Energy in a Simple Harmonic Oscillator. Class 30. Simple Harmonic Motion

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Tracey Sparks
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1 Simple Harmonic Motion Class 30 Here is a simulation of a mass hanging from a spring. This is a case of stable equilibrium in which there is a large extension in which the restoring force is linear in the excursion away from equilibrium. It obeys the force excursion relationship F= kx Note again, how the velocity is the slope of the x -t graph and that the acceleration is the slope of the velocity time graph. Also note that the acceleration function is the negative of the position curve. The fact that sine and cosine function have this relationship is the reason that systems which are governed by F= kx undergo sinusoidal motion. In the below drawing you see the places where the position, velocity, and acceleration have their maxima and their zeroes. This animation shows the connection between simple harmonic motion and the circular functions. Energy in a Simple Harmonic Oscillator The energy of a SHO is the sum of the kinetic and the potential energy terms. The zero of the potential energy is set where the spring is at it natural or outstretched length. E= 1 2 m v2 1 2 x 2 (1.) 1

2 Amplitude --- the amplitude of the SHM is the maximum excursion of the moving particle, that is the maximum value that x takes. When the oscillating object is at its maximum excursion its velocity is zero. Therefore, when x = A the total energy becomes We can combine this equation with (1.) to get We can solve this for v to get E= 1 2 k A2 1 2 m v2 1 2 x 2 = 1 2 k A2 v= k m A2 x 2 This gives us a relationship between the speed and the position of the oscillator. 2

3 Period and Frequency Since simple harmonic motion is based on circular functions the period and frequency relations are the same as for motion in a circle. The y coordinate of an object in SHM is can be described by y t = Asin t The quantity inside the parenthesis of the sine function is called the argument of the sine function. If we look as this motion in terms of the rotation of the hand of the angle from zero degrees to 360, of better put from 0 to 2π, the sine function goes through a full period. In other words, after the hand has revolved around one full revolution, the sine function repeats itself. So, just as with circular motion, the revolution around 2π radians is a full revolution, the evolution of the argument of the sine function, from 0 to 2π is also one period. The relation be tween period, frequency, and Let's calculate the time it takes for the sine function to begin to repeat itself. If it starts at t = 0, then when the quantity t reaches 2π, the cycle with be complete. So we call T, the period, the time it takes for the quantity t to go from zero to 2π. T =2 0 3

T = 2 Since frequency is the reciprocal of the period we have F= 1 T = 2 Above we see a graph made by a pen attached to a spring undergoing SHM, while the paper moves to simulate the passage of time.

4 T = 2 Since frequency is the reciprocal of the period we have F= 1 T = 2 Above we see a graph made by a pen attached to a spring undergoing SHM, while the paper moves to simulate the passage of time. T is the point on the time axis, at which the wave begins to repeat itself. Simple Pendulum The pendulum undergoes simple harmonic motion, but only when the angle of the swing of the pendulum is small. Small angle approximation The reason that SHM describes the pendulum for small angles connected to the fact that for small angles the sin 4

5 This is called the small angle approximation because it only works at small angles. You should try it out on your calculators, but of course you calculators must be in radian mode. Using this approximation it can be shown that the period of the pendulum is where L is the length of the pendulum. T = 1 2 g L 5

6 Damped Harmonic Motion All harmonic motion is subject to friction which causes the oscillations to die away over time. In damped harmonic motion the amplitude of the oscillations descreases exponentially over time, as shown above. 6

7 Resonance The collapse of the Tacoma Narrows bridge is one of the most striking example of the phenomena of resonance. You learned about resonance which you were small children on the swing. You learned that if you wanted to swing higher, you had to pump with your legs and arms. But the pumping only worked at a certain frequency. That frequency is the resonant frequency of the swing. Every harmonic oscillator has a natural or resonant frequency, and if it is driven at the frequency the amplitude of the oscillations will increase with each swing. The degree of increase in the amplitude of the oscillations depends on the amount of damping in the system. Damping as removes energy from the oscillator, so it limits the amplitude of the resonant oscillation. The above graph shows the amplitude of oscillation as a function of frequency for a system with a lot of damping, B, and with smaller damping A. 7

8 Prevention of Resonant Oscillations Tall buildings are like large springs and they have resonant oscillation modes. The above picture shows the large structures which are used to provide damping so that the resonant oscillation of the building remains small. 8

9 The mass of the beam is 27 kg. 1. Calculate the tension in the cable. a. 200 N b. 203 N c 206 N d. 209 N e. none 2. Calculate the magnitude of the force of the wall on the beam in the x direction. a. 156N b 158 N c. 160N d. 162N e. none c. Calculate the direction of the force of the wall on the beam. 9

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Oscillations. Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance

Oscillations. Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance Oscillations Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance 1 Revision problem Please try problem #31 on page 480 A pendulum