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Unformatted text preview: 326 CHAPTER 12 ROLLING, TORQUE, AND ANGULAR MOMENTUM C HAPTER 12 Answer to Checkpoint Questions . ( a ) same; ( b ) less . less . ( a ) z ; ( b ) + y ; ( c ) x . ( a ) and tie, then and tie, then (zero); ( b ) and . ( a ) , , and then and tie (zero); ( b ) . ( a ) all tie (same , same t , thus same L ); ( b ) sphere, disk, hoop (reverse order of I ) . ( a ) decreases; ( b ) same; ( c ) increases Answer to Questions . ( a ) same; ( b ) block; ( c ) block . ( a ) same; ( b ) same . ( a ) greater; ( b ) same . ( a ) tie; ( b ) wood cylinder . ( a ) L ; ( b ) : L . ( a ) or ; ( b ) . b, then c and d tie, then a and e tie (zero) . ( a ) ( r and F are both radial); ( b ) same . a, then b and c tie, then e, d (zero) . ( a ) ; ( b ) ; ( c ) ; ( d ) . ( a ) same; ( b ) increases, because of decrease in rotational inertia . ( a ) same; ( b ) increase; ( c ) decrease; ( d ) same, decrease, increase . ( a ) units clockwise; ( b ) then , then the others; or then , then the others . ( a ) , , , , and then , , and tie (zero); ( b ) , , and CHAPTER 12 ROLLING, TORQUE, AND ANGULAR MOMENTUM 327 . ( a ) spins in place; ( b ) rolls toward you; ( c ) rolls away from you Solutions to Exercises & Problems E Consider the pipe of mass m and radius R rolling at an angular speed ! . The linear speed of the center of mass of the pipe is v = R! . Thus the ratio in question is K t K r = mv I! = mv ( mR )( v=R ) = : : E The work required to stop the hoop is the negative of the initial kinetic energy of the hoop. The initial kinetic energy is given by K = I! + mv , where I is its rotational inertia, m is its mass, ! is its angular velocity about its center of mass, and v is the speed of its center of mass. The rotational inertia of the hoop is given by I = mr , where r is its radius. Since the hoop rolls without slipping the angular velocity and the speed of the center of mass are related by ! = v=r . Thus K = mr v r + mv = mv = ( kg)( : m/s) = : J : The work required is W = : J. E ( a ) The angular speed is ! = v R = ( m/h)( h = s) ( : = ) m = : rad/s : ( b ) Use ! f ! i = : The angular acceleration is = ! f ! i = ( : rad/s) ( )( : ) = : rad/s : 328 CHAPTER 12 ROLLING, TORQUE, AND ANGULAR MOMENTUM ( c ) The stopping distance is ` = R = ( : )( ) : m = : m : E Let M be the mass of the car and v be its speed. Let I be the rotational inertia of one wheel and ! be the angular velocity of each wheel. The total kinetic energy is given by K = Mv + I! ; where the factor appears because there are wheels. The kinetic energy of rotation is K r = I! ; and the fraction of the total energy that is due to rotation is f = K r K = I! Mv + I! : For a uniform wheel I = mr , where r is the radius of a wheel and m is its mass. Since the wheels roll without slipping ! = v=r . Thus I! = mr v =r = mv and f = mv Mv + mv = m M + m = ( kg) kg + ( kg) = : : Notice that the radius of the wheel cancels from the equations. The rotational inertia is proportional to r and when it is multiplied by ! = v =r the result is independent of...
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 Fall '07
 GROUPTEST
 Angular Momentum, Kinetic Energy, Momentum, Moment Of Inertia, KF

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