Slide 1 / 67 Simple Harmonic Motion (SHM)
Slide 2 / 67 SHM and Uniform Circular Motion There is a deep connection between Simple Harmonic Motion (SHM) and Uniform Circular Motion (UCM). Simple Harmonic Motion can be thought of as a one- dimensional projection of Uniform Circular Motion. http://www.physics.uoguelph.ca/tutorials/shm/phase0.html
Slide 3 / 67 SHM and Uniform Circular Motion Imagine we have a ball moving in Uniform Circular Motion and we shine a light on it. Now at the end where the image is cast we have a spring and mass system which will oscillate with the same period. This experiment will show that the shadow of the ball in UCM will match that of the mass in SHM. -x x -x x 0 0 -x 0 x t=0 T T/2
Slide 4 / 67 Period Period is defined as the time it takes for an object to complete one trip around a circular path, or to complete one oscillation. Period is represented by "T" The period of a system is normally measured in seconds (s). Usually we are given the total time it takes for a system to rotate around its central axis n number of times. To find the period we divide the total time by the number of times we completed one revolution.
Slide 5 / 67 1 If it takes 50 seconds for an object to travel around a circle 5 times, what is the period of its motion? 1 s A 5 s B C 10 s D 25 s E 50 s
Slide 6 / 67 2 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? A 7/8 s B 8/7 s C 48 s D 56 s E 112 s
Slide 7 / 67 Frequency 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" Frequency is 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 8 / 67 3 An object travels around a circle 50 times in 10s, what is the frequency (in Hz) of its motion? A 0.2 Hz B 1 Hz C 5 Hz D 25 Hz E 500 Hz
Slide 9 / 67 4 If an object is traveling in circular motion with a frequency of 7.0 Hz, how many revolutions will it make in 20s? A 7/20 B 20/7 C 7 D 140 E 280
Slide 10 / 67 Relating Period and Frequency Before we defined the values of Period and Frequency. Now when we compare them side by side we see that each one is the reciprocal of the other.
Slide 11 / 67 5 An object has a period of 4.0s, what is the frequency of its motion (in Hertz)? A 0.25 Hz B 0.5 Hz C 1 Hz D 2 Hz E 4 Hz
Slide 12 / 67 6 An object is revolving with a frequency of 8.0 Hz, what is its period (in seconds)? A 0.125 s B 0.25 s C 1 s D 4 s E 8 s
Slide 13 / 67 Velocity Also, recall from Uniform Circular Motion.... and http:/ / njc.tl/ hn
Slide 14 / 67 7 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? A B C D E
Slide 15 / 67 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? A B C D E
Slide 16 / 67 · Displacement is measured from the equilibrium point · Amplitude is the maximum displacement (equivalent to the radius, r, in UCM). · A cycle is a full to-and-fro motion (the same as one trip around the circle in UCM) · 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 frequence in UCM)
Slide 17 / 67 9 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? A 0 m B 0.5 m C 0.75 m D 1.5 m E 2.0 m
Slide 18 / 67 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 6.0s? A 0.5 m B 0.75 m C 1.0 m D 3.0 m E 2.5 m
Slide 19 / 67 Simple Harmonic Motion Hooke's Law There is a point where the spring is neither stretched nor compressed; this is the equilibrium position. We measure displacement from that point (x = 0 on the previous figure). The force exerted by the spring depends on the displacement:
Slide 20 / 67 11 A spring whose spring constant is 20N/m is stretched 0.20m from equilibrium; what is the magnitude of the force exerted by the spring? A 0.4 N B 2 N C 4 N D 8 N E 10 N
Slide 21 / 67 12 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 mass from equilibrium? A 0.2 m B 0.3 m C 1 m D 2 m E 3 m
Slide 22 / 67 13 A spring exerts a force of 50N on the mass in a mass-spring system when it is 2.0m from equilibrium. What is the spring's spring constant? A 2.5 N/m B 5 N/m C 15 N/m D 25 N/m E 50 N/m
Slide 23 / 67 Simple Harmonic Motion In Hooke's Law the negative sign indicated that it is a restoring force, meaning the force wants to bring the system back to its original position. k is the spring constant The force is not constant, therefore the acceleration is not constant either.
Slide 24 / 67 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 minimum force exerted on the mass is when the spring is not stretched at all (x = 0) F = 0 (when x = 0)
Slide 25 / 67 14 At which location(s) is the magnitude of the force on the mass in a mass-spring system a maximum? A x=A B x=0 C x=-A D A & C E All of the above
Slide 26 / 67 15 At which location(s) is the magnitude of the force on the mass in a mass-spring system a minimum? A x=A B x=0 C x=-A D A & C E all of the above
Slide 27 / 67 Gravity does not effect the mass-spring system If the spring is hung vertically, the only change is in the equilibrium position, which is at the point where the spring force equals the gravitational force. The effect of gravity is cancelled out by changing to this new equilibrium position.
Slide 28 / 67 Displacement in SHM If we return to the relationship between Uniform Circular Motion and Simple Harmonic Motion we can explain the displacement for the spring system. Which can also be written as:
Slide 29 / 67 Velocity in SHM As stated in previous chapters velocity is the rate of change of position, so by taking the derivative of the position equation for SHM we can calculate the velocity at a specific point in time. Acceleration in SHM Acceleration is defined at the rate of change of velocity, therefore by taking the derivative of the velocity equation we just solved for we can find the acceleration of the system in SHM at any point in time.
Slide 30 / 67 Graphical Representations t t t
Slide 31 / 67 16 What is the acceleration when x = 0? a (acceleration) v (velocity) x (displacement) T/4 T/2 3T/4 T A a<0 B a=0 C a>0 D It varies
Slide 32 / 67 17 What is the acceleration when x=A? a (acceleration) v (velocity) x (displacement) T/4 T/2 3T/4 T A a<0 B a=0 C a>0 D It varies
Slide 33 / 67 18 What is the acceleration when x=-A? a (acceleration) v (velocity) x (displacement) T/4 T/2 3T/4 T A a<0 B a=0 C a>0 D It varies
Slide 34 / 67 19 What is the velocity when x=0? a (acceleration) v (velocity) x (displacement) T/4 T/2 3T/4 T A v<0 B v=0 C v>0 D A or C
Slide 35 / 67 20 What is the velocity when x=A? a (acceleration) v (velocity) x (displacement) T/4 T/2 3T/4 T A v<0 B v=0 C v>0 D A or C
Slide 36 / 67 21 Where is the mass when acceleration is at a maximum? a (acceleration) v (velocity) x (displacement) T/4 T/2 3T/4 T A x=A B x=0 C x=-A D A or C
Slide 37 / 67 22 Where is the mass when velocity is at maximum? a (acceleration) v (velocity) x (displacement) T/4 T/2 3T/4 T A x=A B x=0 C x=-A D A or C
Slide 38 / 67 23 Which of the following represents the position as a function of time? a (acceleration) v (velocity) x (displacement) T/4 T/2 3T/4 T A x=4 cos(2t) C x=8 cos(2t) B x=2 cos(2t) D x=2 sin(2t)
Slide 39 / 67 24 Which of the following represents the velocity as a function of time? a (acceleration) v (velocity) x (displacement) T/4 T/2 3T/4 T C v=-4 sin(2t) A v= -12 sin(2t) D v=-4 cos(2t) B v=-12 cos(2t)
Slide 40 / 67 25 Which of the follwing represents the acceleration as a function of time? a (acceleration) v (velocity) x (displacement) T/2 3T/4 T T/4 A a=-8 sin(2t) C a=-4 sin(2t) B a=-8 cos(2t) D a=-4 cos(2t)
Slide 41 / 67 Energy and Simple Harmonic Motion Any vibrating system where the restoring force is proportional to the negative of the displacement is in simple harmonic motion (SHM), and is often called a simple harmonic oscillator. Also, SHM requires that a system has two forms of energy and a method that allows the energy to go back and forth between those forms.
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