Ans from the previous to previous slide and since it

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ANS: From the previous to previous slide, and Since it is released from the origin The solution to this equation is either A = 0 (then x ( t ) = 0 , not acceptable) or φ = π /2 or 3 π /2 (same as −π /2 ). Using these in v (0) we have or where the upper sign is for φ = π /2 Since we want A > 0 we choose the lower sign, and φ = −π /2: x (0) = A cos( φ ) v (0) = - A ω sin( φ ) x ( t ) = A cos( ω t + φ ) x (0) = 0 A cos( φ ) = 0 v (0) = - A ω sin( ± π 2 ) = A ω A = v 0 / ω x ( t ) = v 0 ω cos( ω t - π 2 ) = v 0 ω sin( ω t )
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Phys 2A - Mechanics As this solution suggests, we could have equally started from We can check that this is also a solution of the differential equation d 2 x dt 2 + ω 2 x = 0 by taking derivatives, as before. Or we can simply shift the value of ϕ by π /2 in the solution, x ( t ) = A sin( ω t + φ ) x ( t ) = A cos( ω t + φ - π 2 ) = A sin( ω t + φ )
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Phys 2A - Mechanics While A and phi are fixed by initial conditions, omega is determined by the force and the mass in the system, so it cannot be changed unless we change the system! Recall ω 2 = k m or ω = k m (we always take the positive root) So what is the meaning of ω ? It controls how often the periodic motion repeats itself.
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Phys 2A - Mechanics t A A x change ω , keep everything else fixed. x ( t ) = A cos( ω t ) x ( t ) = A cos(˜ ω t ) ω < ˜ ω
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Phys 2A - Mechanics sine and cosine are periodic functions with common period 2 π sin( θ + 2 π ) = sin( θ ) cos( θ + 2 π ) = cos( θ ) As time increases from t to some later value t +T , displacement and velocity take on their previous values at time t x ( t + T ) = x ( t ) v ( t + T ) = v ( t ) For simple harmonic motion we have θ + 2 π = ω ( t + T ) + φ θ = ω t + φ which gives ω T = 2 π
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Phys 2A - Mechanics The “period” of oscillation, T , is the time for one oscillation T = 2 π ω = 2 π m k The “frequency,” f , is the number of oscillations per unit time: f = 1 T = ω 2 π = 1 2 π k m ω is the frequency in radians ( 2 π radians for each oscillation): ω = 2 π f We call it “angular frequency.”
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Phys 2A - Mechanics t Ω A Ω A v t A Ω 2 A Ω 2 a t A A x x = x max v = 0 a = a min x = 0 v = v min a = 0 T
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Phys 2A - Mechanics Example: A 5.0 kg mass hangs at rest from a 20 N/m spring.
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