Slide 1 / 70 Slide 2 / 70 New Jersey Center for Teaching and Learning Progressive Science Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written Simple Harmonic Motion permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org www.njctl.org Slide 3 / 70 Slide 4 / 70 Table of Contents Click on the topic to go to that section Period and Frequency Period and Frequency · SHM and UCM · Spring Pendulum · Simple Pendulum · Return to Table of Contents http:/ / njc.tl/ he Slide 5 / 70 Slide 6 / 70 Period SHM and Circular Motion The time it takes for an object to complete one trip around There is a deep connection between Simple Harmonic a circular path is called its Period. Motion (SHM) and Uniform Circular Motion (UCM). The symbol for Period is "T" Simple Harmonic Motion can be thought of as a one- dimensional projection of Uniform Circular Motion. Periods are measured in units of time; we will usually use seconds (s). All the ideas we learned for UCM, can be applied to SHM...we don't have to reinvent them. Often we are given the time (t) it takes for an object to make a number of trips (n) around a circular path. In that case, So, let's review circular motion first, and then extend what we know to SHM. Click here to see how circular motion relates to simple harmonic motion. http:/ / njc.tl/ he http:/ / njc.tl/ he
Slide 7 / 70 Slide 7 (Answer) / 70 1 If it takes 50 seconds for an object to travel around a circle 5 1 If it takes 50 seconds for an object to travel around a circle 5 times, what is the period of its motion? times, what is the period of its motion? Answer Answer [This object is a pull tab] http:/ / njc.tl/ hf http:/ / njc.tl/ hf Slide 8 / 70 Slide 8 (Answer) / 70 2 2 If an object is traveling in circular motion and its period is If an object is traveling in circular motion and its period is 7.0s, how long will it take it to make 8 complete revolutions? 7.0s, how long will it take it to make 8 complete revolutions? Answer Answer [This object is a pull tab] http:/ / njc.tl/ hg http:/ / njc.tl/ hg Slide 9 / 70 Slide 10 / 70 Frequency 3 An object travels around a circle 50 times in ten seconds, what is the frequency (in Hz) of its motion? The number of revolutions that an object completes in a given amount of time is called the frequency of its motion. The symbol for frequency is "f" Periods are measured in units of revolutions per unit time; we will usually use 1/seconds (s -1 ). Another name for s -1 is Hertz (Hz). Frequency can also be measured in revolutions per Answer minute (rpm), etc. Often we are given the time (t) it takes for an object to make a number of revolutions (n). In that case, http:/ / njc.tl/ hh http:/ / njc.tl/ hi
Slide 10 (Answer) / 70 Slide 11 / 70 3 An object travels around a circle 50 times in ten seconds, 4 If an object is traveling in circular motion with a frequency of what is the frequency (in Hz) of its motion? 7.0 Hz, how many revolutions will it make in 20s? Answer Answer [This object is a pull tab] http:/ / njc.tl/ hi http:/ / njc.tl/ hj Slide 11 (Answer) / 70 Slide 12 / 70 Period and Frequency 4 If an object is traveling in circular motion with a frequency of 7.0 Hz, how many revolutions will it make in 20s? Since and then and Answer [This object is a pull tab] http:/ / njc.tl/ hj http:/ / njc.tl/ hk Slide 13 / 70 Slide 13 (Answer) / 70 5 An object has a period of 4.0s, what is the frequency of its 5 An object has a period of 4.0s, what is the frequency of its motion (in Hertz)? motion (in Hertz)? Answer Answer [This object is a pull tab] http:/ / njc.tl/ hl http:/ / njc.tl/ hl
Slide 14 / 70 Slide 14 (Answer) / 70 6 An object is revolving with a frequency of 8.0 Hz, what is its 6 An object is revolving with a frequency of 8.0 Hz, what is its period (in seconds)? period (in seconds)? Answer Answer [This object is a pull tab] http:/ / njc.tl/ hm http:/ / njc.tl/ hm Slide 15 / 70 Slide 16 / 70 7 An object is in circular motion. The radius of its motion is 2.0 Velocity m and its period is 5.0s. What is its velocity? Also, recall from Uniform Circular Motion.... Answer and http:/ / njc.tl/ hn http:/ / njc.tl/ ho Slide 16 (Answer) / 70 Slide 17 / 70 7 An object is in circular motion. The radius of its motion is 2.0 8 An object is in circular motion. The radius of its motion is 2.0 m and its period is 5.0s. What is its velocity? m and its frequency is 8.0 Hz. What is its velocity? Answer Answer [This object is a pull tab] http:/ / njc.tl/ ho http:/ / njc.tl/ hp
Slide 17 (Answer) / 70 Slide 18 / 70 8 An object is in circular motion. The radius of its motion is 2.0 m and its frequency is 8.0 Hz. What is its velocity? SHM and UCM Answer [This object is a pull tab] Return to Table of Contents http:/ / njc.tl/ hp http:/ / njc.tl/ hq Slide 19 / 70 Slide 20 / 70 SHM and Circular Motion 9 It takes 4.0s for a system to complete one cycle of simple harmonic motion. What is the frequency of the system? In UCM, an object completes one circle, or cycle, in every T seconds. That means it returns to its starting position after T seconds. In Simple Harmonic Motion, the object does Answer not go in a circle, but it also returns to its starting position in T seconds. Any motion that repeats over and over again, always returning to the same position is called " periodic". Click here to see how simple harmonic motion relates to circular motion. http:/ / njc.tl/ hq http:/ / njc.tl/ hr Slide 20 (Answer) / 70 Slide 21 / 70 9 It takes 4.0s for a system to complete one cycle of simple 10 The period of a mass-spring system is 4.0s and the harmonic motion. What is the frequency of the system? amplitude of its motion is 0.50m. How far does the mass travel in 4.0s? Answer Answer [This object is a pull tab] http:/ / njc.tl/ hr http:/ / njc.tl/ hs
Slide 21 (Answer) / 70 Slide 22 / 70 11 The period of a mass-spring system is 4.0s and the 10 The period of a mass-spring system is 4.0s and the amplitude of its motion is 0.50m. How far does the mass amplitude of its motion is 0.50m. How far does the mass travel in 6.0s? travel in 4.0s? Answer Answer [This object is a pull tab] http:/ / njc.tl/ hs http:/ / njc.tl/ ht Slide 22 (Answer) / 70 Slide 23 / 70 11 The period of a mass-spring system is 4.0s and the amplitude of its motion is 0.50m. How far does the mass travel in 6.0s? Spring Pendulum Answer [This object is a pull tab] Return to Table of Contents http:/ / njc.tl/ ht http:/ / njc.tl/ hu Slide 24 / 70 Slide 25 / 70 · Displacement is measured from the Simple Harmonic Motion equilibrium point · Amplitude is the maximum There is a point where the spring is neither stretched nor displacement (equivalent to the radius, r, in compressed; this is the equilibrium position. UCM). We measure displacement from that point · A cycle is a full to-and-fro motion (the (x = 0 on the previous figure). same as one trip around the circle in UCM) The force exerted by the spring depends on the displacement: · Period is the time required to complete one cycle (the same as period in UCM) · Frequency is the number of cycles completed per second (the same as frequency in UCM) http:/ / njc.tl/ hu http:/ / njc.tl/ hu
Slide 26 / 70 Slide 26 (Answer) / 70 12 12 A spring whose spring constant is 20N/m is stretched 0.20m A spring whose spring constant is 20N/m is stretched 0.20m from equilibrium; what is the magnitude of the force exerted from equilibrium; what is the magnitude of the force exerted by the spring? by the spring? Answer Answer [This object is a pull tab] http:/ / njc.tl/ hu http:/ / njc.tl/ hu Slide 27 / 70 Slide 27 (Answer) / 70 13 A spring whose spring constant is 150 N/m exerts a force of 13 A spring whose spring constant is 150 N/m exerts a force of 30N on the mass in a mass-spring system. How far is the 30N on the mass in a mass-spring system. How far is the mass from equilibrium? mass from equilibrium? Answer Answer [This object is a pull tab] Slide 28 / 70 Slide 28 (Answer) / 70 14 A spring exerts a force of 50N on the mass in a mass- 14 A spring exerts a force of 50N on the mass in a mass- spring system when it is 2.0m from equilibrium. What is spring system when it is 2.0m from equilibrium. What is the spring's spring constant? the spring's spring constant? Answer Answer [This object is a pull tab]
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