phy293_l02.page1

# phy293_l02.page1 - + K.E. = 1 2 m 2 a 2 = 1 2 sa 2 The...

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PHY293 Oscillations Lecture #2 September 14, 2009 1. Tutorials start this week 2. First problem set is now posted ( http://www.physics.utoronto.ca/ phy293h1f/waves/phy293 ps1.pdf ) 3. Flash demonstration: http://www.upscale.utoronto.ca/PVB/Harrison/Flash/ClassMechanics/SHM/TwoSHM.html Start of material 1. SHO Equation of Motion m ¨ x = - sx or m ¨ x + sx = 0 Guessed the solution: x = a cos( ωt + φ ) If this is solution then: ˙ x = - sin( ωt + φ ) and ¨ x = - 2 cos( ωt + φ ) Plugging this into the equation of motion ﬁnd: m [ - 2 cos( ωt + φ )] + sa cos( ωt + φ ) = 0 This is a solution iff ω = p s/m Often we’ll write: ¨ x + ω 2 0 = 0 Where ω 0 is the natural frequency of the system p s/m 2. Energy Juggling The mass and spring system consist of: A spring: capable of storing elastic potential energy A mass: capable of storing inertial kinetic energy This is a general feature of oscillating systems Potential energy: 1 2 sx 2 . Get this by integrating the force R x 0 s · xds 1 2 sx 2 = 1 2 2 0 a 2 cos 2 ( ωt + φ ) Kinetic energy 1 2 m ˙ x 2 1 2 m ˙ x 2 = 1 2 2 a 2 sin 2 ( ωt + φ ) But since sin 2 θ + cos 2 θ = 1 for any θ Energy = P.E.
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Unformatted text preview: + K.E. = 1 2 m 2 a 2 = 1 2 sa 2 The total energy of a spring/mass system is a constant. Energy is juggled between tension in the spring (potential) and kinetic energy in the mass. We will nd this storage and exchange of energy in all oscillating systems. Note: Could have just integrated x + 2 x = 0 (twice) with respect to time to show that energy would be conserved without nding a solution of x and x . 3. Phase Space Diagram Mechanical systems completely characterised by knowing two variables: position x and velocity (momentum) x (or p = m x ) They are independent because both must be specied as initial conditions to solve F m x . For SHO we have x = a cos( t + ) and x = a sin( t + ) For = 0 these are ellipses in x, x space, with t = 0 starting along the x-axis at ( a, 0)...
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## This note was uploaded on 07/10/2011 for the course PHY 293 taught by Professor Pierresavaria during the Fall '07 term at University of Toronto- Toronto.

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