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Lets first plot it for some specific values eg the

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Let’s first plot it for some specific values, e.g. The motion is “periodic” — the motion repeats itself. There are a maximum and minimum displacements. It starts from some non-zero position at t=0. 1 2 3 4 5 6 7 t 3 2 1 1 2 3 x x ( t ) = 3 cos(2 t + 1 2 )
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Phys 2A - Mechanics x ( t ) = A cos( ω t ) t A A A A x x ( t ) = ˜ A cos( ω t ) Change A , keep everything else fixed.
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Phys 2A - Mechanics The cos function varies between +1 and -1. Then, in x ( t ) = A sin( ω t + φ ) x ( t ) varies between + A and - A , correspond to turning points, x max and x min . For a spring-mass system these correspond to the maximum extension and compression of the spring! U x E K=E-U turning points x min x max x 0 The value of A depends on “initial conditions,” that is, how far we pulled the spring (or whatever else this system may be) so that the total energy is just right. A = x max is called the “amplitude” of the oscillation.
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Phys 2A - Mechanics t A A x change φ , keep everything else fixed. x ( t ) = A cos( ω t ) x ( t ) = A cos( ω t - π 3 )
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Phys 2A - Mechanics Initial position and velocity; φ determines (together with A ) where and with what speed the particle is moving at time t = 0 . This also determines how soon after t=0 there is a maximum of x(t). x (0) = A cos( ω · 0 + φ ) = A cos( φ ) v (0) = - A ω sin( ω · 0 + φ ) = - A ω sin( φ ) φ is called the “phase shift.”
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Phys 2A - Mechanics Example: A particle is released (from rest) from x = x 0 at t =0 . It executes simple harmonic motion ( ). Determine A and φ . ANS: From the previous slide, and Since it is released from rest this gives The solution to this equation is either A = 0 (which implies x ( t ) = 0 , not acceptable) or φ = 0 or π . Using these in x (0) we have x (0) = A or A , for φ = 0 or π So we found two solutions: ( A , φ ) = ( x 0 , 0 ) or ( x 0 , π ) These are equivalent solutions to the problem: x (0) = A cos( φ ) v (0) = - A ω sin( φ ) v (0) = 0 - A ω sin( φ ) = 0 x ( t ) = A cos( ω t + φ ) x ( t ) = x 0 cos( ω t ) = - x 0 cos( ω t + π ) but we prefer the first one so that the amplitude is positive.
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Phys 2A - Mechanics Example: A particle is pushed so that its velocity is v 0 at x = 0 initially (at t =0). It executes simple harmonic motion, with Determine A > 0 and φ . Assume ω / v 0 > 0.
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