3/31/2021 Solved: Two uniform, solid cylinders of radius R and total mass... | Chegg.com https://www.chegg.com/homework-help/two-uniform-solid-cylinders-radius-r-total-mass-m-connected-chapter-14-problem-1bp-solution-978013397804... 1/4 home / study / science / physics / calculus based physics / calculus based physics solutions manuals / university physics with modern physics / 14th edition / chapter 14 / problem 1bp University Physics with Modern Physics (14th Edition) See this solution in the app Problem Two uniform, solid cylinders of radius R and total mass M are connected along their common axis by a short, light rod and rest on a horizontal tabletop ( Fig. 14.29 ). A frictionless ring at the center of the rod is attached to a spring with force constant k; the other end of the spring is fixed. The cylinders are pulled to the left a distance x, stretching the spring, and then released from rest. Due to friction between the tabletop and the cylinders, the cylinders roll without slipping as they oscillate. Show that the motion of the center of mass of the cylinders is simple harmonic, and find its period. SOLUTION GUIDE IDENTIFY and SET UP 1. What condition must be satisfied for the motion of the center of mass of the cylinders to be simple harmonic? 2. Which equations should you use to describe the translational and rotational motions of the cylinders? Which equation should you use to describe the condition that the cylinders roll without slipping? ( Hint: See Section 10.3.) 3. Sketch the situation and choose a coordinate system. List the unknown quantities and decide which is the target variable. EXECUTE 4. Draw a free-body diagram for the cylinders when they are displaced a distance x from equilibrium. 5. Solve the equations to find an expression for the acceleration of the center of mass of the cylinders. What does this expression tell you? 6. Use your result from step 5 to find the period of oscillation of the center of mass of the cylinders. EVALUATE 7. What would be the period of oscillation if there were no friction and the cylinders didn’t roll? Is this period larger or smaller than your result from step 6? Is this reasonable? Step-by-step solution Step 1 of 6 My Textbook Solutions University Physics with... 14th Edition University Physics with... 14th Edition Understand g Operating 7th Edition View all solutions Post a question Answers from our experts for your tough homework questions Enter question Continue to post 20 questions remaining Snap a photo from your phone to post a question We'll send you a one-time download link 888-888-8888 Text me By providing your phone number, you agree to rec a one-time automated text message with a link to the app. Standard messaging rates may apply. Bookmark Show all steps: Chapter 14, Problem 1BP ON Find solutions for your homework Search Textbook Solutions Expert Q&A Practice Study Pack 3/31/2021 Solved: Two uniform, solid cylinders of radius R and total mass... | Chegg.com https://www.chegg.com/homework-help/two-uniform-solid-cylinders-radius-r-total-mass-m-connected-chapter-14-problem-1bp-solution-978013397804... 2/4 The system consisting of the two cylinders and the spring is a system in both rotational and translational motion. Define the origin of the x axis to be at the center of mass’s equilibrium position of 0, positive x to be to the left, and positive torques to be clockwise. In this coordinate scheme, when the cylinders are displaced to the left of zero and then released, the Hooke’s law restoring force of the spring with force constant k acts in the negative x direction because the displacement coordinate x is positive, and the total friction force f on both cylinders together acts in the positive x direction to oppose the motion of the cylinders back toward equilibrium, so find the net force by adding both forces: Since the force of friction acts along the edges of the cylinders while the spring force acts on their center of mass, only friction generates a torque, and this torque is positive at this position since it acts along the bottom of the cylinders towards the left. Therefore, set the torque magnitude produced by friction on both cylinders of radius R equal to the definition of net torque as the product of the center of mass moment of inertia and angular acceleration : Comment In the case of rolling without slipping, the acceleration of the center of mass is equal to the product of the radius R and angular acceleration of the rotating body: Solve this relation for : Comment Substitute for and for in the torque equation Solve the torque equation for the friction force f : Substitute for f in the force equation, along with for since the acceleration of the two cylinders of collective mass M is in the negative x direction at this point: Comment Solve the force equation for the acceleration of the center of mass: Step 2 of 6 Step 3 of 6 Step 4 of 6 Bookmark Show all steps: Chapter 14, Problem 1BP ON Textbook Solutions Expert Q&A Practice Study Pack 3/31/2021 Solved: Two uniform, solid cylinders of radius R and total mass... | Chegg.com https://www.chegg.com/homework-help/two-uniform-solid-cylinders-radius-r-total-mass-m-connected-chapter-14-problem-1bp-solution-978013397804... 3/4 Recommended solutions for you in Chapter 14 Was this solution helpful? This gives the magnitude only of the acceleration experienced by the rolling cylinders. The requirement for simple harmonic motion is that the restoring force and resulting acceleration on the object in motion be directly proportional to and always in the direction opposite of the displacement. This restoring acceleration of the cylinders’ center of mass is directly proportional to the displacement x , and as described in the first paragraph, it is in the direction opposite the displacement. Therefore, the motion of the cylinders’ center of mass is simple harmonic. Comment The definition of the acceleration a in simple harmonic motion is: Here, is the motion’s angular frequency, and the negative sign illustrates that the acceleration is always in the opposite direction as the displacement. Compare this equation to the derived magnitude of , and set the coefficient equal to the derived coefficient : Take the square root of both sides of this relation to obtain the angular frequency: Angular frequency is defined in terms of period T as , so set the cylinders’ derived angular frequency equal to : Comment Solve for T Therefore, the period of the motion is Comment Step 5 of 6 Step 6 of 6 9 0 Bookmark Show all steps: Chapter 14, Problem 1BP ON Textbook Solutions Expert Q&A Practice Study Pack 3/31/2021 Solved: Two uniform, solid cylinders of radius R and total mass... | Chegg.com https://www.chegg.com/homework-help/two-uniform-solid-cylinders-radius-r-total-mass-m-connected-chapter-14-problem-1bp-solution-978013397804... 4/4 COMPANY LEGAL & POLICIES CHEGG PRODUCTS AND SERVICES CHEGG NETWORK CUSTOMER SERVICE © 2003-2021 Chegg Inc. All rights reserved. See more problems in subjects you study Chapter 14, Problem 68P CP A block with mass M rests on a frictionless surface and is connected to a horizontal spring of force constant k. The other... See solution Chapter 14, Problem 47E A building in San Francisco has light fixtures consisting of small 2.35-kg bulbs with shades hanging from the ceiling at the... See solution Bookmark Show all steps: Chapter 14, Problem 1BP ON Textbook Solutions Expert Q&A Practice Study Pack