Chapter 10 Rotational Kinematics and Energy Answers to Even Numbered Conceptual Questions 2 Yes In fact this is the situation whenever you drive in a circular path with constant speed 4 Every point on Earth has the same angular speed Therefore the smallest linear speed occurs where the distance from the axis of rotation is smallest namely at the poles 6 The moment of inertia of an object changes with the position of the axis of rotation because the distance from the axis to all the elements of mass have been changed It is not just the shape of an object that matters but the distribution of mass with respect to the axis of rotation 8 Spin the two spheres with equal angular speeds The one with the larger moment of inertia the hollow sphere has the greater kinetic energy and hence will spin for a longer time before stopping 10 2 a What determines the winner of the race is the ratio I mr as we see in the discussion just before 2 2 Conceptual Checkpoint 10 4 This ratio is MR 2 MR 2 1 for the first hoop and 2 M R 2M R 1 for the second hoop Therefore the two hoops finish the race at the same time b As in part a we can see that 2 the ratio I mr is equal to 1 regardless of the radius Thus all hoops regardless of their mass or radius finish the race in the same time Solutions to Problems and Conceptual Exercises 1 Picture the Problem This is a units conversion problem radians Strategy Multiply the angle in degrees by to get radians 180 rad rad Solution 30 180 6 rad 45 rad 180 4 rad 90 rad 180 2 rad 180 rad 180 Insight The quantity is the circumference of a circle divided by its diameter 3 1415926536 2 Picture the Problem This is a units conversion problem 180 Strategy Multiply the angle in radians by to get degrees radians Solution 180 30 6 180 0 70 126 180 1 5 270 180 5 900 Insight The quantity is the circumference of a circle divided by its diameter 3 1415926536 3 Picture the Problem The minute and hour hands of the clock rotate at constant angular speed Strategy Determine the time for each hand to complete a revolution Multiply the revolutions per time by 2 radians per revolution and convert the time units to seconds to obtain an angular speed in radians second Solution 1 a The minute hand completes a cycle once per hour 1 rev 2 rad 1 hr 1 745 10 3 rad s hr rev 3600 s Copyright 2010 Pearson Education Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced in any form or by any means without permission in writing from the publisher 10 1 James S Walker Physics 4th Edition Chapter 10 Rotational Kinematics and Energy 2 b The hour hand completes two revolutions per day 2 rev 2 rad 1 day 1 hr 1 454 10 4 rad s day rev 24 hr 3600 s Insight The angular speed of the hour hand is 12 times less than the angular speed of the minute hand because it completes a cycle once every 12 hours Copyright 2010 Pearson Education Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced in any form or by any means without permission in writing from the publisher 10 2 James S Walker Physics 4th Edition Chapter 10 Rotational Kinematics and Energy 4 Picture the Problem The second hand completes one revolution every 60 seconds Strategy Multiply 1 rev 60 s by conversion factors to obtain the appropriate units Solution 1 a Convert the units 1 rev 3600 s 60 s h 60 rev h 2 b Convert the units 1 rev 60 s 360 deg 60 s min rev 360 deg min 3 c Convert the units 1 rev 2 rad 60 s rev 30 rad s 0 1047 rad s Insight Another common unit of angular speed is rev min or rpm The second hand rotates at 1 00 rev min 5 Picture the Problem This is a units conversion problem Strategy Convert the given angular speeds into units of rad s and compare their magnitudes Solution 1 Find for the tire 2 00 103 deg rad tire 180 deg 34 9 rad s s 2 Find for the drill 400 0 rev 2 rad 1 min drill 41 89 rad s min rev 60 s 3 Find for the propeller prop 40 0 rad s 4 We can now rank the angular speeds from least to greatest tire propeller drill Insight In order to compare any quantities you must always ensure they have been measured in the same units 6 Picture the Problem The tire rotates about its axis through a certain angle Strategy Use equation 10 2 to find the angular displacement s 1 95 m 5 9 rad r 0 33 m Insight This angular distance corresponds to 339 or 94 of a complete revolution Solution Solve equation 10 2 for 7 Picture the Problem The Earth travels in a nearly circular path around the Sun completing one revolution per year Strategy Convert the known angular speed of 1 rev yr into units of rev min Solution Convert the units 1 rev 1 yr 1 day 1 h 1 90 10 6 rev min yr 365 days 24 h 60 min Insight This angular speed corresponds to about 0 986 deg day or 1 99 10 7 rad s A good rule of thumb in astronomy is that the Sun appears to move 1 day against the background of the fixed stars 8 Picture the Problem The Earth rotates once on its axis every 24 hours Strategy Convert the known angular speed of 1 rev day into units of radians per second Solution Convert the units 1rev 2 rad 1 day 1 h 7 27 10 5 rad s day rev 24h 3600s Insight This angular speed corresponds to about 15 hour A rule of thumb in astronomy is that the fixed stars will move across the sky at this rate 1 every 4 minutes or 15 arcsec s due to Earth s rotation 9 Picture the Problem The pulsar rotates about its axis completing 1 revolution in 0 33 s Strategy Divide one revolution or 2 radians by the period in seconds to find the angular speed Solution Calculate using equation 10 3 2 rad 2 rad 190 rad s t T 0 033s Copyright 2010 Pearson Education Inc All rights reserved This material is protected under all copyright laws as they currently exist No portion of this material may be reproduced in any form or by any means without permission in writing from the publisher 10 3 James S Walker Physics 4th Edition Chapter 10 Rotational Kinematics and Energy Insight The rotation rate of the pulsar can also be described as 1800 rev min 10 Picture the …
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