Simple Harmonic Motion UNIT PACKET

Student: Mr. Khalilian AP Physics Due Date:

SIMPLE HARMONIC MOTION UNIT PACKET INCLUDES GUIDES FOR NOTES, HW, AND CW Student: Mr. Khalilian AP Physics Due Date: UNIT 6 AGENDA

______Day 1: Introduction to Simple Harmonic Motion  Do Now, p. 19  Simple Harmonic Motion Introduction Lab, p. 3  Note-taking

Expectations  HW: Read Sections 14.1 – 14.4. Take notes in the format of the Notes Outline on p. 4

______Day 2: Spring SHM  Do Now, p. 18  Notes Review  Simple Harmonic Motion I and II, p. 6-7  HW: Read sections 14.5 – 14.6. Take notes in the format of the Notes Outline on p. 4; Finish CW if necessary

______Day 3: Pendulum SHM  Do Now, p. 17  Simple Harmonic Motion III, p. 8  HW: Pendulums – Maximum Speed of the Bob, p. 9

______Day 4: Final Exam Review  Final Exam, p. 15  Introduction to the SHM Lab  HW: SHM Practice and Review, p. 11: #1-4; Answer each question on a sheet of notebook paper, with an explanation.

______Day 5: SHM Lab, Day 1  SHM HW Check  End of Unit Lab, Day 1, p. 10  HW: SHM Practice and Review, p. 11-12: #5 – 8; Answer each question on a sheet of notebook paper, with an explanation.

______Day 6: SHM Lab, Day 2  SHM HW Check  End of Unit Lab, Day 2, p. 10  HW: SHM Practice and Review, p. 13: #9; Show your thought process clearly.

______Day 7: SHM Review  Do Now, p. 16 Student: Mr. Khalilian AP Physics Due Date:  SHM HW Check  SHM Practice and Review, p. 14: #10.  HW: Study for your exam.

______Day 8: SHM Exam

Recommended Book Problems for the Unit: p. 464: 23; p. 465: #1, 5, 7a, 19; p. 466: 25, 29 Student: Mr. Khalilian AP Physics Due Date: SIMPLE HARMONIC MOTION INTRODUCTION LAB

In order to investigate simple harmonic motion of a mass on a spring, graphs of force, position, and velocity as a function of time were made. A spring was hung from a force probe at the top of a support rod, as is show in Figure 1. A 100 g mass was attached to the bottom of the spring. Students created simple harmonic motion by pulling the mass approximately 5 cm down and releasing it. As the mass oscillated up and down, students used LoggerPro to create a graph of Force as a function of time. This procedure was repeated until the students had a clean graph. The vertical axis was adjusted, where needed, by clicking a number on the axis, typing the new one, and hitting return. That graph was used to answer the questions in Part I.

Figure 1

To analyze motion graphs of the mass on the spring, students opened a new file in LoggerPro. In the folder “_Physics with Vernier,” they opened the file entitled “15 Simple Harmonic Motion.” Students then repeated the procedure above until they had clean graphs. These graphs were used to answer the questions in Part II.

Part I. 1. Where is the mass in its motion when the force is at a maximum? ______(e.g. top, bottom, middle? On the way up? On the way down?) 2. Where is the mass in its motion when the force is at a minimum? ______3. Clearly sketch the graph below, labeling your axes. Use your answers to #1-2 to label the locations of the mass at those points.

Part II 4. Where is the mass in its motion when the position is at a maximum? ______(e.g. top, bottom, middle? On the way up? On the way down?) 5. Where is the mass in its motion when the position graph is at a minimum? ______6. Where is the mass in its motion when the velocity is at a maximum? ______7. Where is the mass in its motion when the velocity is crossing the horizontal axis? ______8. Clearly sketch both the position and velocity graphs below. Use your answers to #4-7 to label the locations of the mass at those points. Student: Mr. Khalilian AP Physics Due Date: NOTES OUTLINE Notes for this unit should be taken in your notebook. You may use either Cornell or Outline style. Both are presented below. (P = paragraph of that section) Cornell Style Outline Style 14.1 Equilibrium and  P1 Connection: I. 14.1 Equilibrium and Oscillation Oscillation  P1 Concept: a. Intro section  P2 Connection: a.i. P1 Connection:  P2 Concept: a.ii. P1 Concept: a.iii. P2 Connection: Frequency and Period  Rewrite “In general, a.iv. P2 Concept: …” from P2 in own b. Frequency and Period words. b.i. Rewrite “In general,…” from  Connection: P2 in own words.  Concept: b.ii. Connection: Oscillatory Motion b.iii. Concept:  Translate paragraph  c. Oscillatory Motion Picture  Graph c.i. Translate paragraph  Picture  P1 Connection:  Graph  P1 Concept: c.ii. P1 Connection:  Examples of SHM c.iii. P1 Concept: Concept: c.iv. Examples of SHM Concept: II. 14.2 Linear Restoring Forces and SHM 14.2 Linear Restoring  Connection: a. Intro section Forces and SHM  Concept: a.i. Connection: a.ii. Concept: Motion of a Mass on a  Connection: b. Motion of a Mass on a Spring Spring  Concept: b.i. Connection: b.ii. Concept: Vertical Mass on a Spring  Answer: How is this c. Vertical Mass on a Spring section different from the c.i. Answer: How is this section last section? different from the last section?  End of p. 441 c.ii. End of p. 441 Concept: Concept: c.iii. End of section Connection:  End of section c.iv. End of section Concept: Connection: d. The Pendulum  End of section d.i. Connection: The Pendulum Concept: d.ii. Concept: III. 14.4 Energy in Simple Harmonic Motion  Connection: a. Intro section SKIP 14.3  Concept: a.i. Translate P2  Figure 14.14 a.ii. Connection: 14.4 Energy in Simple a.iii. Concept: Harmonic Motion b. Finding the Frequency for Simple  Translate P2  Figure Harmonic Motion 14.14 b.i. Connection:  Connection: b.ii. Concept:  Concept: IV. 14.5 Pendulum Motion a.i. End of p.453 Connection: Student: Mr. Khalilian AP Physics Due Date: Finding the Frequency for a.ii. End of p. 453 Concept: Simple Harmonic Motion  Connection: V. 14.6 Damped Oscillations  Concept: a.i. P1-3 Connection: SKIP p. 451 – 452 a.ii. P1-3 Concept:

14.5 Pendulum Motion  End of p. 453 Connection:  End of p. 453 SKIP Physical Pendulums Concept: and Locomotion

14.6 Damped Oscillations

 P1-3 Connection: SKIP p. 456 – 462  P1-3 Concept: Student: Mr. Khalilian AP Physics Due Date: SIMPLE HARMONIC MOTION I – HORIZONTAL SPRING A spring (k = 200 N/m) is mounted horizontally on a frictionless surface. One end of the spring is attached to a fixed wall, and the other end is attached to a block of mass 2 kg. The block is pulled aside to a distance of 0.04 m and released from rest.

1. Calculate the period and frequency of oscillation of the mass.

2. Compute the maximum velocity of the vibrating mass.

3. Calculate the maximum acceleration.

4. Compute the velocity and acceleration when the body has moved halfway into the center from its initial position.

5. Graph the a) total, b) kinetic, and c) elastic potential energy of the system as a function of position. Graph all of these on the same axis. What is the relationship between the kinetic and potential energies? Student: Mr. Khalilian AP Physics Due Date:

SIMPLE HARMONIC MOTION II – VERTICAL SPRING A body of mass 5kg is suspended by a spring which stretches 0.1 m when the body is attached. The body is then displaced downward an additional 0.05 m and released from rest. 1. Calculate the period, frequency, and amplitude of the motion.

2. What is the new equilibrium position of this mass?

3. What is the maximum speed of this mass?

4. Calculate the energy of the system… a. when the mass is at its lowest point. b. when the mass is at the equilibrium position.

5. Graph the position, velocity, and acceleration of the mass as a function of time for the first two complete oscillations. Student: Mr. Khalilian AP Physics Due Date:

SIMPLE HARMONIC MOTION III – PENDULUM 1. A simple pendulum 4 m long swings with an amplitude of 0.20 m. a. Compute the period and frequency of the pendulum.

b. Compute the linear velocity at its lowest point.

c. Compute the linear acceleration of the pendulum at either end of its path.

d. If the pendulum is shortened to a length of 2.0 m, what happens to your answers in (a), (b), and (c)? Student: Mr. Khalilian AP Physics Due Date: 2. A certain pendulum has a period on Earth of 2.0 seconds. What is the period on the moon, where the acceleration due to gravity is roughly 1/6th of its value on Earth?

3. A grandfather clock, which keeps time on Earth by means of a simple pendulum, is taken to the moon. a. If the clock is operated on the moon in the same fashion, will the clock run slow or fast?

b. How much time passes on Earth while the hands of the grandfather clock on the moon move through 24 hours? Student: Mr. Khalilian AP Physics Due Date: Student: Mr. Khalilian AP Physics Due Date:

Due Date: Student Name: rd February 3 Course Name: AP Physics at 11:59 Period: 3 pm1 Teacher Name: Mr. Khalilian Assignment Title: End of Unit Lab: Hooke’s Law and Simple Harmonic Motion for Elastic Materials Assignment Your job is to use simple harmonic motion to create a claim which answers the Summary: following question: What are the spring constants for a given rubber band and spring?

You will have 2 class days. The recommended course of action is:  Decide how you want to create your collisions and what measuring devices you want.  Make measurements using your collisions and decide how to account for experimental error.  Create your claim. Format: 5 – 10  MLA Format heading  Section headers for each section from the deductions  Title: The SHM Lab Write-Up rubric. possible for any  Submitted to Turnitin.com.  You do NOT need to submit a paper copy. infraction (Without this, the lab report is a 0.) Procedure and Before writing your paper, you should: Helpful Hints: 1. Determine what measurements need to be taken. 2. Take the necessary measurements. 3. Write the lab report, carefully.

1 Computer errors reported the day before will not be excused. Start early. Student: Mr. Khalilian AP Physics Due Date: SHM PRACTICE AND REVIEW

1. A mass m is attached to a spring with a spring constant k. If the mass is set into simple harmonic motion by a displacement d from its equilibrium position, what would be the speed, v, of the mass when it returns to the equilibrium position?

Questions 2-3: A block oscillates without friction on the end of a spring as shown. The minimum and maximum lengths of the spring as it oscillates are, respectively, xmin and xmax. The graphs below can represent quantities associated with the oscillation as functions of the length x of the spring.

(A) (B) (C) (D)

2. Which graph can represent the total mechanical energy of the blockspring system as a function of x ? (A) A (B) B (C) C (D) D

3. Which graph can represent the kinetic energy of the block as a function of x ? (A) A (B) B (C) C (D) D

4. An object is attached to a spring and oscillates with amplitude A and period T, as represented on the graph. The nature of the velocity v and acceleration a of the object at time T/4 is best represented by which of the following?

(A) v > 0, a > 0 (B) v > 0, a < 0 (C) v > 0, a = 0 (D) v = 0, a < 0

Questions 5-6

A sphere of mass m1, which is attached to a spring, is displaced downward from its equilibrium position as shown above left and released from rest. A sphere of mass m2, which is suspended from a string of length L, is displaced to the right as shown above right and released from rest so that it swings as a simple pendulum with small amplitude. Assume that both spheres undergo simple harmonic motion Student: Mr. Khalilian AP Physics Due Date: 5. Which of the following is true for both spheres? (A) The maximum kinetic energy is attained as the sphere passes through its equilibrium position (B) The maximum kinetic energy is attained as the sphere reaches its point of release. (C) The minimum gravitational potential energy is attained as the sphere passes through its equilibrium position. (D) The maximum gravitational potential energy is attained when the sphere reaches its point of release. (E) The maximum total energy is attained only as the sphere passes through its equilibrium position.

6. If both spheres have the same period of oscillation, which of the following is an expression for the spring constant? (A) L / m1g (B) g / m2L (C) m2g / L (D) m1g / L

Question 7 refers to the graph below of the displacement x versus time t for a particle in simple harmonic motion.

7. Which of the following graphs shows the kinetic energy K of the particle as a function of time t for one cycle of motion?

8. An ideal massless spring is fixed to the wall at one end, as shown above. A block of mass M attached to the other end of the spring oscillates with amplitude A on a frictionless, horizontal surface. The maximum speed of the block is vm. The force constant of the spring is (A) (B) (C) (D) Student: Mr. Khalilian AP Physics Due Date:

9. 1983B2. A block of mass M is resting on a horizontal, frictionless table and is attached as shown above to a relaxed spring of spring constant k. A second block of mass 2M and initial speed vo collides with and sticks to the first block Develop expressions for the following quantities in terms of M, k, and vo a. v, the speed of the blocks immediately after impact

b. x, the maximum distance the spring is compressed

c. T, the period of the subsequent simple harmonic motion Student: Mr. Khalilian AP Physics Due Date:

10. 2005B2B. A simple pendulum consists of a bob of mass 0.085 kg attached to a string of length 1.5 m. The pendulum is raised to point Q, which is 0.08 m above its lowest position, and released so that it oscillates with small amplitude θ between the points P and Q as shown below.

(a) In the space below, draw free-body diagrams showing and labeling the forces acting on the bob in each of the situations described. i. When it is at point P ii. When it is in motion at its lowest position

(b) Calculate the speed v of the bob at its lowest position.

(c) Calculate the tension in the string when the bob is passing through its lowest position.

(d) Describe one modification that could be made to double the period of oscillation. Student: Mr. Khalilian AP Physics Due Date:

FINALS REVIEW 1. 2009B1. (15 points) In an experiment, students are to calculate the spring constant k of a vertical spring in a small jumping toy that initially rests on a table. When the spring in the toy is compressed a distance x from its uncompressed length L0 and the toy is released, the top of the toy rises to a maximum height h above the point of maximum compression. The students repeat the experiment several times, measuring h with objects of various masses taped to the top of the toy so that the combined mass of the toy and added objects is m. The bottom of the toy and the spring each have negligible mass compared to the top of the toy and the objects taped to it. (a) Derive an expression for the height h in terms of m, x, k, and fundamental constants.

With the spring compressed a distance x = 0.020 m in each trial, the students obtained the following data for different values of m.

(b) i. What quantities should be graphed so that the slope of a best-fit straight line through the data points can be used to calculate the spring constant k ?

ii. Fill in one or both of the blank columns in the table with calculated values of your quantities, including units. (c) On the axes below, plot your data and draw a best-fit straight line. Label the axes and indicate the scale.

(d) Using your best-fit line, calculate the numerical value of the spring constant. Student: Mr. Khalilian AP Physics Due Date:

(e) Describe a procedure for measuring the height h in the experiment, given that the toy is only momentarily at that maximum height.

FOUR IN FIVE 1. A person throws a marble straight up into the air, releasing it a short height above the ground and catching it at that same height. If air resistance is negligible, which of the following graphs of position y versus time t is correct for the motion of the marble as it Questions 3-4 goes up and then comes down? An object is thrown with an initial speed v near the surface of Earth. Assume that air resistance is negligible and the gravitational field is constant.

3. If the object is thrown vertically upward, the direction and magnitude of its acceleration while it is in the air is (A) upward and decreasing (B) upward and constant (C) downward and decreasing (D) downward and increasing (E) downward and constant

4. If the object is thrown horizontally, the direction and magnitude of its acceleration while it is in the air is (A) upward and decreasing 2. A rocket lifts a payload upward from the (B) upward and constant surface of Earth. The radius of Earth is R, (C) downward and decreasing and the weight of the payload on the surface (D) downward and increasing of Earth is W. The force of Earth's gravity (E) downward and constant on the payload is W/2 when the rocket's distance from the center of Earth is (A) R (B) (C) 2R (D)

(E) 4R Student: Mr. Khalilian AP Physics Due Date:

FOUR IN FIVE 1. Inertia can be best described as the (A) force that keeps an object in motion with constant velocity (B) force that keeps an object at rest (C) force that overcomes friction (D) property responsible for an object's resistance to changes in motion (E) property responsible for slowing down an object

Questions 2-3

A block released from rest at position A slides with negligible friction down an inclined track, around a vertical loop, and then along a horizontal portion of the track, as shown above. The block never leaves the track.

2. After the block is released, in which of the following sequences of positions is the speed of the block ordered from fastest to slowest? (A) B C D E (B) B E C D (C) D C E B (D) E B C D (E) E D C B

3. Which of the following is true of the net force on the block when it is at position C? (A) It is directed vertically downward only. (B) It is directed vertically upward only. (C) It is directed to the left only. (D) It is directed to the right only. (E) It has components both to the left and vertically downward.

4. The gravitational potential energy and the kinetic energy of the block are most nearly equal at which position? (Consider the potential energy to be zero at position B.) (A) A (B) B (C) C (D) D (E) E Student: Mr. Khalilian AP Physics Due Date:

FOUR IN FIVE 1. Balls 1 and 2 are each thrown horizontally 14. from the same height above level ground, but ball 2 has a greater initial velocity after leaving the thrower's hand. If air resistance is negligible, how do the accelerations of the balls and the times it takes them to hit the ground compare? Acceleration a Time to Hit Ground 15. A railroad car of mass m is moving with (A) Greater for ball 2Greater for ball 2 speed v when it collides with and connects (B) Greater for ball 2 Equal to a second railroad car of mass 3m, initially (C) Equal Greater for ball 2 at rest, as shown above. How do the speed (D) Equal Less for ball 2 and kinetic energy of the connected cars (E) Equal Equal compare to those of the single car of mass m 2. before the collision? 3. Two objects, X and Y, accelerate from rest Speed Kinetic Energy with the same constant acceleration. Object (A) Less Less X accelerates for twice the time as object Y. (B) Less The same Which of the following is true of these (C)The same Less objects at the end of their respective periods (D) The same The same of acceleration? (E) Greater The same (A) Object X is moving at the same speed as 16. A ball of mass m and momentum p has object Y. kinetic energy equal to which of the (B) Object X is moving four times faster following? than object Y. (A) (C) Object X has traveled the same distance (B) as object Y. (C) (D) Object X has traveled twice as far as (D) object Y. (E) (E) Object X has traveled four times as far as 17. object Y. 18. 4. 19. 5. 20. 6. 21. 7. 22. 8. 23. 9. 24. 10. 25. 11. 26. 12. 27. 13. Student: Mr. Khalilian AP Physics Due Date: 28. 29. 30. 31. 32. FOUR IN FIVE 33. 1. A simple pendulum consisting of a small 40. Questions 3-4 object of mass m attached to a string of 41. length has a period T. A pendulum with which of the following combinations of object mass and string length will also have period T ? Object Mass String Length (A) m/2 (B) m /4 (C) 42. A block of mass 3 kg slides along a (D) 2m 4 horizontal surface that has negligible friction (E) 4m 2 except for one section, as shown above. The 34. block arrives at the rough section with a speed 35. of 5 m/s and leaves it 0.5s later with a speed of 36. 3 m/s. 37. 43. 38. 3. What is the magnitude of the work done by 2. A planet of mass m orbits a star of mass M, the frictional force exerted on the block by where m<