328/2018 PHYS 201 Assignment Thirteen-Chapter Fifteen 6. -/1 pointsHRW10 15.P.04

Question

328/2018 PHYS 201 Assignment Thirteen-Chapter Fifteen 6. -/1 pointsHRW10 15.P.040. A physical pendulum consists of a meter stick that is pivoted at a small hole drilled through the stick a distance x from the 50 cm mark. The period of oscillation is observed to be 5.0 s. Find the distance x. Additional Materials 7. -/1 pointsHRW10 15.P.058. In the figure below, a damped simple harmonic oscillator has mass m = 240 g, k-95 N/m, and b-65 g/s. Assume all other components have negligible mass. What is the ratio of the amplitude of the damped oscillations to the initial amplitude at the end of 20 cycles? Adamped/Ainitial xRigid support Springiness, k Mass m Vane Damping,b Additional Materials

Explanation / Answer

6)

equivalent length

Leq = I / mx

where I is the moment of inertia about the pivot

and m is the mass

and x is the distance from the pivot to the center of mass

By the parallel axis theorem,

I = mL²/12 + mx², so

Leq = (mL²/12 + mx²) / md = L²/12x + x

period T = 2??(Leq / g) ? square everything

T² = 4?²(L²/12x + x) / g

gT² / 4?² = L²/12x + x ? plug in g and T and drop units for ease (x is in meters):

9.8*(5)² / 4?² = 1/12d + x

6.21 = 1/12x + x ? multiply by x

6.21x = 1/12 + x²

0 = x² - 6.21x + 1/12

This quadratic has roots at

x = 0.01345 m ? solution

(and x = 6.2 m ? greater than 1 m; disregard)

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