**Problem Statement:**4. Use conservation of energy to determine the angular speed of the spool shown in the figure below after the 3.00-kg bucket has fallen 4.00 m, starting from rest. The light string attached to the bucket is wrapped around the spool and does not slip as it unwinds. The mass of the spool is \( M = 5.00 \, \text{kg} \) and the radius is \( r = 0.600 \, \text{m} \). The moment of inertia for the spool is \( I = \frac{1}{2} M r^2 \).**Diagram Explanation:**- The diagram shows a cylindrical spool with a mass of 5.00 kg and a radius of 0.600 m. It is horizontally positioned and capable of rotating.- A string is wound around the spool and is attached to a 3.00-kg bucket.- The bucket is hanging vertically and will fall, unwinding the string from the spool.- As the bucket falls 4.00 meters from rest, the spool rotates due to the tension in the string.- There is a rotational arrow indicating the direction of the spool's rotation caused by the falling bucket.

Given that: Mass of bucket, m= 3 kg Height of descend of the bucket, h=4m Initial speed of the bucket, u=0 Mass of spool, M=5 kg Radius of the spool, r=0.6 m Let the angular speed of the spool is ω and the velocity of bucket is v. Initial kinetic energy of the bucket, Kb1=0Intital kinetic energy of the spool, Ks1=0Initial potential energy of the bucket, U1=mgyiFinal kinetic energy of the bucket, Kb2=12mv2Final kinetic energy of the spool, Ks2=12Iω2Final potential energy of the bucket, U2=mgyfApplying the conservation of energy:Kb1+ Ks1+ U1=Kb2+ Ks2+U20+0+mgyi=12mv2+12Iω2+mgyf12mv2+12Iω2=mgyi-mgyfNow we know that:v=ωrThen:12m(ωr)2+12Iω2=mgyi-mgyf12mω2r2+12×12Mr2ω2=mg(yi-yf)12mω2r2+14Mr2ω2=mghω2(12mr2+14Mr2)=mghω2=mgh(12mr2+14Mr2)ω2=(3×9.81×4)(12×3×0.62+14×5×0.62)ω2=117.72(0.54+0.45)ω=10.9 rad/s The angular speed of the spool will be 10.9 rad/s

Science

Physics

Use conservation of energy to determine the angular speed of the spool shown in figure below after the 3.00-kg bucket has fallen 4.00 m, starting from rest. The light string attached to the bucket is wrapped around the spool and does not slip as it unwinds. The mass of the stool is M = 5.00kg and the radius is r 0.600m. The moment of inertia for %3D 1 the stool is I = -Mr2. %3D

Use conservation of energy to determine the angular speed of the spool shown in figure below after the 3.00-kg bucket has fallen 4.00 m, starting from rest. The light string attached to the bucket is wrapped around the spool and does not slip as it unwinds. The mass of the stool is M = 5.00kg and the radius is r 0.600m. The moment of inertia for %3D 1 the stool is I = -Mr2. %3D

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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