Slide 1 / 88 Slide 2 / 88 AP Physics 1 Simple Harmonic Motion 2015-12-21 www.njctl.org Slide 3 / 88 Slide 4 / 88 Table of Contents Click on the topic to go to that section Period and Frequency · SHM and UCM · Spring Pendulum · Simple Pendulum Period and Frequency · Sinusoidal Nature of SHM · Return to Table of Contents Slide 5 / 88 Slide 6 / 88 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.
Slide 7 / 88 Slide 7 (Answer) / 88 1 1 If it takes 50 seconds for an object to travel around If it takes 50 seconds for an object to travel around a circle 5 times, what is the period of its motion? a circle 5 times, what is the period of its motion? Answer [This object is a pull tab] Slide 8 / 88 Slide 8 (Answer) / 88 2 If an object is traveling in circular motion and its 2 If an object is traveling in circular motion and its period is 7.0s, how long will it take it to make 8 period is 7.0s, how long will it take it to make 8 complete revolutions? complete revolutions? Answer [This object is a pull tab] Slide 9 / 88 Slide 10 / 88 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 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,
Slide 10 (Answer) / 88 Slide 11 / 88 3 An object travels around a circle 50 times in ten 4 If an object is traveling in circular motion with a seconds, what is the frequency (in Hz) of its frequency of 7.0 Hz, how many revolutions will it motion? make in 20s? Answer [This object is a pull tab] Slide 11 (Answer) / 88 Slide 12 / 88 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] Slide 13 / 88 Slide 13 (Answer) / 88 5 An object has a period of 4.0s, what is the frequency of its motion (in Hertz)?
Slide 14 / 88 Slide 14 (Answer) / 88 6 An object is revolving with a frequency of 8.0 Hz, what is its period (in seconds)? Slide 15 / 88 Slide 16 / 88 7 An object is in circular motion. The radius of its Velocity motion is 2.0 m and its period is 5.0s. What is its velocity? Also, recall from Uniform Circular Motion.... and Slide 16 (Answer) / 88 Slide 17 / 88 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?
Slide 17 (Answer) / 88 Slide 18 / 88 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? Answer SHM and UCM [This object is a pull tab] Return to Table of Contents Slide 19 / 88 Slide 20 / 88 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 In UCM, an object completes one circle, or the system? cycle, in every T seconds. That means it returns to its starting position after T seconds. In Simple Harmonic Motion, the object does 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. Slide 20 (Answer) / 88 Slide 21 / 88 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 travel in 4.0s?
Slide 21 (Answer) / 88 Slide 22 / 88 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 amplitude of its motion is 0.50m. How far does the the amplitude of its motion is 0.50m. How far does mass travel in 6.0s? the mass travel in 4.0s? Answer [This object is a pull tab] Slide 22 (Answer) / 88 Slide 23 / 88 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 Slide 24 / 88 Slide 25 / 88 · 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 The force exerted by the spring depends on the UCM) 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)
Slide 26 / 88 Slide 26 (Answer) / 88 12 A spring whose spring constant is 20N/m is 12 A spring whose spring constant is 20N/m is stretched 0.20m from equilibrium; what is the stretched 0.20m from equilibrium; what is the magnitude of the force exerted by the spring? magnitude of the force exerted by the spring? Answer [This object is a pull tab] Slide 27 / 88 Slide 27 (Answer) / 88 13 A spring whose spring constant is 150 N/m exerts a 13 A spring whose spring constant is 150 N/m exerts a force of 30N on the mass in a mass-spring system. force of 30N on the mass in a mass-spring system. How far is the mass from equilibrium? How far is the mass from equilibrium? Answer [This object is a pull tab] Slide 28 / 88 Slide 28 (Answer) / 88 14 A spring exerts a force of 50N on the mass in a 14 A spring exerts a force of 50N on the mass in a mass-spring system when it is 2.0m from mass-spring system when it is 2.0m from equilibrium. What is the spring's spring constant? equilibrium. What is the spring's spring constant? Answer [This object is a pull tab]
Slide 29 / 88 Slide 30 / 88 Simple Harmonic Motion Simple Harmonic Motion The maximum force exerted on the mass is when the spring is most stretched or compressed (x = -A or +A): F = -kA (when x = -A or +A) The minus sign indicates that it is a restoring force – it is directed to restore the mass to its equilibrium position. The minimum force exerted on the mass is when the spring is not stretched at all (x = 0) k is the spring constant F = 0 (when x = 0) The force is not constant, so the acceleration is not constant either Slide 31 / 88 Slide 32 / 88 Simple Harmonic Motion Simple Harmonic Motion When the spring is at equilibrium and When the spring is all the way compressed: heading in the positive direction: The displacement is at the negative amplitude. The displacement is zero. · · The force of the spring is in the positive direction. The force of the spring is zero. · · The acceleration is in the positive direction. The acceleration is zero. · · The velocity is zero. The velocity is positive and at a maximum. · · Slide 33 / 88 Slide 34 / 88 Simple Harmonic Motion Simple Harmonic Motion When the spring is at equilibrium and heading When the spring is all the way stretched: in the negative direction: The displacement is at the positive amplitude. · The displacement is zero. · The force of the spring is in the negative direction. · The force of the spring is zero. · The acceleration is in the negative direction. · The acceleration is zero. · The velocity is zero. · The velocity is negative and at a maximum. ·
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