6bsgmidterm1 - Study Guide for Midterm 1 Simple Harmonic...

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Study Guide for Midterm 1 Simple Harmonic Motion (undamped) An object of mass m is connected to a spring with spring constant k . x = 0 is the position of equilibrium. At t = 0 s, the object has initial position x (0) and initial velocity v (0). The equation of simple harmonic motion is d 2 x dt 2 + ω 2 x = 0 Other important quantities and equations are x ( t ) = A cos( ωt + φ ) v ( t ) = dx dt = - ωA sin( ωt + φ ) a ( t ) = dv dt = - ω 2 A cos( ωt + φ ) ω = r k m = 2 πf period : T = 1 f = 2 π ω potential energy : U ( t ) = 1 2 kx ( t ) 2 kinetic energy : K ( t ) = 1 2 mv ( t ) 2 U ( t ) + K ( t ) = E total = 1 2 kA 2 = constant a ( t ) = - ω 2 x ( t ) x ( t ) 2 + v ( t ) 2 ω 2 = A 2 amplitude : A = r x (0) 2 + v (0) 2 ω 2 A cos φ = x (0) A sin φ = - v (0) ω φ = arctan - v (0) ωx (0) , see below ! Here is an important message regarding the last equation above. If you take the arctan of a NEGATIVE number, the calculator will give you an answer (call it θ ) that’s either 1
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in quadrant 2 or quadrant 4. It’s in quadrant 2 if π/ 2 < θ < π . It’s in quadrant 4 if - π/ 2 < θ < 0, or 3 π/ 2 < θ < 2 π , etc.. The other answer the calculator doesn’t give you is θ + π . So you have two angles, θ and θ + π . One of these is in quadrant 2, and the other is in quadrant 4. If you want the sine to be positive and the cosine to be negative, then you choose the value that’s in quadrant 2. If you want the sine to be negative and the cosine to be positive, you choose the value that’s in quadrant 4. If you take the arctan of a POSITIVE number, the calculator will give you an answer (call it θ ) that’s either in quadrant 1 or quadrant 3. It’s in quadrant 1 if 0 < θ < π/ 2. It’s in quadrant 3 if - π < θ < - π/ 2, or π < θ < 3 π/ 2, etc.. The other answer the calculator doesn’t give you is θ + π . So you have two angles, θ and θ + pi . One of these is in quadrant 1, and the other is in quadrant 3. If you want the sine and cosine to both be positive, then you choose the value that’s in quadrant 1. If you want the sine and cosine to both be negative, you choose the value that’s in quadrant 3. Remember that θ + 2 π , θ + 4 π , etc., is the same angle as θ . Also recall that π is about 3.14, so π/ 2 is about 1.57. Here are the equations of motion for a few typical initial situations, each occurring at t = 0 s. 1: The mass is displaced by amount A in the positive direction, and released from rest. 1: x ( t ) = A cos( ωt ) 2: The mass is displaced by amount A in the negative direction, and released from rest. 2: x ( t ) = - A cos( ωt ) = A cos( ωt + π ) 3: The mass is moving through equilibrium in the positive direction. 3: x ( t ) = A sin( ωt ) = A cos( ωt - π/ 2) 4: The mass is moving through equilibrium in the negative direction.
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