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Unformatted text preview: ROTATIONAL VECTORS AND ANGULAR MOMENTUM E XERCISES Section 11.1 Angular Velocity and Acceleration Vectors 12. If we assume that the wheels are rolling without slipping (see Section 10.5), the magnitude of the angular velocity is 1 cm v / (70 m/3.6 s)/(0.31 m) 62.7 s . r  = = = With the car going north, the axis of rotation of the wheels is eastwest. Since the top of a wheel is going in the same direction as the car, the righthand rule gives the direction of as west. 13. I NTERPRET The problem asks about the angular acceleration of the wheels as the car traveling north with a speed of 70 km/h makes a 90 left turn that lasts for 25 s. D EVELOP The speed of the car is cm 70 km/h 19.4 m/s. v = = Assuming that the wheels are rolling without slipping, the magnitude of the initial angular velocity is cm 19.4 m/s 62.7 rad/s 0.31 m v r = = = With the car going north, the axis of rotation of the wheels is eastwest. Since the top of a wheel is going in the same direction as the car, the righthand rule gives the direction of i r as west. In unitvector notation, we write . i i =  r After making a left turn, the angular speed remains unchanged, but the direction of f r is now south (see sketch). In unitvector notation, we write . f j =  r E VALUATE Using Equation 10.4, we find the angular acceleration to be av ( ) ( ) f i j i i j t t t t    = = = = r r r r The magnitude of av r is 2 av 2 2(62.7 rad/s)   3.55 rad/s 25 s t = = = r and av r points in the southeast direction (in the direction of the vector ). i j A SSESS Angular acceleration av r points in the same direction as . r 14. Suppose that the xaxis is horizontal in the direction of the final angular velocity ( (60 rpm) ) f i = and the yaxis is vertical in the direction of the initial angular velocity ( (45 rpm) ). j = i Equation 11.1 implies that av ( )/ f i t = = (60 45 ) rpm/15 s (4 3 ) rpm/s. i j i j = Its magnitude is 2 2 av (4) ( 3) rpm/s = +  = 2 2 5 rpm/s 5( /30) s 0.524 s ,  = = at an angle 1 3 4 tan ( ) 36.9  = =  to the x axis (i.e., below the horizontal). 11.1 11 11.2 Chapter 11 15. I NTERPRET The problem asks about the angular velocity of the wheels after an angular acceleration has been applied within a time interval. D EVELOP Take the xaxis east and the yaxis north, with positive angles measured CCW from the xaxis. In unit vector notation, the initial angular velocity i r and the angular acceleration r can be expressed as 2 2 2 (140 rad/s) (cos sin ) (35 rad/s )[cos(90 68 ) sin(90 68 ) ] ( 32.45 rad/s ) (13.11 rad/s ) i i i i i j i j i j = = = + = + + + =  + r r The final angular velocity can be found by using Equation 10.8....
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This note was uploaded on 09/12/2009 for the course PH PH101 taught by Professor Prof.kim during the Spring '09 term at Korea Advanced Institute of Science and Technology.
 Spring '09
 Prof.Kim
 Physics, Acceleration, Angular Momentum, Momentum

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