The damped oscillation of a vibrating block is given by x = Re(z); where z = 0.25 + 7i. a) Find x(t). b) Find the velocity of the block by evaluating dx/dt. c) Find the velocity of the block by evaluating dRe(z)/dt. d) Determine the first time t where x = 0 using both (b) and (c) and show that the answers are identical.
Added by Robert G.
Step 1
We can write z in terms of its magnitude and phase as z = Aeiθ, where A is the amplitude and θ is the phase angle. Then, we have: ~025+79} x = Re(Aeiθ) Using Euler's formula, we can write: Aeiθ = Acos(θ) + iAsin(θ) Taking the real part, we get: x = Show more…
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For the damped oscillator system, the block has a mass of 1.50 kg and the spring constant is 8.00 N/m. The damping force is given by -b(dx/dt), where b = 230 g/s. The block is pulled down 12.0 cm and released. (a) Calculate the time required for the amplitude of the resulting oscillations to fall to one-third of its initial value. (b) How many oscillations are made by the block in this time?
Krishna G.
For the damped oscillator system shown in Fig. $15-16,$ the block has a mass of 1.50 $\mathrm{kg}$ and the spring constant is 8.00 $\mathrm{N} / \mathrm{m} .$ The damping force is given by $-b(d x / d t),$ where $b=230$ $\mathrm{g} / \mathrm{s}$ . The block is pulled down 12.0 $\mathrm{cm}$ and released. (a) Calculate the time required for the amplitude of the resulting oscillations to fall to one-third of its initial value. (b) How many oscillations are made by the block in this time?
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