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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 east-west. Since the top of a wheel is going in the same direction as the car, the right-hand 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 east-west. Since the top of a wheel is going in the same direction as the car, the right-hand rule gives the direction of i r as west. In unit-vector 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 unit-vector 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 south-east 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 x-axis is horizontal in the direction of the final angular velocity ( (60 rpm) ) f i = and the y-axis 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 x-axis east and the y-axis north, with positive angles measured CCW from the x-axis. 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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