Simple Harmonic Motion Test Tuesday 11/7 Chapter 11 Vibrations and Waves 1
If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system is a useful model for a periodic system. We assume that the surface is frictionless. 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). 2
Displacement is measured from the equilibrium point Amplitude is the maximum displacement A cycle is a full to-and-fro motion; this figure shows half a cycle Period is the time required to complete one cycle Frequency is the number of cycles completed per second The force exerted by the spring depends on the displacement: The minus sign on the force indicates that it is a restoring force it is directed to restore the mass to its equilibrium position. k is the spring constant The force is not constant, so the acceleration is not constant either 3
Example The spring constant of the spring is 320 N/m and the bar indicator extends 2.0 cm. What force does the air in the tire apply to the spring? F kx F (320 N / m)( 0.02 m) F 6.4N 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. 4
A spring has a length of 15.4 cm and is hanging vertically from a support point above. A weight of 0.200 kg is then attached to the spring, causing it to extend to a length of 28.6 cm. What is the value of the spring constant? How much force is then needed to lift this weight 4.6 cm from that position? x.286.154 0.132m 2 F (0.200 kg)(10 m / s ) 2.0N 2.0 N k(.132) k 15.2 N / m F (15.2 N / m)(.046 m) F 0.7N f F (.60)(20 N) 12N f s s kx N 12 N (50 N / m) x x.24m 24cm 24cm time 4.8sec 5.0 cm / s 5
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. We already know that the potential energy of a spring is given by: The total mechanical energy of a spring system is: The total mechanical energy will be conserved, as we are assuming the system is frictionless. 6
If the mass is at the limits of its motion, the energy is all potential. If the mass is at the equilibrium point, the energy is all kinetic. 1 The total energy is, therefore: ka 2 2 The energy equation for the system is 1 1 1 mv + kx = ka 2 2 2 2 2 2 7
1 1 kx mv 2 2 2 2 kx mv kx m 2 v 2 2 v 2 kx (10 N / m)(.1 m) m (.01 kg) v 3.2 m / s 2 2 Assignment Read pg. 292-297 Do pg. 316-317 Questions #2,5 Problems #1,3,5,13 8
Simple Harmonic Motion Test Tuesday 11/7 Spring virtual lab phet simulation lab is on the class website 9
Simple Harmonic Motion Test Tuesday 11/7 What is the value of the spring constant of a spring that is stretched a distance of 0.5 m if the restoring force is 24 N? a) 12 N/m b) 18 N/m c) 24 N/m d) 48 N/m 10
An object in simple harmonic motion is observed to move between a maximum position and a minimum position. The minimum time that elapses between the object being at its maximum position and when it returns to that maximum position is equal to which of the following parameters? a) frequency b) angular frequency c) period d) amplitude A block is attached to the end of a spring. The block is then displaced from its equilibrium position and released. Subsequently, the block moves back and forth on a frictionless surface without any losses due to friction. Which one of the following statements concerning the total mechanical energy of the block-spring system this situation is true? a) The total mechanical energy is dependent on the maximum displacement during the motion. b) The total mechanical energy is at its maximum when the block is at its equilibrium position. c) The total mechanical energy is constant as the block moves back and forth. d) The total mechanical energy is only dependent on the spring constant and the mass of the block. 11
Period is the time (seconds) required to complete one cycle The period of a spring system can be found using Frequency is the number of cycles completed per second and is the reciprocal of the period Frequency is measured in Hertz (Hz) The Period and Sinusoidal Nature of SHM v a max max 2 f A 2 A 12
The frequency of motion is 1.0 KHz and the amplitude is 0.20 mm. (a)what is the maximum speed of the diaphragm? (b)where in the motion does this maximum speed occur? a) 2 f v A A(2 f ) max v (.0002 m)(6.28)(1000 Hz) v max max occurs at x 0 1.3 m / s The displacement of an object is described by the following equation, where x is in meters and t is in seconds: x = (0.30m) cos (8.0 t) Determine the oscillating object s (a) amplitude, (b) frequency, (c) period, (d) max speed, and (e) max acceleration a) amp 0.30m b)8 6.28 f f 1.3Hz 1 ct ).77sec f d) v (.3)(8) 2.4 m / s max e) a (.3)(8 ) 19.2 m / s max 2 2 13
The Simple Pendulum A simple pendulum consists of a mass at the end of a lightweight cord. We assume that the cord does not stretch, and that its mass is negligible. The Simple Pendulum In order to be in SHM, the restoring force must be proportional to the negative of the displacement. Here we have: which is proportional to sin θ and not to θ itself. However, if the angle is small, sin θ θ. 14
The Simple Pendulum Therefore, for small angles, we have: where The Simple Pendulum So, as long as the cord can be considered massless and the amplitude is small, the period does not depend on the mass. 15
Determine the length of a simple pendulum that will swing back and forth in simple harmonic motion with a period of 1.00 s. T 2 L g L 1 6.28 10 L.1592 10 L.02536 10 L 0.254m Pendulums and Energy Conservation Energy goes back and forth between KE and PE. At highest point, all energy is PE. As it drops, PE goes to KE. At the bottom, energy is all KE. 16
Pendulum Energy ½mv max 2 = mgh For minimum and maximum points of swing A mass of 1.4kg is attached to a 3.2m long string to make a simple pendulum. a)what is the period of the pendulum? b)if the pendulum is pulled back to an angle of 15 o and released, what is the maximum speed of the pendulum? 17
Assignment Do pg. 317-318 Problems #9,16,21,24,28,30,32 Simple Harmonic Motion Test Tuesday 11/7 18
Pendulum virtual lab phet simulation lab is on the class website due tomorrow Simple Harmonic Motion Test Tuesday 11/7 19
Which one of the following units is used for frequency? a) ohm b) second c) farad d) hertz Which one of the following statements concerning the total mechanical energy of a harmonic oscillator at a particular point in its motion is true? a) The total mechanical energy depends on the acceleration at that point. b) The total mechanical energy depends on the velocity at that point. c) The total mechanical energy depends on the position of that point. d) The total mechanical energy does not vary during the motion 20
A simple pendulum consists of a ball of mass m suspended from the ceiling using a string of length L. The ball is displaced from its equilibrium position by a small angle and released. Which one of the following statements concerning this situation is correct? a) If the mass were increased, the period of the pendulum would increase. b) The frequency of the pendulum does not depend on the acceleration due to gravity. c) If the length of the pendulum were increased, the period of the pendulum would increase. d) The period of the pendulum does not depend on the length of the pendulum. A block of mass M is attached to one end of a spring that has a spring constant k. The other end of the spring is attached to a wall. The block is free to slide on a frictionless floor. The block is displaced from the position where the spring is neither stretched nor compressed and released. It is observed to oscillate with a frequency f. Which one of the following actions would increase the frequency of the motion? a) Decrease the mass of the block. b) Increase the length of the spring. c) Reduce the spring constant. d) Reduce the distance that the spring is initially stretched. 21
In simple harmonic motion, an object oscillated with a constant amplitude. In reality, friction or some other energy dissipating mechanism is always present and the amplitude decreases as time passes. This is referred to as damped harmonic motion. Damped Harmonic Motion However, if the damping is large, it no longer resembles SHM at all. A: underdamping: there are a few small oscillations before the oscillator comes to rest. B: critical damping: this is the fastest way to get to equilibrium. C: overdamping: the system is slowed so much that it takes a long time to get to equilibrium. 22
1)simple harmonic motion 2&3) underdamped 4)critically damped 5) overdamped Damped Harmonic Motion There are systems where damping is unwanted, such as clocks and watches. Then there are systems in which it is wanted, and often needs to be as close to critical damping as possible, such as automobile shock absorbers and earthquake protection for buildings. 23
Forced Vibrations; Resonance Forced vibrations occur when there is a periodic driving force. This force may or may not have the same period as the natural frequency of the system. RESONANCE Resonance is the condition in which a time-dependent force can transmit large amounts of energy to an oscillating object, leading to a large amplitude motion. Resonance occurs when the frequency of the force matches a natural frequency at which the object will oscillate. Forced Vibrations; Resonance The sharpness of the resonant peak depends on the damping. If the damping is small (A), it can be quite sharp; if the damping is larger (B), it is less sharp. Like damping, resonance can be wanted or unwanted. Musical instruments and TV/radio receivers depend on it. 24
When a force is applied to an oscillating system at all times, the result is driven harmonic motion. Here, the driving force has the same frequency as the spring system and always points in the direction of the object s velocity. Assignment Damped Harmonic Motion assignment on the class website 25