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Chapter 11 - Vibrations and Waves

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McGill University
PHYS 101
Kenneth Ragan

Chapter 11 – Vibration and Waves 11.1 – Simple Harmonic Motion When an object vibrates or oscillates back and forth, the motion is periodic.  Example: a spring o Any spring has a natural length at which it exerts no force on the mass, m  the position of the mass at this point is called equilibrium position Hooke’s Law: Force Exerted by spring: F = - kx Restoring Force: the restoring force is always in the direction opposite to the displacement Clicker Question: When x is at equilibrium position, then…speed is maximum (F=0) at equilibrium x = 0 and F = -kx therefore F = 0 Spring constant or spring stiffness constant: constant k  The greater the value of k, the greater the force needed to stretch a spring a given distance. Therefore, the stiffer the spring, the greater the spring constant k  The force F is not a constant, but varies with position Clicker Question: Maximum acceleration occurs at: maximum compression of a spring Displacement: distance x of the mass from the equilibrium point Amplitude: greatest distance from the equilibrium point Period: (T) defined as the time required to complete one cycle Frequency: (f) the number of complete cycles per second (1 Hz = 1 cycle/sec) One Cycle: refers to the complete to-and-fro motion from some initial point back to the same point 1 1 𝑚 f = and T = and T = 2𝜋√ 𝑇 𝑓 𝑘 Simple Harmonic Motion: any vibrating system for which the restoring force is directly proportional to the negative of the displacement  The system is often called a simple harmonic oscillator Clicker Question: A ball drops from a height h, and in a series of perfectly elastic collisions, rebounds to h. Is this simple harmonic motion? No, this is free fall projectile motion (parabola if you draw the position vs time graph) Chapter 11 – Vibration and Waves 11.2 – Energy in Simple Harmonic Oscillators To stretch or compress a spring, work has to be done. Hence potential energy is stored in a stretched or compressed spring. The total energy in a harmonic oscillator is proportional to the square of the amplitude A of the oscillation. 2 2 E xmax= Eo= ½ kx max = ½ kA When looking for a velocity at a particular distance, use: 2 2 V = vmaxsqrt (1-x /A )) 𝑘 2𝜋𝐴 V max= A √ 𝑚 and vmax= 𝑇 11.3 – The Period and Sinusoidal Nature of SHM Uniform Circular Motion is Simple Harmonic Motion (view from the side) Period and Frequency of SHM: T= 2𝜋𝐴 and f =1 √( ) 𝑣𝑚𝑎𝑥 2𝜋 𝑚 The simple harmonic oscillator depends on the stiffness of the spring and also on the mass m that is oscillating. (PERIOD DOES NOT DEPEND ON THE AMPLITUDE) Clicker Question: compared to 0.25 g fly, the new insect is more massive (0.50 g). Is the new frequency of vibration higher or lower? Lower Curves of displacement in SHM are sinusoids! Chapter 11 – Vibration and Waves 2𝜋𝑡 2𝜋𝑡 x = Asin x = Acos 𝑇 𝑇 These curves have different phases. The both represent displacement in SHM; the phase difference is simply a result of different initial conditions. Displacement, velocity and acceleration are all sinusoidal functions with different phases. 2 Amax = kA/m or = w A 11.4 – The Simple Pendulum A simple pendulum consists of a small object suspended from the end of a lightweight cord (represents one of the simplest examples of SHM). The period of the pendulum depends on: its length. (does not depend on amplitude or mass) Restoring force: F = -mg sin𝜃 What would happen if you could dig a hole through the center of the Earth and drop something into it? The object would get to the far side and drop back in like an oscillator when it gets to the middle it is at max velocity; no net force but still moving fast. Force is linear with the position, resulting in SHM. Period and Frequency of a Pendulum: 1 𝑔 𝐿 f = √ and T = 2𝜋√( ) 2𝜋 𝐿 𝑔 11.6 – Forced Vibrations; Resonance Resonance: (nudging the system) we drive it by applying a periodic impulse or force to an arbitrary frequency. Then, it will oscillate at that frequency f, even if it is different from 0 . The amplitude of the oscillation will depend on the difference
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