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Consider a graphical representation (Fig. 12.3) of
Figure 12.3 (Quick Quiz 12.2)
An x–t graph for a particle undergoing simple harmonic motion. At a particular time, the particle’s position is indicated by Ⓐ in the graph.
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Principles of Physics: A Calculus-Based Text
- We do not need the analogy in Equation 16.30 to write expressions for the translational displacement of a pendulum bob along the circular arc s(t), translational speed v(t), and translational acceleration a(t). Show that they are given by s(t) = smax cos (smpt + ) v(t) = vmax sin (smpt + ) a(t) = amax cos(smpt + ) respectively, where smax = max with being the length of the pendulum, vmax = smax smp, and amax = smax smp2.arrow_forwardA simple harmonic oscillator has amplitude A and period T. Find the minimum time required for its position to change from x = A to x = A/2 in terms of the period T.arrow_forwardthe general solution to a harmonic oscillator are related. There are two common forms for the general solution for the position of a harmonic oscillator as a function of time t: 1. x(t) = A cos (wt + p) and 2. x(t) = C cos (wt) + S sin (wt). Either of these equations is a general solution of a second-order differential equation (F= mā); hence both must have at least two--arbitrary constants--parameters that can be adjusted to fit the solution to the particular motion at hand. (Some texts refer to these arbitrary constants as boundary values.) Part D Find analytic expressions for the arbitrary constants A and in Equation 1 (found in Part A) in terms of the constants C and Sin Equation 2 (found in Part B), which are now considered as given parameters. Express the amplitude A and phase (separated by a comma) in terms of C and S. ► View Available Hint(s) Α, φ = V ΑΣΦ ?arrow_forward
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- Modeling with periodic functions: Cyclical exponential decay. A spring is attached to the ceiling and pulled 5 cm down from equilibrium and released. After 2 seconds the amplitude has decreased to 3 cm. Also, the spring oscillates 13 times each second. Assuming that the amplitude is decreasing exponentially, below is a model for the distance, D the end of the spring is below equilibrium, in terms of seconds, t: D(t) = 5(0.7746)* · cos(26rt) . [Note: Here we define t as the number of seconds elapsed. Also, locations below the resting position have negative values for D.] 7 seconds, how far from equilibrium is the end of the spring? If your answer is a decimal, please round to no more than 3 decimal places. After t cmarrow_forwardA buoy floating in the ocean is bobbing in simple harmonic motion with amplitude 6 ft and period 8 seconds. Its displacement d from sea level at time t=0 seconds is -6 ft, and initially it moves upward. (Note that upward is the positive direction.) Give the equation modeling the displacement d as a function of time t. d = 0 T X 3 0/0 sin cosarrow_forwardThe oscillatory movement of a simple pendulum is a characteristic of the regular repetition of displacements around an equilibrium position. The pendulum swings due to the restored force of gravity, which seeks to bring the mass back to the equilibrium point. The oscillatory behavior is described by a trigonometric solution, which relates the pendulum's position to time, considering its amplitude, frequency and initial phase. Statement: A simple pendulum moves according to the following question:y(t) = A.sin(ωt + φ)In this question, y(t) is the horizontal position of the pendulum, A is its amplitude, ω is its angular velocity, given by ω = 2πf, and φ is the initial phase of the movement, in radians. Since the initial phase of the movement is equal to 0 and its angular velocity is π/2 rad/s, the oscillation frequency of this pendulum is correctly given by the alternative: a) 2.0Hz b) 1.5Hz c) 1.0Hz d) 0.5Hz e) 0.25Hzarrow_forward
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