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The block is slowly pulled from its equilibrium position to some position æinit > 0 along the x axis. At time t = 0 , the block is released with zero initial velocity. The goal is to determine the position of the block æ (t) as a function of time in terms of w and æinit - It is known that a general solution for the displacement from equilibrium of a harmonic oscillator is æ(t) = C cos (wt) + S sin (wt), where C, S, and w are constants. (Figure 2) Your task, therefore, is to determine the values of C and S in terms of w and ¤init -

The block is slowly pulled from its equilibrium position to some position æinit > 0 along the x axis. At time t = 0 , the block is released with zero initial velocity. The goal is to determine the position of the block æ (t) as a function of time in terms of w and æinit - It is known that a general solution for the displacement from equilibrium of a harmonic oscillator is æ(t) = C cos (wt) + S sin (wt), where C, S, and w are constants. (Figure 2) Your task, therefore, is to determine the values of C and S in terms of w and ¤init -

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The block is slowly pulled from its equilibrium position to some position xinit > 0 along the x axis. At time t = 0, the block is released with zero initial velocity. Constants I Periodic Table Find the value of S using the given condition that the initial velocity of the block is zero: v(0) = 0. The goal is to determine the position of the block x (t) as a function of time in terms of w and xinit - View Available Hint(s) It is known that a general solution for the displacement from equilibrium of a harmonic oscillator is Hint 1. How to approach the problem x(t) = C cos (wt) + S sin (wt), Using the general equation x (t), obtain the expression for the block's velocity v (t) in terms of C, S, w, and t. Then evaluate the general expression for v (t) when t = 0. where C, S, and w are constants. (Figure 2) Your task, therefore, is to determine the values of C and S in terms of w and xinit. Hint 2. Differentiating harmonic functions Recall that d cos wt -w sin wt dt Figure 1 of 3 > and d sin wt = w COS Wt. dt Note the negative sign in the first formula. L Xinit Xinit tan(wt) win- Xinit w Xinit x = 0
Transcribed Image Text:The block is slowly pulled from its equilibrium position to some position xinit > 0 along the x axis. At time t = 0, the block is released with zero initial velocity. Constants I Periodic Table Find the value of S using the given condition that the initial velocity of the block is zero: v(0) = 0. The goal is to determine the position of the block x (t) as a function of time in terms of w and xinit - View Available Hint(s) It is known that a general solution for the displacement from equilibrium of a harmonic oscillator is Hint 1. How to approach the problem x(t) = C cos (wt) + S sin (wt), Using the general equation x (t), obtain the expression for the block's velocity v (t) in terms of C, S, w, and t. Then evaluate the general expression for v (t) when t = 0. where C, S, and w are constants. (Figure 2) Your task, therefore, is to determine the values of C and S in terms of w and xinit. Hint 2. Differentiating harmonic functions Recall that d cos wt -w sin wt dt Figure 1 of 3 > and d sin wt = w COS Wt. dt Note the negative sign in the first formula. L Xinit Xinit tan(wt) win- Xinit w Xinit x = 0
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