, Hooke's Law states that the force required
to stretch a spring is directly proportional to the distance stretched, as
long as the elastic limit is not exceeded.
9. The minus sign in F = -kx means that F is to the left when the
displacement x is positive, whereas F is to the right when x is negative.
In other words, the direction of the force F is always towards the
equilibrium position.
10. If a graph of F versus x is plotted, the slope will be k, the elastic
constant. It also should be apparent that the area under the graph
represents the work done in stretching the spring. In fact,
W = ∫F(x) dx = ∫kx dx =
½kx2 . Since energy is the ability to do work,W = U, the elastic potential energy of the spring, and U = ½kx2.
11. You should study carefully the comparison between the motion of
the mass-spring system and that of the simple pendulum. A simple
pendulum consists of a mass m attached to a light string of length L.
12. In particular, notice that when the displacement is a maximum, the
energy of the system is entirely potential energy; whereas, when the
displacement is zero, the energy is entirely kinetic energy.
13. This is consistent with the fact that v = 0 when x = A, the
amplitude, or maximum displacement from the equilibrium position.
And it follows that v = vmax , when x = 0. For an arbitrary value of x,
the energy is the sum of K and U.
14. When the angular displacement is small during the entire motion (less
than about 10 degrees), the pendulum exhibits SHM. In this case, the
resultant force acting on the mass m equals the component of weight
tangent to the arc, and has a magnitude F = mg·sin(θ).
15. Since this force is always directed towards θ = 0, it corresponds to a
restoring force. For small θ, we use the approximation that sin(θ) = θ.
16. In other words, the period depends only on the length of the pendulum
and the acceleration of gravity. The period does not depend on mass, so we
conclude that all simple pendula of equal length oscillate with the same
frequency, f, and period, T. For the simple pendulum T = 2π√(L/g) , while
other pendula have equation T = 2π√(I/mgL) .
17. Period and frequency are reciprocal quantities. T = 1/f . And using the
equations v = 2πr/T and v = ωr , we can show that ω= 2πf . We call ω
the angular frequency.
18. Similarly, it can be shown that the maximum speed of an object in SHM
is given by vmax = Aω . Also the maximum acceleration is, amax = Aω2 .
Another way to express angular frequency is ω = √(k/m) .
19. The position x of a simple harmonic oscillator varies periodically with
time according to the expression x(t) = Acos(ωt+φ).
20. In the previous equation, A is the amplitude of the motion, ω is the
angular frequency, and φ is the phase constant.
21. The velocity of a simple harmonic oscillator is v(t) = -ωAsin(ωt+φ).
The acceleration is a(t) = -ω2Acos(ωt+φ). It also can be shown that
v = ±ω√(A2-x2).
22. The Kinetic and Potential Energy of a simple harmonic oscillator vary
with time and are given by K = ½mv2 = ½mω2A2sin2(ωt+φ) and
U = ½kx2 = ½kA2cos2(ωt+φ). The Total Energy is E = ½kA2 .
23. If an oscillator experiences a damping force R = -bv, its position for
small damping is given by x = Ae-(b/2m)tcos(ωt+φ), where we now have
ω = √(k/m - (b/2m)2) .
24. If the oscillator is subject to a sinusoidal driving force F(t) = Fosin(ωt),
it exhibits resonance, in which the amplitude is largest when the driving
frequency matches the natural frequency of the oscillator.
And still, we need these steps to solve any problem in Physics:
(i) read the problem and identify the given variables
(ii) determine what you are asked to solve for
(iii) find the correct equation to use
(iv) use Algebra, Trigonometry, and/or Calculus to isolate the unknown
(v) substitute-in the given information and simplify.
Click for Your Semester Exam Review.
THE
AP PHYSICS C SEMESTER 1 ARCHIVES
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And Always Remember... "From Newtonian Mechanics, Through Quantum Theory, Without Knowledge of Physics, Life Would Be Dreary." |
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