Question

Learning Goal: To practice Problem-SolvingStrategy 9.1 Rotational energy.

A string is attached to the rim of a small hoop of radius = 8.00×10⁻² and mass = 0.180 and then wrapped several times around therim. If thefree end of the string is held in place and the hoop is releasedfrom rest and allowed to drop, as shown in the figure (Intro 1figure), calculate the angularspeed and the translational speed of the rotating hoop after it hasdescended = 0.750 . Use = 9.80 for the acceleration due to gravity.

Problem-Solving Strategy 9.1 Rotational energy

1. Identify the system youwill analyze, and decide on the initial and final states (positionsand velocities) you will use in solving the problem. Draw oneormore sketches showing the initial and finalstates.
2. Define your coordinatesystem, particularly the zero points for gravitational and elasticpotential energies (the point at which $y=0$ in the case of gravitational potential energy or at whichthe spring is relaxedin the case of elastic potentialenergy).
3. List the initial andfinal kinetic and potential energies, that is, $K_i$ , , $U_i$ , and $U_f$ . Some of these will be known and some unknown. Usealgebraicsymbols for any unknown coordinates or velocities. Notethat includes both gravitational and elastic potential energy, andincludes both translational and rotational kinetic energy. Recallthat for an object that rolls without slipping, the speed $v_cm$ of the center of mass is given by .

4. Writeexpressions for the total initial mechanical energy and for thetotal final mechanical energy, equate them, and solve to findwhatever unknownquantity is required.

5. Take a hard look atyour results to see whether they make sense. Are they within thegeneral range of magnitudes you expected? If you change one of thegivenquantities, do the results change in a way you canpredict?

SET UPBefore writing any equations, organizeyour information and draw appropriate diagrams.Part AConsider the sketch of the situationdescribed in the problem introduction. What type of velocity doesthe hoop have in its initial state?What type of velocity doesthehoop have in its final state?

1. translational velocity
2. angular velocity
3. zero velocity

Enter the letters that correspond tothe types of velocities present in the initial state and theletters that correspond to the types of velocities present inthefinal state of this system separated by a comma. Enter each set ofletters in alphabetical order. For example, if the letters A and Crepresent the types ofvelocities present in the initial state andA is the type of velocity in the final state, enterAC,A.