CH2_oscillation

CH2_oscillation - CHAPTER 2 OSCILLATIONS 2.1 Quick Review...

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1 CHAPTER 2 OSCILLATIONS 2.1 Quick Review of Simple Harmonic Motion (SHM) 2.1.1 Equation of Motion for SHM The equation of motion for the SHM is: or (2.1) where . The general solutions of this equation of motion are : (1) x(t) = A cos ( ω t + φ ); (2) x(t) = A sin ( t + ); (3) x(t) = A cos t + B sin t (4) x(t) = A exp( i t ) + B exp(- i t ) (2.2) All these four solutions are equivalent. 2.1.2 Energy of SHM 1) Kinetic energy (2.3) 2) Potential Energy (2.4)
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2 At the equilibrium position EP (i.e. x = 0), take PE=0, Therefore, total Energy E T = KE + PE (2.5) If there is no energy dissipation, E T is a constant. 2.1.3 Phasor Diagram Construct a rotating vector (or the position phasor) such that , where A is the amplitude of the SHM. The phasor is initially at position as shown in the figure and then rotating anti-clockwise with an angular velocity of ω . Similarly, we can also construct the velocity and the acceleration phasor. =Phasor of Position, =Phasor of Velocity and =Phasor of Acceleration Notice that: , ,
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3 Example A 1kg mass is placed on a horizontal smooth table and linked a wall by a spring with force constant of 1Nm -1 . Let the equilibrium position of the particle be x=0. If the mass is placed at a position of x=-1m at t=0 and then released, (i) Prove that the subsequent motion is a SHM. (ii) Find the frequency and the phase of the SHM. Then find the functions x, v and a with respect to t. (iii) Repeat (ii) if initially the mass is placed at x=+0.5m and has an initial velocity of +0.1ms -1 . 2.2 Damped Oscillation O ther than the force that drives the particle in SHM, there exists another damping force which is against the motion of the particle. We write the damping force as , which is against the velocity. (2.6) or in a one dimensional case, ˙ ˙ x + 2 γ ˙ x + ω 2 x = 0 (2.7) where and = k m . Notice that γ is a measure of the effectiveness of the damping force. ω 0 is determined only by the oscillator and is called the natural frequency of the oscillator. The solution of equation (2.7) is . The characteristic equation is: λ 2 + 2 γλ + 2 = 0 Therefore, = ± 2 2 . The solution has three possibilities, namely,: (i) γ > ω , overdamped, λ 1 and λ 2 are real roots (ii) γ = ω , critically damped, real and equal roots (iii) γ < ω , underdamped, λ 1 and λ 2 are imaginary roots.
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This note was uploaded on 08/17/2011 for the course PHYS 230 taught by Professor Harris during the Winter '07 term at McGill.

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CH2_oscillation - CHAPTER 2 OSCILLATIONS 2.1 Quick Review...

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